The Learning and Teaching of Calculus: Ideas, Insights and Activities (IMPACT: Interweaving Mathematics Pedagogy and Content for Teaching) [1 ed.] 1032069732, 9781032069739

Table of contents :
Cover
Endorsements
Half Title
Series
Title
Copyright
Contents
Series foreword
Acknowledgement
1 Introduction
2 Calculus across time and over countries
3 Making sense of limits and continuity
4 Making sense of differentiation
5 Integration and the fundamental theorem of calculus
6 Interlude: the ordering of chapters 3, 4 and 5
7 Calculus applications: differential equations and integration
8 Beyond elementary calculus
Index

Citation preview

“When we teach calculus, we need to know about differential and infinitesimal ways to understand calculus and about how calculus ideas evolved over time. We need to know how calculus curricula look around the world and about the rich tradition of calculus education research. We need to consider alternatives to the classic order of a first calculus course. And, we must have answers to questions about the role of calculus in solving real-world problems. Somewhat miraculously, this essential volume in the IMPACT series does quite a bit of all of that.” Elena Nardi, Professor of Mathematics Education (UEA, UK)

“John, Rob, Márcia, and Mike have accomplished a delicate balance among broad historical, pedagogical, mathematical, cultural, and curricular perspectives on matters related to the teaching and learning of calculus. Their book will be important reading for teachers of calculus to understand the historical evolution of the calculus curriculum they teach. Teachers will also become aware of the vibrant discussions within mathematics education of how the calculus might be reshaped to maintain intellectual integrity while simultaneously making central ideas more accessible to students. Their book will also be important reading for graduate students and researchers who are interested in inquiry into calculus learning and teaching. The most impressive aspect of their book is that the authors cover so much ground without taking a stance on any of the controversies they so carefully explain.” Pat Thompson, Arizona State University

“The Learning and Teaching of Calculus presents a fascinating and compelling discussion of calculus and the teaching of calculus. The authors have done an exceptional job of writing in a way that is accessible and educative for a wide range of readers, including teacher educators, those that teach calculus, and those that are interested in conducting research on the learning and teaching of calculus. The book is full of thought-provoking and engaging sample problems for students as well as clear exposition of key theoretical ideas that is, paradoxically, both comprehensive and concise. I highly recommend this book. You will undoubtably be inspired and awed at the beauty of the ideas presented.” Chris Rasmussen, San Diego State University

THE LEARNING AND TEACHING OF CALCULUS This book is for people who teach calculus – and especially for people who teach student teachers, who will in turn teach calculus. The calculus considered is elementary calculus of a single variable. The book interweaves ideas for teaching with calculus content and provides a reader-friendly overview of research on learning and teaching calculus along with questions on educational and mathematical discussion topics. Written by a group of international authors with extensive experience in teaching and research on learning/teaching calculus both at the school and university levels, the book offers a variety of approaches to the teaching of calculus so that you can decide the approach for you. Topics covered include

• A history of calculus and how calculus differs over countries today • Making sense of limits and continuity, differentiation, integration and the fundamental theorem of calculus (chapters on these areas form the bulk of the book)

• The ordering of calculus concepts (should limits come first?) • Applications of calculus (including differential equations). The final chapter looks beyond elementary calculus. Recurring themes across chapters include whether to take a limit or a differential/infinitesimal approach to calculus and the use of digital technology in the learning and teaching of calculus. This book is essential reading for mathematics teacher trainers everywhere. John Monaghan is a professor at the University of Agder, Norway and an emeritus professor at the University of Leeds, UK. He has taught in schools and universities, and the learning and teaching of calculus has been a research interest throughout his career. Robert Ely is a professor of mathematics education at the University of Idaho, United States. He studies the reasoning of students with infinitesimals, integrals, variables, and argumentation, and he is particularly interested in the perspectives that history can bring to such reasoning. Márcia M.F. Pinto is Associate Professor at a public university in Brazil. She has experience teaching mathematics to prospective teachers, mathematicians and engineers and co-authoring textbooks for distance learning courses on calculus. Michael O. J. Thomas is Professor Emeritus in the Mathematics Department at Auckland University, New Zealand. His research explores advanced mathematical thinking at school and university, including the role of representations, versatility and digital technology.

IMPACT (Interweaving Mathematics Pedagogy and Content for Teaching)

The Learning and Teaching of Calculus Ideas, Insights and Activities John Monaghan, Robert Ely, Márcia M.F. Pinto and Michael O. J. Thomas The Learning and Teaching of Number Paths Less Travelled Through Well-Trodden Terrain Rina Zazkis, John Mason and Igor’ Kontorovich The Learning and Teaching of Mathematical Modelling Mogens Niss and Werner Blum The Learning and Teaching of Geometry in Secondary Schools A Modeling Perspective Pat Herbst, Taro Fujita, Stefan Halverscheid and Michael Weiss The Learning and Teaching of Algebra Ideas, Insights and Activities Abraham Acravi, Paul Drijvers and Kaye Stacey

THE LEARNING AND TEACHING OF CALCULUS Ideas, Insights and Activities

John Monaghan, Robert Ely,

Márcia M.F. Pinto and Michael O.J. Thomas

Cover image: © Shutterstock First published 2024 by Routledge 4 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 605 Third Avenue, New York, NY 10158 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2024 John Monaghan, Robert Ely, Márcia M.F. Pinto and Michael O. J. Thomas The right of John Monaghan, Robert Ely, Márcia M.F. Pinto and Michael O. J. Thomas to be identified as authors of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-032-06972-2 (hbk) ISBN: 978-1-032-06973-9 (pbk) ISBN: 978-1-003-20480-0 (ebk) DOI: 10.4324/9781003204800 Typeset in Bembo by Apex CoVantage, LLC

CONTENTS

Series foreword Acknowledgement

viii

x

1

Introduction

2

Calculus across time and over countries

19

3

Making sense of limits and continuity

54

4

Making sense of differentiation

91

5

Integration and the fundamental theorem of calculus

133

6

Interlude: the ordering of chapters 3, 4 and 5

178

7

Calculus applications: differential equations and integration

183

8

Beyond elementary calculus

248

Index

1

286

SERIES FOREWORD

IMPACT, an acronym for lnterweaving Mathematics Pedagogy and Content for Teaching, is a series of textbooks dedicated to mathematics education and suitable for teacher education. The leading principle of the series is the integration of mathematics content with topics from research on mathematics learning and teaching. Elements from the history and the philosophy of mathematics, as well as curricular issues, are integrated as appropriate. In mathematics, there are many textbook series representing internationally ac­ cepted canonical curricula, but such a series has so far been lacking in mathematics education. It is the intention of IMPACT to fill this gap. The books in the series will focus on fundamental conceptual understanding of the central ideas and relationships, while often compromising on the breadth of coverage. These central ideas and relationships will serve as organizers for the structure of each book. Beyond being an integrated presentation of the central ideas of mathematics and its learning and teaching, the volumes will serve as guides to further resources. Historically, the field of calculus has been one of the central areas of mathemat­ ics and is the basis for many other disciplines, especially in applied mathematics. Its development was marked by a fruitful fundamental dispute between Newton and Leibniz, which we can understand scientifically today. Again, it was Felix Klein who successfully campaigned for Calculus to be established as part of the mathematical school canon at the beginning of the 20th century. It has remained so to this day. Moreover, both in terms of content and methods, Calculus is indispensable for many university degrees. Hence Calculus constitutes a bridge from high school to university mathematics, and this is why a Calculus book belongs into the IMPACT series.

Series foreword

ix

Series editors Tommy Dreyfus (Israel), Ghislaine Gueudet (France), Nathalie M. Sinclair (Canada) and Günter Törner (Germany) Series Advisory Board Abraham Arcavi (Israel), Michèle Artigue (France), Jo Boaler (USA), Hugh Burkhardt (Great Britain), Willi Dörfler (Austria), Francesca Ferrara (Italy), Koeno Gravemeijer (The Netherlands), Angel Gutiérrez (Spain), Eva Jablonka (Germany). Gabriele Kaiser (Germany), Carolyn Kieran (Canada), Kyeong-Hwa Lee (South Korea), Frank K. Lester (USA), Fou-Lai Lin (Republic of China Taiwan), John Monaghan (Great Britain/ Norway), Mogens Niss (Denmark), Alan H. Schoenfeld (USA), Peter Sullivan (Australia), Michael O. Thomas (New Zealand) and Patrick W. Thompson (USA).

ACKNOWLEDGEMENT

We thank the editors and two reviewers for their formative comments on drafts of this book.

1 INTRODUCTION

1.1 Introduction

Welcome to our book on calculus.1 We are four mathematics education researchers, but we have clocked-up many decades of practical teaching of calculus at school and university level in Brazil, New Zealand, the United Kingdom and the United States. This book is about teaching calculus, it is not a calculus textbook. This book concerns introductory courses in calculus (elementary calculus) of a sin­ gle variable, which may be in the latter years of high school or the first year of uni­ versity, depending on country and/or institution. It is primarily a book for teacher educators, i.e. people who teach teachers, secondly for (school or university) teach­ ers who wish to know more about ways to introduce their students to calculus and thirdly for people, such as PhD students, who are contemplating doing mathematics education research on an aspect of calculus. This book offers a variety of methods to approach the teaching of calculus, provides a reader-friendly overview of research on the learning and teaching of calculus and presents educational and mathematical mat­ ters for consideration (Edumatters and Mathematters) at various points in the chapters. There are eight chapters: 1 2 3* 4* 5*

Introduction Calculus across time and over countries Making sense of limits and continuity Making sense of differentiation Integration and the fundamental theorem of calculus

1 Or ‘analysis’ in some countries. The literature on calculus sometimes writes ‘the calculus’ or ‘Calculus’. We have opted for ‘calculus’ to keep the writing style plain. DOI: 10.4324/9781003204800-1

2

Introduction

6 Interlude: the ordering of Chapters 3, 4 and 5 7* Calculus applications: differential equations and integration 8 Beyond elementary calculus. The chapters marked with a star (*) address the substance of elementary calculus. We now briefly describe each chapter. Chapter 1, which you are reading, introduces the book. This introduction is followed by three sections: mathematical prerequisites for the study of calculus, theoretical approaches mentioned in this book and a mathematical overview of dif­ ferential and infinitesimal calculus. Mathematical prerequisites are important for teaching any course, so we thought we’d put our considerations at the beginning of the book. The section on theoretical approaches is there to help readers who do not work in the field of mathematics education. You can be a brilliant calculus teacher without knowing anything about the social-linguistic theory of commognition, but if we mention, say, a commognitive study on teaching the derivative, we want a quick way for you to find out what the theory of commognition is. The last sec­ tion provides a mathematical overview of differential and infinitesimal calculus. This overview is needed to fully understand parts of the following chapters as we often present differential and/or infinitesimal ways to understand calculus ideas. Chapter 2 has two sections: a brief account of the history of calculus and a brief account of calculus around the world. Both sections could be books in themselves, so there is a need to focus on what is needed for the rest of the book. The history of calculus section outlines major landmarks from Archimedes to Leibniz and Newton to the arithmetisation of calculus in the 19th century to the invention of nonstand­ ard analysis in the 20th century. The calculus around the world section considers school and beginning university calculus curricula from a number of countries. The approach is synchronic not diachronic; it would be nice to present the development of calculus curricula over time, but this is unrealistic given the length of the book. Chapter 3 has four sections. The first looks at the place of limits and continuity in elementary calculus curricula and one well know curriculum, Advanced Placement Cal­ culus, in particular. The second section provides an overview of education research on limits and continuity. This is followed by a short history of limits and continuity. This history is important in understanding that limits have not always been a part of calcu­ lus and that continuity was a central construct in the early days of calculus. The final section looks at ways that limits and continuity can be introduced in your classroom. Chapter 4 presents ways that (parts of) a first course in differential calculus can be taught and learnt meaningfully. It does not tell you how to teach differentiation but, rather, presents you with different ways to do this. The chapter discusses prior knowledge and curricula matters, ways that differentiation can be introduced, the rules for differentiation, special functions, what derivatives tell us about functions and their graphs, tasks and an overview of education research on differentiation. Chapter 5 presents the main ideas of integral calculus and the fundamental theorem of calculus (FTC). The focus is on significant mathematical and conceptual elements involved in making sense of these ideas. Different approaches to teaching integration

Introduction

3

and the FTC are presented along with notes on the pros and cons of these approaches. Throughout the chapter, descriptions of research about student reasoning with integrals and the FTC are presented to augment the discussion of concepts and ways of teaching. Chapter 6 considers, and questions, the classic ordering of a first calculus course: limits, differentiation and integration. It is a very short chapter, an interlude that reflects on the previous three chapters. Chapter 7 discusses some applications of calculus to real-world problems such as the spread of viral infections and simple harmonic motion. It begins with a consideration of first and second order ordinary differential equations and covers the nature of their solu­ tions as functions, how these may be approximated and their graphical representation us­ ing slope fields. There is also an overview of educational research on differential equations. The second part of the chapter looks at applications of integration that involve finding volumes of revolution, surface area and lengths of curves. Finally, some methods of ap­ proximating the values of integrals are considered with applications such as ship stability. Chapter 8 considers aspects of calculus and/or real analysis courses that come after an elementary course in calculus. It does not attempt to describe the content of these courses but raises matters that teachers of elementary calculus should, in our opinion, note with regard to what calculus related things their students may do (or not!) in their future studies. 1.2 Mathematical prerequisites for the study of calculus

It could be argued that, apart from giving them practical arithmetic skills, one of the primary purposes of the secondary mathematics curriculum is to prepare students for the study of calculus. In which case, describing the prerequisites for the study of calculus ought to be easy. However, our goal here is not to give a list of school mathematics topics, such as arithmetic techniques, that would have little value in this context but rather to try to say what are some key constructs that will contribute to student understanding of calculus. In the preceding sentence, we have deliberately avoided use of the word ‘skills’ in favour of constructs, whatever they are. We should explain why. One way to divide mathematics content is into skills and processes, and another way to divide math­ ematics content is into objects, concepts and constructs. Of course, these two groups are not totally disjoint but are related in a fundamental way, which will be explained in Section 1.3 of this chapter. Suffice it to say at this point that mathematical pro­ cesses can undergo a cognitive encapsulation into mathematical objects (Dubinsky & McDonald, 2001; Tall et al., 2000). We would argue, with the support of consider­ able research evidence (see, for example, Kieran, 2007, and the review of Rakes et al., 2010) that there has been, in many countries, a greater emphasis on the former than on the latter. For example, students may be able to add or multiply decimals without really understanding the concept of place value or be able to factorise bino­ mials without understanding what factors or quadratic functions are. Hence, in the main we will place an emphasis here on what objects, concepts and constructs students would be well advised to understand in order to do well in

4

Introduction

calculus. Occasionally we may also mention a skill or process that would also help. It will be difficult to present them in the order that would usually be met in school since this will vary from country to country, and even within countries, from school to school. In addition, the approach employed and the level of formality or rigour used will also vary considerably. So the key point is that they need to be understood in the context of what that may mean in the local curriculum. 1.2.1 Important mathematical constructs 1.2.1.1 Number

Clearly the concept of number underpins much of mathematics, and while the set of real numbers as a complete ordered field and the corresponding real number line will be a step too far for the vast majority of secondary school students, some idea of the relationships between sets of numbers and how they might be represented will be of benefit. Further, representing these using both interval and set notation is recommended. So, for example, there is value in having the sets  = {1, 2, 3,.},  = {. - 3, -2, -1, 0,1, 2, 3.. and relationships N ⊂ Z ⊂ Q ⊂ R (without needing x : 1}, a formal definition of ), along with interval notation, ( --,1] = {x : x E R,x [– 1, 3), and so on. A good understanding of proportionality will be beneficial along 1 1 1 with some basic familiarity with convergent sequences (e.g. that 1, , , , . . . gets 2 4 8 nearer and nearer to 0). 1.2.1.2

Variable

We have known for a number of years that many students’ understanding of letter or symbolic literal use in algebra does not extend to generalised number or variable (Küchemann, 1981). This may be because greater emphasis is placed in classrooms on the use of variables rather than on understanding what they are or that many school texts are silent on the definition of a variable. A key part of the difficulty here is to know what kind of definition to give to students. For example, both Schoenfeld and Arcavi (1988) and Wagner (1981) give a number of possible examples of symbol usage in mathematical statements. We would, of course, expect variables to vary, but describing the manner in which they do is not so straightforward. Skemp (1979) says “In mathematics, an unspecified element of a given set is called a variable” (p. 228). So the statement ‘Let x ∈ ’ is used in this way to say that x is an unspecified real number. However, this can appear to be a rather static idea that says nothing about exactly how x might vary. Thompson and Harel (2021) stress the point that in order to understand both rate of change and accumulation, two fundamental ideas in calculus, students need to be conversant with the process of covarying two quantities. Addressing the thought of how variables vary, they are unconvinced by the idea that it means replacing one value with another. Instead they espouse the construct of ‘smooth variation’ and

Introduction

5

say that “If calculus students are to understand something akin to instantaneous rate of change, they must envision that smooth variation happens in bits” (ibid, p. 512). Thus, students would benefit from experiencing quantities that do indeed vary. For example, if a conical container is filling with water such that the height of the water in the container is given by x and the corresponding volume by V(x) then V varies simultaneously with x, and it is this covariation that can provide valuable experience (Thompson & Harel, 2021). Assisting students to see, for example, the smooth vari­ ation of the height quantity from x to x + h as the volume varies smoothly from V(x) to V(x + h) and representing these on Cartesian axes as a continuous graph can help build understanding in preparation for calculus. Other examples that could prove useful are to think of how variables change over time. If students are familiar with basic kinematics (formulas and graphs linking distance, velocity, acceleration and average velocity), then this can also be useful for seeing this kind of variation, as well as providing good applications for later differentiation and integration. For example, if a velocity is changing with respect to time, then we may have V (t ) = 30 + 10t and, if δ t is a small change in t, and the corresponding small change in V is δV , then V + 8V = 30 + 10 (t + 8 t ) So

8V = 108 t. 1.2.1.3

Function

Of course, we have already used function notation in the previous section, and we stress the point that what is provided is not an intended order for studying these concepts and neither should they be thought of as disjoint entities, but rather there is some overlap between them. While function is a fundamental concept of mathemat­ ics, there is a tendency, as with variable, for it to be used in many school classrooms without being defined. As a consequence many students may think of a function as an equation, a graph or an input-output process rather than as a correspondence, a mapping, a set of ordered pairs or a rule (Williams, 1998). This leads to erroneous ideas such as a function must have a formula, its graph needs to be smooth and con­ tinuous, or even that due to stressing the vertical line test, the graph, rather than any algebra, is the function (Thomas, 2003). In addition, often student experiences with functions, and their graphs, has been limited to construction of pointwise and global perspectives (Vandebrouck, 2011). Hence, they may evaluate the value of a function at a specific point, finding f ( a ), say, deal with a function globally or on an inter­ val by translating its graph or finding its concavity, but they rarely consider a local

6

Introduction

perspective that involves small intervals such as [ x - h,x + h ] (Thomas et al., 2017), where the behaviour of the function on small intervals of decreasing size is a focus. This becomes important for a study of, for example, when a function is continuous. Working with functions is also important in calculus. We mention just a few key aspects of this here. One is that a working understanding of simple polynomial func­ tions (linear, quadratic and cubic) and their Cartesian graphs seems essential, along with the concept of gradient (or slope) and its application to linear functions (e.g. an ability to identify positive and negative gradients and to ascribe approximate nu­ merical values to the gradients) and estimating gradients of quadratic functions using tangents. It is important to define a polynomial (function) since research has shown that a number of students do not think of examples such as 0, 5, 2x + 1 as polynomials (since, they may reason, poly means greater than 1) but may also think that x -1 + x is one. Another important concept is the domain of a function, the values of the inde­ pendent variable for which the function is defined. This becomes important when thinking about differentiating functions given by a formula such as f ( x ) = ln ( g ( x ) ), in order to consider when g(x) might be zero or negative, and hence f is not defined, 1 as well as thinking about the values of x for which functions such as y = can be x integrated. We note in passing that both notations, y = . and f ( x ) = . will be of value in different areas of calculus2 and so making students familiar with both is a good idea (see Chapter 2 for historical information on why both are used). Know­ ing and understanding a wide range of different functions is also valuable. Some other examples include trigonometric, rational, exponential and power functions. See Section 4.2.5 for a fuller discussion of some of these functions. A second consideration is the concept of an inverse function and when it exists. This is when the function, possibly on a restricted domain, is 1–1 (injective) and onto (surjective), also called a bijection. In this case if y = f ( x ) then x = f -1 ( y ), when it exists. This, of course, is why the standard process for finding an inverse function, to make x the subject of the equation, works. Students may be taught that the graphs of a function and its inverse are symmetric about the line y = x (see Figure 1.1). If this is used then it is important, of course, to stress that this is only the case if the x- and y-axes have the same scales. A third useful idea is that of composition of functions, and this is best accom­ plished with the f ( x ) notation. A composite function of f and g, written f  g can be defined3 as

(f

 g )( x ) = f ( g ( x ) )

2 It is worth noting here that we should use, say, f as the notation to represent a function, whereas f(x). is the value of the function f at some point x. 3 g  f is also a composite function.

Introduction

FIGURE 1.1

7

Illustrating the relationship between f ( a ) and f -1 ( a )

providing we also make sure that the domain of f contains the range of g so that f ( g ( x ) ) is defined. Then the domain of f  g is the domain of g (or a subset of it) and the range of f  g is a subset of (or equal to) the range of f. It is worth not­ ing that although real functions are strictly defined between two given sets (such as f :  \ {0} - ) it is common practice in mathematics to give a rule for a function and assume that the (natural) domain is the largest subset of the real numbers for which the function is defined. Thus, instead of, say, h : [4, o ) -  defined such that h (x ) = x - 4 we will often be given just h (x ) = x - 4 without the domain information, and we have to supply this ourselves. This makes the composition of functions trickier. For example, we might ask whether our

8

Introduction

students could say what the natural domains would be for the functions f  g and g  f where f (x ) =

( 2x - 1)2 x +1

and g (x ) =

2x - 1 1 - x2

.

In addition, it may also be useful if students have some experience working with the binomial theorem, but it is possible to introduce this topic/concept during the study of differentiation. 1.2.1.4

Geometry

An understanding of geometrical concepts can be very useful in calculus. One important example is the measurement of angles using radians (see also Section 4.4). If we define a radian to be the angle subtended at the centre of a unit circle by an arc 2n of unit length, then since the circumference of a unit circle is 2π.1 there are = 2n 1 radians in a circle. Thus 2π radians are equal to 360˚, and so, dividing by 2 and using c to represent radians,

t c = 180. and 1c =

180° n

1o =

nc . 180

or

Thus the length of an arc subtending an angle of θ c at the centre of a circle of radius r is

e • 2n r = re 2n And the corresponding area of the sector is

e 1 • n r 2 = r 2e 2n 2 These ideas become essential when we start to differentiate trigonometric func­ tions such as f ( x ) = sin ( x ) and g ( x ) = cos ( x ) in order for the answers to be easier t cos ( xº ), rather than cos ( x ) when x is functions, since, for example f ' ( x º ) = 180

Introduction

9

in radians. This is because the limit arising in the formal definition of derivative (see Chapter 4) (h( sin | | # 2 lim ( ) = h->0 ( h ( 180 |2| ( ) when h is measured in degrees, but 1 when h is in radians. Mathematter Questions to ponder – concepts versus procedures (i)

(ii)

A cylinder has radius r and height h, while a sphere has radius r. If the radius of each is increasing at the same rate, which is increasing faster, the volume of the cylinder or the surface area of the sphere? Explain. Why should a student think about before they begin a process to solve 2x + 3 = 1? 6 + 4x If they go ahead and use a procedure to ‘solve’ it, what would you say to them about the answer x = -1.5 ?

(iii) Which of these do your students think are equations? y = 2x + 1, 9 = 3 + 6, 2 ( x - 3) = 2x - 5, a - 3b + 5a - 8b, 3 ( x + 1) = 3x + 3 What reasons can you think of for why they may think that way? (iv) Which of these represent functions? Which polynomials? Why or why not? 1 x -1

a)

y=

c)

2x 2 + 3y - x = 5

b)

h (t ) = #

d)

#) ( p (m ) = sin-1 | m - | 4) (

If 2x - 1 = 0 and 3x + 1 = 0 then 2x - 1 = 3x + 1, since both are equal to 0, and hence x = -2. How would you discuss this reasoning with a student? (vi) How would we get students to solve these equations? (v)

a) (i) 2x - 5 = 9 (ii) 2 ( x + 3) - 5 = 9 b) (i) 2x 2 - 5x - 7 = x 2 + 2x + 1 (ii) 2x 2 - x - 7 = x 2 + 6x + 1 2

(vii) Does the function f : (1, 2 ) -> R where f ( x ) = ( x - 1) value?

have a maximum

10

Introduction

1.3 Theoretical approaches mentioned in this book

As part of the IMPACT series of books, this book aims to integrate mathemat­ ics content teaching with the broader research and theoretical base of mathematics education. In particular we refer, at times, to approaches and ideas current in math­ ematics education research. We are aware that some readers will not be familiar with these. This section is written to provide such readers with an overview of approaches and ideas we mention. Our overview errs on the side of brevity; fuller descriptions of approaches and ideas are available in the Encyclopedia of Mathematics Education (EME, https://link.springer.com/referencework/10.1007/978-94-007-4978-8). We start with constructivism.4 This is a theory of knowledge development that emerged, alongside other influences, from Jean Piaget’s developmental psychology. Piaget considered knowledge as cumulative and cognitive development as mov­ ing through four stages: sensorimotor (0 to 2 years), preoperational (2 to 7 years), concrete operational (7 to 11 years) and formal operational (12+ years). He posited, amongst other things, three constructs in cognitive development:

• Schemas – cognitive structures representing a person’s knowledge about some entity or situation, including its qualities and the relationships between these5 • Assimilation – fitting new information into existing schemas, e.g. negative numbers • Accommodation – modification of existing schemas in the light of new informa­ tion assimilated. Constructivist mathematics educators apply these constructs to all aspects of learn­ ers’ mathematical development. For example, Taback (1975) found that, at each Piagetian stage, limit-related schemas held by the children were inherently contra­ dictory. In the late 20th century, two versions of constructivism emerged: radical constructivism, cognition is the sole driver of knowledge development; and social constructivism, the cultural-historical development of knowledge precedes individ­ ual knowledge development. Social cultural (socio-cultural) approaches, including social constructivism and activity theory,6 view individual knowledge development within social structures, historical development and the use of language and tools. Many social culturalists subscribe to Vygotsky’s (1978, p. 57) statement, “Every function in the child’s cul­ tural development appears twice: first, on the social level, and later, on the individual level; first between people . . . then inside the child”. The context of learning is not a side issue for social culturalists as the who with, where, with what (tools) and why of learning cannot be separated from the learning itself. For example, mathematics done by a student using a computer algebra system (CAS) cannot, for a social cultur­ alist, be reduced to what the student alone can do or to what the CAS can do; the 4 EME: Constructivism in Mathematics Education 5 https://dictionary.apa.org/schema 6 EME: Activity Theory in Mathematics Education

Introduction

11

mathematics is done by the student-with-CAS. Bingolbali and Monaghan (2008) is an example of a social cultural study on derivatives as it views students’ understand­ ings through students’ positional identities as engineers or as mathematicians. Realistic Mathematics Education7 (RME) is an approach to the design of teaching mathematics that emerged in the Netherlands in the 1970s and is still developing. The word ‘realistic’ in the title does relate to real-world situations, but it is also intended to convey that the mathematical teaching sequences designed should connect with the real-life experiences and imaginations of the students by offering them problem situ­ ations for the guided reinvention of mathematics. The word ‘mathematising’ was coined by RME didacticians to emphasise the verb of doing mathematics rather than just learning facts. RME distinguishes between horizontal and vertical mathematisations. The former involves mathematising extra-mathematical phenomena from the real world. Vertical mathematisation is inter-mathematical and involves building new (for the student at a particular stage in their mathematical development) mathematical connections between prior knowledge. RME has strong links with (and influenced the development of) design research and stresses: starting from problems which are meaningful to students, learning by doing and gradual mathematisation. The Anthropological Theory of the Didactic8 (ATD) was initiated by Yves Chevallard in the 1980s and focuses on institutional aspects of mathematics education. A central construct is the didactical transposition which traces the movement, over institutions, of scholarly knowledge (produced by mathematicians, e.g. the limit notion) to curricula knowledge (knowledge to be taught) to knowledge taught and knowledge learnt (by stu­ dents). ATD posits two knowledge blocks praxis and logos. Praxis consists of tasks and techniques (e.g. integration by parts) to solve the tasks. Logos concerns the underlying rationale for the praxis and has two levels: technology, which concerns the discourse used in describing techniques; and theory, which provides the basis for the techno­ logical discourse. The theory is supposed to justify the technology by linking histori­ cally accumulated mathematical knowledge to knowledge taught but ATD analyses often show that this linkage is often nebulous. For example, in an ATD study of limits at high school, Barbé et al. (2005) reveal constraints that significantly determine the teacher’s practice and the mathematics taught. The development of ATD is ongoing but the aforementioned description suffices for references to ATD in this book. Commognition9 is a word made from the words ‘communication’ and ‘cogni­ tion’ and is a socio-cultural approach which views cognition and communication as two sides of the same coin: thinking is communication with oneself, and learning mathematics is participation in mathematical communication. Discourses are types of communication which include words (e.g. functions); visual mediators (e.g. graphs); narratives, stories about the objects of the discourse and routines, repetitive patterns in the discourse. Routines in mathematics classes include explorations aimed at endorsing narratives, deeds which involve practical action and rituals which are things done to 7 EME: Realistic Mathematics Education 8 EME: Anthropological Theory of the Didactic 9 EME: Commognition

12

Introduction

create a common purpose in mathematics lessons (e.g. algebraic actions to establish a point of inflection). Commognition views learning as ‘change in discourse’ which can occur at the object-level or the meta-level. At the object-level, change concerns the logical development of previously endorsed narratives of the discourse. At the meta-level, learning change does not follow logically from previously endorsed nar­ ratives but involves discursants (participants) making choices. Commognitive research on the teaching of learning of calculus thus pays close attention to what is said and done in classrooms with respect to the many constructs it introduces. Embodied cognition10 is a branch of cognitive psychology, but in mathematics edu­ cation it represents a view that learning mathematics is a mind-body activity, not just a mental act. This is not a new idea, but embodied cognition is a relatively new term (late 20th century). This is easy to appreciate in elementary mathematics – a child using her fingers to count – but is it relevant to higher mathematics, calculus in particular? Carry out the following thought experiment: you are observing a student teacher teaching a first lesson on the derivative and introducing the gradient of the tangent to a function at a point – does the student teacher put her hand to graph at the point and incline the hand in the direction of the tangent at the point? Probably. There are now many different views on the extent and importance of embodied cognition but reference to it in this book will keep to this basic idea. Before introducing the next two theoretical approaches, we introduce the phrase/ construct, process-object encapsulation, which refers to making an object out of a pro­ cess. For example, the teaching and learning of functions usually begins with an input-output process, e.g. x f (x )

–2 4

–1 1

0 0

1 1

2

4

Students usually need to spend a considerable amount of time before this concep­ tion moves to viewing f ( x ) as a single entity, an object. Process-object encapsula­ tion permeates much of school mathematics number and algebra curriculum. Gray and Tall (1994) introduce the term ‘procept’ (process-concept) for process-object encapsulations where the student can move back and forth between process and object conceptions. Actions, Processes, Objects, Schemas (APOS)11 is a branch of constructivism which embraces process-object encapsulation. Schemas are the end-point in learning a con­ cept, which starts with actions and moves to processes and then to objects and finally schemas for the concept. Students are not always successful in reaching object or schema conceptions. APOS is used widely in studies of advanced mathematics. Asiala et al. (2001) is an example of an APOS study on students’ graphical under­ standings of the derivative. David Tall (with others) developed the metaphor of the three worlds of mathematics and has brought this into many accounts of learning calculus (Tall, 2013). The three 10 EME: Embodied Cognition 11 EME: Actions, Processes, Objects, Schemas (APOS) in Mathematics Education

Introduction

13

worlds are: an embodied world, a symbolic world (developed from the embod­ ied world) of actions into symbolic procedures (procepts) and a formal (axiomatic) world where concepts are defined and their properties are deduced. He views these three worlds as developing over time in the life of an individual and in the history of mathematics. 1.3.1 Teacher knowledge12

An import construct, pedagogical content knowledge (PCK), in teacher education was introduced in Shulman (1986, p. 8). Good teachers, he argued, must not only possess content knowledge and pedagogical knowledge; there must be interaction (intersection in terms of Venn diagrams) between these two forms of knowledge – pedagogical content knowledge. In the field of mathematics education this idea was refined, by D. Ball and H. Bass (2002), to mathematical knowledge for teaching (MKT). They sub-divide content (mathematical) knowledge into ‘common’, ‘specialised’ and ‘horizon’ content knowledge. Common content knowledge is required by everyone; specialised content knowledge is unique to mathematics teaching; horizon content knowledge is knowledge that would benefit teaching but may not be taught because it is beyond the level being taught. The topology of the real number line is an exam­ ple of horizon content knowledge in the teaching of calculus at the school level. We end this section with a short account of conversions and treatments of math­ ematical representations, e.g. algebraic, graphic and numeric forms of a mathemati­ cal construct such as the derivative. The following does not present a theoretical approach but introduces two constructs used in mathematics education research. A seminal work on representations in mathematics is Duval (2006) where the difference between conversions and treatments is considered. “Treatments are transformations of representations that happen within the same register”13 (ibid., p. 111), for example, changing y - x - 1 = 0 to y = x + 1. “Conversions are transformations of representa­ tion that consist of changing a register without changing the objects being denoted” (ibid., p.  112), for example, changing y = x + 1 to a Cartesian graph. Duval (ibid.) goes on to note “Conversion is more complex than treatment because any change of register first requires recognition of the same represented object between two rep­ resentations whose contents have very often nothing in common”. In the following chapters of this book, we often note that multiple representations are important for learning calculus, but Duval’s conversion/treatment distinction reminds us that alter­ native representations of the objects of calculus also add a level of difficulty. 1.4 A mathematical overview of differential and infinitesimal calculus

When it was developed in the 17th century, calculus was ‘the infinitesimal calculus,’ – a set of methods for working with infinitesimal quantities. In the 19th century, 12 EME: Subject Matter Knowledge Within “Mathematical Knowledge for Teaching”

13 Duval (2006) defines ‘registers’ as semiotic systems that permit a transformation of representations.

14

Introduction

infinitesimals were jettisoned from the subject because they were seen to be not rigorously defined, and calculus was reconceptualised in terms of limits. Calculus classes followed suit, and today the vast majority of classes avoid infinitesimals and instead use limits when defining the principal ideas of calculus such as derivatives, integrals and continuity. In the 1960s, Abraham Robinson developed the field of nonstandard analysis (Robinson, 1966), which allows infinitesimals to be formally defined and for calculus to be conducted rigorously using them. He also proved that essentially all the same things can be done with infinitesimal calculus as with limitsbased calculus. Some calculus classes around the globe now use infinitesimals, and some researchers have argued for the benefits of such approaches (for a survey, see Ely, 2021). One potential benefit is that notations originally invented with infini­ tesimals in mind can more directly refer to quantities, rather than serving as token dy short hands for limit processes. Using infinitesimals, dx has meaning on its own, dx b really is a quotient of small differences, and f ( x ) dx really is a sum of little bits.

f

a

We refer to infinitesimal approaches periodically in this book, since one of our goals is to stimulate the reader to conceptualise familiar elements of calculus in multiple ways and with multiple meanings. The purpose of this section is to provide some technical background for such references and to describe how infinitesimals can be rigorously defined in such a way that calculus can be performed using them. Of course, an infinitesimals-based calculus course need not formalise infinitesimals at all, just as a standard calculus course need not formalise limits with the epsilondelta definition. This section summarises the conceptual basis of such a formalisa­ tion, based on Keisler’s (1976) treatment, which should be consulted for further details, and which, at the time of this book’s publication, is available online for free on https://people.math.wisc.edu/~keisler/calc.html. Robinson’s development of nonstandard analysis is one way of formalising Leib­ nizian infinitesimals. It allows one to imagine a continuum where it is possible to ‘zoom in’ infinitely to reveal differences between points that at a finite scale appear identical. This continuum, the hyperreal numbers, is an extension of the real num­ ber line, and nonstandard analysis is the theory that works with this continuum. By saying that nonstandard analysis ‘formalises’ the idea of infinitesimal quantities, we mean that it grounds these in the regular ZFC axiomatisation of modern math­ ematics. Robinson proved the transfer principle, that all (first-order logical) theo­ rems true in the hyperreal numbers (*) are true in  and vice versa, which means that standard analysis and nonstandard analysis are equivalent in scope, consistency and power. This actually allows current-day mathematicians to pursue results in * or , whichever they find handier, without sacrificing rigour. For instance, Terence Tao uses nonstandard analysis to avoid excessively complicated manage­ ment of epsilons, and he notes that “non-standard analysis is not a totally ‘alien’ piece of mathematics”, but is “basically only ‘one ultrafilter away’ from standard analysis” (2007). Shortly we shall see what Tao means by “ultrafilter”.

Introduction

15

The image Robinson provides us for extending the reals to the hyperreals is to start by picturing an infinitesimal hyperreal number as a sequence of real numbers that converges to 0. Sequences that converge faster to 0 are imagined to be smaller infinitesimals than those that converge slower. Thus we get an array of infinitesimals and comparing them amounts to comparing sequences of reals. The first technical­ ity is that we ought to view the two sequences {1, ½, ¼, 1/8, . . ., 1/2n . . .} and {3, ½, ¼, 1/8, . . ., 1/2n . . .} as the same hyperreal number, since they both converge to 0 the same way. So Robinson begins by describing a hyperreal number not as a sequence but as an equivalence class of sequences of real numbers. The idea is to consider two sequences to be equivalent, (an) ~ (bn), if an = bn for “most” indices n. Likewise, we want (an) > (bn) if an > bn for “most” n. But how do we know what “most” means? In other words, how can we decide if an index set S = {n: an = bn} or Q = {n: an > bn} is “large”? One criterion for sets to be “large” is that it should allow the relation = (or >) to be transitive. We want (an) ~ (bn) and (bn) ~ (cn) to imply (an) ~ (cn). Thus, if S = {n: an = bn} is large, and T = {n: bn = cn} is large, then we need S∩T to be large too, because it might be that an = cn only for indices n∈S∩T. A second criterion for large sets is that for any set P, exactly one of P or \P should be large. This is because we want (an) = (bn) when P = {n: an = bn} is large, and we want (an) ≠ (bn) when Q = {n: an ≠ bn} is large. Two more straightforward criteria for “largeness”:  is large and ∅ is not, and if A is large and A ⊆ B, then B is large. An ultrafilter is a collection of subsets of  that satisfy these four criteria for largeness. In other words, an ultrafilter is a collection of “large” sets of indices n, and on any of these index sets it is possible to coherently compare two sequences an = bn. In order to define the hyperreal numbers, we need to pick an ultrafilter that meets an additional criterion to make sure the hyperreals do not end up being exactly the same as the reals: the ultrafilter must be nonprincipal. This means that every finite set is small and every cofinite set is large. Proving that a nonprincipal ultrafilter even exists requires the axiom of choice, and ‘picking’ such an ultrafilter is a non-constructive endeavour. Nonetheless, this enables the hyperreals to be de­ fined as follows: fix a nonprincipal ultrafilter. Let R N be the set of all sequences of real numbers and say (an) ~ (bn) if {n: an = bn} is large. The hyperreals are the set of equivalence classes * = R N /∼. The hyperreals are an extension of the reals, since we can identify any real number r with the sequence {r, r, r, . . .}. It is worth checking that there indeed exist some infinitesimal hyperreals that are not just 0. Consider ε = {1, 1/2, 1/3, 1/4, . . .}. This hyperreal number ε < 1/k for any integer k, because the set S of indices on which {1, 1/2, 1/3, 1/4, . . .} < {1/k, 1/k, 1/k, . . .} is cofinite, hence large. But ε > 0, since the set of indices for which {1, 1/2, 1/3, 1/4, . . .} > {0, 0, 0, . . .} is , which is large. Likewise, ∂ = {1, 1/2, 1/4, 1/8, . . .} is an even smaller infinitesimal, and it is not difficult to show that there are uncountably many more. The reciprocals of all these infinitesimal hyperreals are all infinite numbers, which are also hyperreal.

16

Introduction

An operation on real numbers has its analogue in * by just performing the same operation coordinate by coordinate. It can be shown that * is a field under standard arithmetic operations *+ and *×. This means some of the heuristics Leib­ niz used with infinitesimal and infinite numbers can be proven as theorems in the hyperreals, e.g. “an infinitesimal number times an infinitesimal number is infinitesi­ mal” and “the reciprocal of an infinitesimal number is infinite”. In general any statement about real numbers has a ‘starred’ statement for hy­ perreals, which means that it is true on a ‘large’ set of coordinates. The transfer principle establishes that this relationship goes both ways: a (first-order) statement is true for all real numbers if and only if its starred version is true for all hyperreal numbers. An example is the following statement, which is true in : “For any positive x, there is a natural number n (∈) such that x > 1/n”. This sounds like it is not true in *, since there exist infinitesimals there. But the way to ‘star’ this statement is “for any positive (hyper)real x, there is a (hyper)natural number n (∈*) such that x > 1/n”. This is true because * contains infinite numbers. Another very important example is the Archimedean axiom for the real numbers: for any positive a and b, there exists n (∈) s.t. na > b. The spirit of this state­ ment is that a magnitude can always be iterated some number of times to exceed another given magnitude. This statement is not true in the hyperreals, which are a nonarchimedean field. Nonetheless, the statement becomes true when you allow n to be an infinite hypernatural (∈*). Using the least upper bound property of the reals, it is not hard to show that each finite hyperreal number p is infinitely close to exactly one real number r. This real r is called the shadow of p (“sh(p)”), or standard part of p (“st(p)”). Likewise, each real number r has a cloud or monad14 of hyperreal numbers that are infinitely close to it. Using ≈ to mean infinitely close, p ≈ st(p). The monad of hyperreals around the real number p formalises the idea that there are numbers that become visible only when you zoom in infinitely on p. When you prove a theorem in the hyperreals, and then ‘unstar’ it to get the corre­ sponding statement in the reals, this unstarring often entails ‘rounding’ finite hyperreal numbers to their shadow or standard part. For example, consider the function y = x2 in *. Choose an infinitesimal non-zero increment of x, calling it “dx”. We can find the corresponding increment of y, represented as dy, as follows: dy = ( x + dx ) - ( x ) 2

2

dy = x 2 + 2xdx + dx 2 - x 2 dy = 2xdx + dx 2

14 Robinson picked this term as a tribute to Leibniz. For Leibniz, a monad was a fundamental meta­ physical particle, not a mathematical entity used in calculus.

Introduction

17

We can go further if we wish, and divide both sides by the infinitesimal dx, since division works normally in the field *: dy = 2x + dx dx ( dy J = st ( 2x + dx ) = 2x . dx This is an example of how transferring between * and  is done not by “pre­ tending infinitesimals are 0” (a complaint against infinitesimal techniques levelled by some philosophers in the early days of calculus) but rather by taking standard parts when we wish to create a statement in . Typically transferring to  by taking standard parts does the same work as taking a limit in standard analysis. Defining the definite integral provides another example of how this unstarring Thus we can define the derivative function f ' ( x ) = st

n

replaces the use of a limit. First, we note that if a sum n

n, by the transfer principle we can define

n *r

k

Lr

k

is defined for any natural

k=0

for any hypernatural n (includ­

k =0

ing infinite n), and this will have all the same properties as an ordinary sum. Now suppose we wish to define

J

b a

f ( x ) dx . First, we can find an infinitesimal dx and an

infinite hypernatural n so that b = a + n·dx. Then we partition the interval [a, b] into n increments of size dx. On each increment, find an x, and calculate f(x)dx. The integral

(

b a

f ( x ) dx is the standard part of the infinite sum of these f(x)dx.

References Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. E. (2001). The development of students’ graphical understanding of the derivative. Journal of Mathematical Behavior, 16, 399–431. Ball, D. L., & Bass, H. (2002). Toward a practice-based theory of mathematical knowledge for teaching. In E. Simmt & B. Davis (Eds.), Proceedings of the 22nd annual meeting of the Canadian mathematics education study group (pp. 3–14). CMESG. Barbé, Q., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s practice: The case of limits of functions in Spanish high school. Educational Studies in Mathematics, 59, 235–268. Bingolbali, E., & Monaghan, J. (2008). Concept image revisited. Educational Studies in Math­ ematics, 68(1), 19–35. Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 275–282). Kluwer Academic Publishers. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of math­ ematics. Educational Studies in Mathematics, 61(1–2), 103–131. Ely, R. (2021). Teaching calculus with infinitesimals and differentials. ZDM Mathematics Edu­ cation, 53, 591–604.

18

Introduction

Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. The Journal for Research in Mathematics Education, 26(2), 115–141. Keisler, H. J. (1976). Elementary calculus. Prindle, Weber & Schmidt. Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). National Council of Teachers of Mathematics. Küchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s understanding of mathematics: 11–16. John Murray. Rakes, C. R., Valentine, J. C., McGatha, M. B., & Ronau, R. N. (2010). Methods of instructional improvement in algebra: A Systematic review and meta-analysis. Review of Educational Research, 80(3), 372–400. https://doi.org/10.3102/0034654310374880 Robinson, A. (1966). Non-standard analysis. North-Holland Publ. Comp. Schoenfeld, A. H., & Arcavi, A. (1988). On the meaning of variable. Mathematics Teacher, 81, 420–427. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Skemp, R. R. (1979). Intelligence, learning and action: A foundation for theory and practice in educa­ tion. Wiley. Taback, S. (1975). The child’s concept of limit. In H. Roskopf (Ed.), Children’s mathematical concepts. Teachers’ College Press. Tall, D. O. (2013). How humans learn to think mathematically: Exploring the three worlds of math­ ematics. Cambridge University Press. Tall, D. O., Thomas, M. O. J., Davis, G., Gray, E., & Simpson, A. (2000). What is the object of the encapsulation of a process? Journal of Mathematical Behavior, 18(2), 223–241. Tao, T. (2007). Ultrafilters, nonstandard analysis, and epsilon management. https://terrytao.wordpress. com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ Thomas, M. O. J. (2003). The role of representation in teacher understanding of function. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 291–298). University of Hawai’i. Thomas, M. O. J., Hong, Y. Y., & Oates, G. (2017). Innovative uses of digital technology in undergraduate mathematics. In E. Faggiano, A. Montone, & F. Ferrara (Eds.), Innovation and technology enhancing mathematics education (pp. 109–136). Springer. Thompson, P. W., & Harel, G. (2021). Ideas foundational to calculus learning and their links to students’ difficulties. ZDM – Mathematics Education, 53, 507–519. https://doi. org/10.1007/s11858-021-01270-1 Vandebrouck, F. (2011). Perspectives et domaines de travail pour l’étude des fonctions. Annales de Didactiques et de Sciences Cognitives, 16, 149–185. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University Press. Wagner, S. (1981). An analytical framework for mathematical variables. Proceedings of the 5th international conference of psychology in mathematics education (pp. 165–170). Grenoble, France. Williams, C. G. (1998). Using concept maps to assess conceptual knowledge of function. Journal for Research in Mathematics Education, 29(4), 414–421.

2 CALCULUS ACROSS TIME AND OVER COUNTRIES

This chapter has two sections which position calculus over time and at a point in time (2022). It provides a base for events and concepts that we refer to in the follow­ ing chapters. Both sections could be books in themselves, but this is a short book, so we apologise for the brevity. The first section visits the history of calculus and outlines major landmarks from Archimedes to Leibniz and Newton to the arith­ metisation of calculus in the 19th century. This section does not include the devel­ opment of nonstandard analysis in the 1960s as this has been covered in Section 1.4. The second section considers school and beginning university calculus curricula in a number of countries. 2.1 Where did calculus come from?

Our purpose in this part is not to go through the historical development of calculus; there are books, e.g. Edwards (1979), that already do this. Instead, our goal in this section is to tell a short story about the problems and questions that people worked on for many centuries that informed the development of calculus. We partition this story into three sub-sections: the area problem, tangents and optimisation and the calculus of Newton and Leibniz. 2.1.1 The area problem

For millennia, mathematicians around the globe tackled the general problem of how to find the area of regions with curved edges. Before seeing examples, it is worth asking what exactly it means to find the area of a curvy shape. For example, for ancient Greek mathematicians it was a problem of construction: given a curved region, construct a rectilinear region that takes up the same amount of space. The DOI: 10.4324/9781003204800-2

20

Calculus across time and over countries

term quadrature for area-finding reflects this idea – you ‘quadrate’ a region if you make it a quadrilateral without changing its area. But in general, how would you know if you have succeeded in accomplishing this?

Mathematter What is the area of this blob? How would you go about finding it?

2.1.1.1 Area of a circle

How do you find the area of a circle? Today that is a simple question that many school students will be able to answer immediately. But that was not always the case. Mathematicians in Babylon, Greece, Egypt, China, India and other parts of the ancient world tried in various ways. In so doing, all of them developed ways to estimate the value of the circumference of a circle of diameter 1, a value that we now call π. In some of these ancient cases, this estimation was probably done through careful measurement (see Figure 2.1). In other cases, the estimate of π was done by trap­ ping a circle inside and outside of polygons with many sides and then using other geometric facts to determine the areas of these polygons. For instance, Archimedes (c.287–c.212 BC) began with hexagons inside and outside a circle (see Figure 2.2), reasoning that the true area of the circle must lie between the areas of the two hexa­ gons. He then doubled the number of sides to get 12-gons, using known formulas for areas of isosceles triangles. He kept doubling the number of sides, eventually

FIGURE 2.1

Some estimates of π in the ancient world

Calculus across time and over countries

FIGURE 2.2

21

Trapping a circle between polygons with more and more sides

stopped with 96-gons. With some other reasoning, he could show that the circum­ 1 10 ference of a circle of diameter 1 must be between 3 and 3 , which is actually 7 71 within 0.0002 of the value of π as we know it today. This illustrates the true area of a circle as the limit of these upper and lower bounds as the number of sides in the polygons increased to infinity. This technique for estimating the circle’s area was used throughout the ancient world. Chinese mathematician Liu Hui also used 96-gons to estimate π in c.263 AD. Two centuries later in India, Aryabhatta used a 384-gon and in China, Zu Chongzhi used a 12288-gon (!) to estimate π ≈ 355/113, correct to seven digits. 2.1.1.2 Area of a parabola

Archimedes also tackled the problem of finding the area of a piece of a parabola. In his Method, he determined the exact area of a piece of a parabola. He first sur­ rounded the parabola piece ABV with a triangle ABC (as in Figure 2.3) and then imagined chopping it into infinitely many indivisible slices (e.g., XX‴). Then he imagined a lever PB, whose midpoint and fulcrum D was also the midpoint of AC. He used some geometric facts about parabolas along with the law of lever moments (which he had earlier in his life discovered) to deduce the following: each indivisible slice XX′ of the parabola, when moved to the end of the lever at point P, balances with its corresponding slice XX‴ of the triangle, left where it is. Thus the entire parabola sector ABV, hanging at P, balances with the entire triangle ABC left where it is. Since the triangle’s centre of mass is at its centroid, which is 1/3 of the way from the fulcrum D to the lever end B, this means the triangle must be three times bigger than the parabola sector. In Quadrature of the Parabola, Archimedes used a different technique to prove this answer was correct; he made a double exhaustion argument that relied on approxi­ mating the parabolic region’s area as closely as wanted by packing enough triangles inside the region (see Figure 2.4). This is quite similar to the method he used for the circle’s area. Archimedes’ Method was lost for at least a millennium until the Archimedes Pal­ impsest was found in the 20th century. Without knowing about it, Bonaventura

22

Calculus across time and over countries

FIGURE 2.3 Archimedes’ Method used to determine the area of a parabola sector, from Boyer (1949, pp. 49–50)

FIGURE 2.4

Archimedes’ proof involves filling a parabolic region with triangles

Cavalieri (1598–1647) invented in the 1630s a similar technique for slicing a region into indivisible pieces and putting these into one-to-one correspondence with slices of another known figure to determine the region’s area (or volume). Cavalieri’s col­ league Torricelli, in his work de Dimensione Parabolae, presents 11 different ways to find the area of the parabola using Cavalieri’s method of indivisibles. In one of these, Proposition 20, he even hangs pieces on an imagined lever! At almost the same time, Fermat and Roberval were writing letters back and forth in 1636 generalising Archimedes’ quadrature question to “parabolas” of the form y = x k . Instead of Archimedes’ triangles, Fermat and Roberval put N rectangles of

Calculus across time and over countries

23

uniform width inside the region (see Figure 2.5). Since the curve is increasing, the right side of each subinterval gives a height for an over-estimating rectangle, and the left side gives a height for an underestimating rectangle. This means that the area A is bounded between the left-hand sum and the right-hand sum: k

k

k

( 0 J 1 ( 1J 1 + + N N N N k

+

( 2J 1 + N N

+

k

( N ;1J 1 ( 1J 1 0 for every point in the interval. We now consider graphs over intervals further and introduce two terms: ‘concave downwards’ and ‘concave upwards’.19 A function is concave upwards (respec­

17 We say ‘almost’ because of ≥/≤ in (ii)/(iii) and >/< in (v)/(vi). 18 We are still in the domain of ‘well-behaved’ functions, so we do not make stipulations regarding continuity or differentiability at a point. For functions in general, stipulations must be made. 19 Some texts use the terms ‘convex’ and ‘concave’ for ‘concave upwards’ and ‘concave downwards’.

120

Making sense of differentiation

tively downwards) if the line segment between any two points on the graph of the function lies above (respectively to follow) or on the graph. A graphical introduction to these terms could mention that concave upwards (respectively downwards) refers to the shape ∪ (respectively ∩) of the curve in an interval.20 For example, the function in Figure 4.17a is concave upwards; it is ∪-shaped and the line segment between the two points on the graph of the function lies above the graph. Figure 4.17b shows the same function and it appears that the graph of the function is above the tangent lines except, of course, at the point of contact of the function and a tangent line; this can be proved but we leave this as an exercise. Figure 4.18 shows two tangent lines through points on the same graph; they are there to help show that the slope of the tangent lines at points increase, as x-values increase. This diagram suggests a proposition (for an interval containing a and b): If f is concave upwards and a < b, then f ' ( a ) < f ’( b ).

FIGURES 4.17

The line segment between the two points (a) and tangents (b) on the graph of the concave upwards function

20 Indeed, there is an argument that (i) this graphical introduction should precede a definition; (ii) for some classes, the graphical introduction alone is sufficient.

Making sense of differentiation

FIGURE 4.18

121

The slope of the tangent lines at points increase, as x-values increase

Mathematter Prove the two statements: If f is concave upwards, then: (i) the graph of the function is above the tangent lines except at the point of contact of the function and a tangent line. (ii) if a < b, then f ' ( a ) < f ' ( b ). Note that we could repeat this for concave downwards to get the proposition: If f is concave downwards and a > b, then f ' ( a ) > f ' ( b ).

The points earlier, on increasing functions and concave graphs, give students further means to interpret graphs on the basis of their derivatives. It is probably a good idea for teachers to pause the development of theory at this point and give students tasks that can help consolidate these points. We consider student-led and teacher-led tasks. A student-led task is for students, in pairs, to sketch (by hand) graphs of functions (no equations needed) and to sketch the derivative functions. A possible teacher-led task is to present a graph (no equation is needed) such as Figure 4.18 and ask students to give descriptions of the graph in terms of the sign of the 1st derivative, the sign of the 2nd derivative and regions where the graph is concave upwards and downwards.

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Making sense of differentiation

FIGURE 4.19

The graph of a function with tangent lines at five points on the curve

FIGURE 4.20

Local maximum and minimum and points of inflection

Figure 4.19 can be used to introduce (or consolidate) local maximum and minimum and points of inflection – see Figure 4.20. Figure 4.20 marks the local maximum and minimum (points ( a, f ( a ) ) at which ' f ( a ) = 0 and f "" ( a ) 0 or f "" ( a ) 0) and the point of inflection (points at which the concave upwards part of the curve meets the concave downward part). This last point should be carefully explored with students for many harbour the misconcep­ tion that the following is true for (well-behaved) functions: If f ' ( a ) = 0 and f "" ( a ) < 0 , then ( a, f ( a ) ) is a local maximum. If f ' ( a ) = 0 and f "" ( a ) > 0 , then ( a, f ( a ) ) is a local minimum.

If f ' ( a ) = 0 and f "" ( a ) = 0 , then ( a, f ( a ) ) is a point of inflection. The misconception above is the third statement – given conditions ( a, f ( a ) ) may or may not be a point of inflection – a simple example is the function f ( x ) = x 4 and the point x = 0. Once these ideas have been consolidated (which takes time) students should be equipped to interpret graphs on the basis of their 1st and 2nd derivatives, and it is a good idea for teachers to celebrate this equipment by giving students tasks which can help consolidate what they have learnt. We provide one example, but (i) teach­ ers can tailor their presentation to their own styles and (ii) a range of examples may aid students’ consolidation of knowledge.

Making sense of differentiation

123

x4 (i) Find all maxima, minima and points of inflection for the function y = - x2 2 for -2 < x < 2. (ii) Sketch the function and mark the maxima, minima and points of inflection. (iii) Mark all interval where the function is increasing or decreasing. (iv) Mark all intervals where the function is concave up or concave down. We end this section with consideration of rules and tasks. The textbooks and exami­ nations that support many calculus courses focus on the algebraic properties (apply­ ing the 2nd derivative test) of what derivatives tell us about functions and their graphs. For example:

(

)

Find any point(s) of inflection for the curve with equation y = e x 2x 2 + 5x + 4 . In your answer, give coordinates accurate to 3 decimal places.21

But curricula are developing to go further than this. As noted in Section 2.2, Dreyfus et al. (2021) reports on an Israeli curriculum which values fundamental mathematical ideas and mathematical reasoning and offers open and closed reason­ ing tasks for this area of calculus.

In each statement, choose the correct option and justify your claim: (i) A second degree polynomial has a point of inflection – always/sometimes/ never (ii) A third degree polynomial has a point of inflection – always/sometimes/ never (iii) A fourth degree polynomial has a point of inflection – always/sometimes/ never Is the following claim true or not? Justify! You may justify by sketching graphs. Given a function f(x) whose graph passes through the point P(2, 6). If the second derivative changes sign at x = 2, then the equation of the tangent to the graph of the function at P cannot be y = 3x.

4.6 Tasks

Setting appropriate tasks (teacher designed student activities for introducing, extending and consolidating student knowledge) is, of course, essential for student 21 This example was found at a UK examination board revision website.

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Making sense of differentiation

learning. But what are appropriate tasks in the area of differential calculus? Well, typical tasks that appear in examinations that your students will take are obviously important – both you and your students want them to do well in their exams. But tasks set for high stakes examinations are usually designed to be done in conditions where students have limited time and cannot talk to each other; surely classroom tasks should go beyond such tasks. We have introduced some exploratory and rea­ soning tasks previously. We now suggest a few more. dy Some tasks hit you in the face with calculus, e.g. “. . . Find ”. Others are not dx obviously calculus tasks, such as

Find the number that most exceeds its square.

When the focus is on using algebraic techniques to find f ′, there are various things we can keep in mind when designing tasks such as: (i) Rather than make up a random list of functions for your students to differentiate, systematically add variation, for example

(

) ( 4

) ( 5

) ( 6

)

7

Differentiate22 with respect to x: 3x 2 + 1 ; 3x 2 + a ; mx 2 + 1 ; px 2 + q .

Variation here has two aspects: the order of the polynomials, the use of constants and parameters. d tan(sec(cos x ))’. Instead dx of making up such ridiculous tasks, we can turn the symbol manipulation part (repeated applications of the chain in this case) into an investigation, such as

(ii)In Section 4.3, we considered ‘Find the derivative of

d d d sin x ; sin(sin x ); sin(sin(sin x )); . . . dx dx dx Stop when you see a pattern. Can you explain this pattern?

Find:

22 The functions which follow are not actually functions, they are algebraic expressions. We were going to rewrite these as functions and then we thought that this abbreviation that many (most?) teachers use sometimes is worth commenting on.

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125

Tasks on the rules for differentiation can be designed so that it is the student rather than the teacher who suggests the rule. Here is a possible task for this: Explore whether the following statements appear always, sometimes or never true: (i) (ii) (iii)

(kf ( x ) ) ' = kf ' ( x ) ( f ( x ) + g ( x ) ) ' = f ' ( x ) + g' ( x ) ( f ( x ) . g ( x ) ) ' = f ' ( x ) . g' ( x )

(iv) If f and g are inverse functions then f ' ( x ) =

1 g' ( x )

Statements (i) to (iv) could be replaced by other statements related to the rules for differentiation. This warmup exercise can pave the way for discussion of what the rules mean (including conditions on the functions). We have suggested using mathematical software (MSW) in several tasks above but the extent that MSW can help students’ understanding depends on the task. For example, consider the following task: Find the dimensions of the cylinder of maximal volume that can be inscribed in a given sphere.

The expectation in this task is that students will assign variables, say R for the radius of the sphere and r and h for the radius and height of the cylinder. Use Pythagoras’ 2 [ h3 ] (hJ theorem to obtain r 2 = R 2 . Use this to obtain V = n r 2h = n hR 2 2

4

and then differentiate this expression, set the differentiated expression to 0 and solve (performing suitable tests to ensure that this is a maximum). Using MSW (a computer algebra system) could be useful here as the focus of the task is to find the dimension [ h3 } of the cylinder. But, for the isolated task ‘differentiate V = n r 2h = n hR 2 , 4 using a computer algebra system seems pointless button pushing as the focus of such a task is on the proficient use of the rules for differentiation. MSW can be used, amongst other tools, in exploratory tasks, investigations, to reduce the complexity in some of the steps of a task (as in the earlier exam­ ple) and to check paper and pen work. In the previous investigation, starting ‘Find d d sin x; sin(sin x )’, MSW can be used to generate output after the first couple dx dx of derivatives are found by hand, and MSW can also be used to check paper and pen work. The following is a template for a range of tasks that can be explored

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Making sense of differentiation

( (( ---) )x)xx( (x(xx+++11)1))

algebraically (computer algebra), graphically (graph plotter) or tabularly (a difference table in a spreadsheet): Input a polynomial in the form f ( x ) = ( x ± a )( x ± b ) ..., e.g. f ( x ) = ( x - 3) ( x - 1) x ( x + 1)

(ff()xx=) = x-(-3 ( x) - 1) x ( x + 1) and repeatedly differentiate it.

What happens? Explain why this happens.

4.7 What education research says about teaching and learning differentiation

We have referred to the research literature at relevant points. In this section we augment this by presenting selective research on the teaching and learning differ­ entiation (i) over time (1980 to 2020) and (ii) that illustrate a variety of theoretical approaches used in education research (which are briefly explained in Section 1.3). Tall (1981) and Orton (1983) are early reports of difficulties with differentiation. Both papers are short and do not require a lot of theory to follow. Strictly speaking Tall (1981) is a scholarly reflection rather than research. It considers affordances and constraints, for the learner, of six approaches to differentiation: (i) The original Leibniz method, the variable x is incremented by an infini­ tesimal quantity dx and the dependent variable y is then incremented to y + dy = f ( x + dx ) (ii) The dynamic limit method, compute the ratio ( f ( x + h ) - f ( x )) / h as h gets closer and closer to zero (iii) The numerical method, where f ' ( a ) is approximated by tabulating specific values of h against ( f ( x + h ) - f ( x )) / h (ix) The computer drawing method, plotting the graph of f over an interval ( x - a, x + a ) and zooming in (x) The epsilon-delta method (xi) The modern infinitesimal method. Orton (1983) examines students’ performance, and the types of errors students make, on a variety of differentiation tasks. The findings show that these students found applications of differentiation relatively easy; for example, the following questions

• Calculate the rate of change of y = x 2 - 4x + 1 at x = 2 • Find the turning or stationary points of y = x 3 - 3x 2 + 4 and determine what kind of points they are. However, the students found questions which (i) asked students to explain things and (ii) concerned graphical approaches to rate of change to be difficult; for example

Making sense of differentiation

• Explain the meaning of the following symbols: δ x; δ y; • What is the relationship between

δy dy and ? δx dx

127

δy dy ; dx; dy; δx dx

• The graph that follows represents y = 3x 2 + 1 (graph not shown here): • What is the average rate of change in y in the x-interval from a to a + h? The tasks in Orton (1983) are presented as an appendix; you could give some of them to your students and see if you get similar results to Orton. Asiala et al. (2001) consider students’ graphical understandings of the derivative. The paper reports on a specific US calculus course from the 1990s, Calculus, Concepts, Computers and Cooperative Learning (C4L), and adopts APOS (described in Section 1.3) as its theoretical framework. The research leads them to propose the following stages to take note of in the design of learning paths for the derivative:

• Actions (graphical): Connecting two points on a curve to form a chord and the • • • • • •

action of computing the slope of the secant Actions (analytical): The action of computing the average rate of change via the difference quotient Processes (graphical): Internalising the two graphical actions above into a single process as the two points get closer together Processes (analytical): Internalising the individual average rate of change actions into a single process Objects (graphical): Being able to regard the above graphical process as producing the tangent line as the limiting position of the secant lines and to be able to show the tangent line at a point on the curve Objects (analytical): Being able to regard the aforementioned analytical process as producing the instantaneous rate of change of dependent to independent variables Schema: Being able to regard the aforementioned graphical and analytical process as producing the derivative at a point as the limit of the difference quotients at that point.

Kendal and Stacey (2001) report on a curriculum and teacher development programme linked to the use of a handheld symbolic calculator (i.e. it had computer algebra system) in a 20 lesson introductory calculus unit at high school level in Australia. The paper examines how two teachers (A and B) taught differentiation using this calculator, which made numerical, graphical and symbolic representations of the derivative available to their classes. The two teachers taught in quite different ways. This was not expected, and the paper focuses on these differences. The paper uses the term ‘privileging’ to describe the impact of teachers’ mathematical emphases on students’ learning of differentiation. Although the two teachers planned their lessons

128

Making sense of differentiation

together, they made differing pedagogical choices regarding aspects of differentia­ tion to emphasise: Teacher A privileged rules and exhibited a strong preference for symbolic representation while teacher B privileged conceptual understanding and student construction of meaning. Teacher A privileged graphical-numerical connec­ tions while teacher B privileged graphical-symbolic connections. The results show that students’ performances were strongly influenced by the aspect of the derivative privileged by their teachers. This last statement, that student performances (and hence, presumably, their understandings) are strongly influenced by the aspect of the derivative privileged by their teachers, puts context and teaching (mediation) to the fore and so this research can be called socio-cultural; i.e. student understanding of differentiation is not an individually constructed phenomenon. This statement, that student understandings are linked to teachers’ privileging, may come as no surprise to the reader but it raises important issues in teaching differentiation which we raise in the Edumatter.

Edumatter Kendal and Stacey (2001) call attention to the different ways that colleagues in the same department may approach differentiation with their students. What are the similarities and differences in your department? Investigating this could form take the form of mutual lesson observations and follow up discussions.

Bingolbali and Monaghan (2008) adopt, what we called in Section 1.3, a socio-cultural framework in as much as it argues that student thinking about the derivative is influenced by their departmental affiliations. The paper focuses on first year mechanical engineering (ME) and mathematics (M) undergraduate students’ understanding of the derivative. The paper presents four tasks given to the students (reproduced in the Appendix to the paper), two focused on tan­ gents and two focused on rate of change. The ME students achieved statistically significantly higher marks than the M students in the rate of change items and the opposite applied to the tangent questions. This phenomenon was explored further in two questions which asked students about their understanding of the meaning of the derivative. For example, the second question has comments from two students, A and B: Student A says “The derivative tells us how quickly and at what rate something is changing since it is related to a moving object”. Student B says “The derivative is a mathematical concept and it can be described as the slope of the tangent line of a graph of y against x”.

Making sense of differentiation

129

The question asks: “If you had to support just one student, which one would you support and why”? Responses suggest that many students positioned themselves with regard to their perceptions of their departments’ ways of thinking about the derivative – their iden­ tities as students: M student: B interprets the derivative from a mathematician’s perspective, and A interprets it from a physicist standpoint. At the end of the day, since I too am from mathematics department, I find B’s explanation closer to myself. ME student: Calculating rates of change seems to me more real . . . Because I would be the one who makes mathematics concrete. The themes raised in the paper are worthy of discussions in departments teaching both mathematics and engineering students. Park (2013) focuses on how students, who took a calculus course as part of their engineering or natural science degree at a US university, talk about the derivative. Students were presented with nine questions on the derivative. The paper employs constructs from the commognitive theoretical framework23 (see Section 1.3): word use, visual metaphors, narratives and routines. Their use of words and visuals showed that most students did not appreciate the derivative at a point as a number and the derivative function as a function. Specific comments on the four constructs include Some students used the derivative as ‘the tangent line’ instead of its ‘slope’; stu­ dents used a tangent line as their visual mediator . . . (but) . . . written nota­ tions indicated their use of the tangent line as both the derivative at a point or the derivative of a function; the most frequently identified endorsed narrative was ‘the derivative increases/decreases iff24 a function increases/decreases’; when problems involved algebraic notations of a function, they tended to apply differ­ entiation rules. However, they used the same routine when graphs were given. They tried to find the equation of the graph and differentiate it. (p. 636) As for Orton (1983), the tasks are presented as an Appendix, and you could give some of them to your students and see if you get similar results to Park.

23 This paper is not a full commognitive study, but we don’t dwell on such issues here. 24 iff is short for ‘if and only if ’.

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Making sense of differentiation

Zengin (2018) examines how a class of 33 preservice mathematics student teach­ ers construct the relationship between the concepts of differential and derivative with the mathematical software (GeoGebra) and a particular teaching approach. We select it for inclusion here because it links these three aspects, and there are few stud­ ies examining the relationship between derivatives and differentials. The teaching approach combines individual and group work and includes five classroom stages: individual work, group work, debate, self-reflection and teacher-led whole class plenary discussion. The paper examines students’ work on four multi-part tasks (pre­ sented in the Appendix). For example, the first (of three) part to the second task is: Explain how a secant line passing through points P and Q on the curve can be tangent to the curve at point P. Explain your claim with the help of GeoGebra. The examination of student work is conducted over a series of lessons using Toul­ min’s model of argumentation. This model was first proposed in 1958 but is still used (and respected) in mathematics education research.25 The basic model has three elements: claim, data and warrant. The warrant acts as a bridge between the data and the claim. For instance, in task 2, a student claim was “When Q approaches P along the curve, the tangent to the curve at P occurs”. The data supporting this claim was a GeoGebra figure and the words “The dynamic behaviour of the secant through the points P and Q in GeoGebra”. The warrant was “As h → 0, the secant line becomes the tangent line”. The author claims that the integration of GeoGebra and the teach­ ing approach enabled the student teachers: To generate appropriate conjectures and arguments for validation . . . when they constructed the relationship between the concepts of differential and deriva­ tive. . . . The participants remarked that their misconceptions about the relation­ ships between derivative and differential were eliminated due to self-reflection and systematic learning, and they also explored the relationships between the concepts. (ibid., p. 330) The paper is noteworthy, in our opinion, in as much as the teaching approach does not try to steer a smooth course in concept development but allows initial concep­ tions to be refined through individual and group work, debate and self-reflection; when the subject matter is difficult, as in differentiation, there may be much to be said for ‘rough roads’ to concept development.

25 See, for example, Simpson (2015).

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4.8 Final words

This ends our discussion of differentiation in an elementary calculus course. There is so much more we could have introduced but we did not have space to do so within the page constraints. This turns our minds to time constraints on the teacher and raises questions which you could investigate:

• How many teaching hours are you expected to devote to differentiation? • How many private study hours are your students expected to devote to differentiation? Estimating these hours and thinking about what can be meaningfully covered in this time may help in the design of courses which, in the language of Dreyfus et al. (2021) (see Section 2.2), celebrate: fundamental mathematical ideas and mathemati­ cal reasoning, connecting mathematics to everyday life and science and the cultiva­ tion of fertile intellectual ground from which new concepts may emerge. References Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. E. (2001). The development of students’ graphical understanding of the derivative. Journal of Mathematical Behavior, 16, 399–431. Bingolbali, E., & Monaghan, J. (2008). Concept image revisited. Educational Studies in Math­ ematics, 68(1), 19–35. Bressoud, D. M. (2019). Calculus reordered: A history of the big ideas. Princeton University Press. Clark, J. M., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D. S., John, D., et al. (1997). Constructing a schema: The case of the chain rule? Journal of Mathematical Behavior, 16, 345–364. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26, 135–164. Dreyfus, T., Kouropatov, A., & Ron, G. (2021). Research as a resource in a high-school cal­ culus curriculum. ZDM – Mathematics Education, 1–15. Drijvers, P. (2001). The concept of parameter in a computer algebra environment. In Proceed­ ings of the 25th conference of the International Group for the psychology of mathematics education (Vol. 2, pp. 385–393). Utrecht, The Netherlands. Ely, R. (2021). Teaching calculus with infinitesimals and differentials. ZDM – Mathematics Education, 53(3), 591–604. Faulkner, B., Earl, K., & Herman, G. (2019). Mathematical maturity for engineering stu­ dents. International Journal of Research in Undergraduate Mathematics Education, 5(1), 97–128. Keisler, H. J. (1976). Elementary calculus. Prindle, Weber & Schmidt Inc. Kendal, M., & Stacey, K. (2001). The impact of teacher privileging on learning differentiation with technology. International Journal of Computers for Mathematical Learning, 6(2), 143–165. Lim, K. F. (2008). Differentiation from first principles using spreadsheets. Australian Senior Mathematics Journal, 22(2), 41–48. Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathemat­ ics, 14, 235–250.

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Park, J. (2013). Is the derivative a function? If so, how do students talk about it? International Journal of Mathematical Education in Science and Technology, 44(5), 624–640. Simpson, A. (2015). The anatomy of a mathematical proof: Implications for analyses with Toulmin’s scheme. Educational Studies in Mathematics, 90(1), 1–17. Spivak, M. (1967). Calculus. W A Benjamin. Tall, D. O. (1981). Comments on the difficulty and validity of various approaches to the cal­ culus. For the Learning of Mathematics, 2(2), 16–21. Tall, D. O. (1991). Intuition and rigour: The role of visualization in the calculus. In W. Zimmermann & S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 105–119). Mathematical Association of America. Vandebrouck, F. (2011). Students’ conceptions of functions at the transition between second­ ary school and university. In Proceedings of the 7th conference of European researchers in math­ ematics education (pp. 2093–2102). Rzeszow. Weber, E., Tallman, M., Byerley, C., & Thompson, P. W. (2012). Introducing derivative via the calculus triangle. Mathematics Teacher, 104(4), 274–278. Zengin, Y. (2018). Examination of the constructed dynamic bridge between the concepts of differential and derivative with the integration of GeoGebra and the ACODESA method. Educational Studies in Mathematics, 99(3), 311–333.

5 INTEGRATION AND THE FUNDAMENTAL THEOREM OF CALCULUS

5.1 Introduction

This chapter discusses the main ideas of integral calculus (Section 5.2) and the fundamental theorem of calculus (FTC) (Section 5.3). Our focus is on the sig­ nificant mathematical and conceptual elements involved in making sense of these ideas. In light of these elements, we discuss a few different approaches to teach­ ing these main ideas, noting their pros and cons. We also include throughout the chapter descriptions of research about student reasoning with integrals and the FTC, since that research bears upon how we discuss the content and ways of teaching it. Throughout the chapter are interspersed mathematical and educa­ tional questions for reflection and discussion, with a few notes about these ques­ tions in Section 5.4. Also interspersed through the chapter are some problems from calculus classes that provide helpful contexts through which to discuss and illustrate content ideas. Like we did in Chapter 4, we can find a helpful summary of learning goals that relate to this chapter’s content in the AP Calculus introduction section for “Big Idea 3: Integrals and the FTC”:1 Integrals are used in a wide variety of practical and theoretical applications. AP Calculus students should understand the definition of a definite integral involving a Riemann sum, be able to approximate a definite integral using

1 See https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap-calculus-ab-and-bc-course-and­ exam-description.pdf

DOI: 10.4324/9781003204800-5

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Integration and the fundamental theorem of calculus

different methods, and be able to compute definite integrals using geometry. They should be familiar with basic techniques of integration and properties of integrals. The interpretation of a definite integral is an important skill, and students should be familiar with area, volume, and motion applications, as well as with the use of the definite integral as an accumulation function. It is criti­ cal that students grasp the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus – a central idea in AP Calculus. Students should be able to work with and analyze functions defined by an integral. Our purpose is to explore the reasoning students need in order to meet such learning goals, and some pedagogical approaches that can develop that reasoning. 5.2 Integrals

This section has five sub-sections: Why do we need integrals? Defining the (definite) integral Modelling with integrals Student reasoning with integrals Evaluating integrals – Part 1: front-door approaches 5.2.1 Why do we need integrals?

We start with a context that can help to develop an intellectual need for the idea of the integral. 5.2.1.1 Situation 1 – bar and ball (force of attraction)

For two point masses ma and mb located x metres apart, the law of gravitation g · ma · mb kg · m newtons ( ), states that the attractive force between them is F = x2 s2 3 m where g is a gravitational constant (with units ). Can you use this to kg· s2 determine the force of attraction between the 18 kg bar and the 4 kg ball (point mass) in Figure 5.1?

Integration and the fundamental theorem of calculus

FIGURE 5.1

135

Bar and ball problem

Before reading on, it is worth considering this question: On the first day of a calcu­ lus class on integration, what approaches would you expect students to use on this question? The following is from an article by Michèle Artigue (1999). She discusses why its inventor, M. Legrand, and others have found it effective for motivating the idea of integrals for new calculus students. We have used this problem on the first day of calculus classes too; students happily tackle it and argue about it even if they know nothing about integrals. Approach 1: Pretend the bar is a single point mass. This is what most students start with, but where should the point mass should be located? Most students contend that it ought to be at the bar’s centre of mass, so x = 6. A few will say it should be elsewhere, such as at the close end of the bar, making x = 3. But there are always a few students who worry that this approach does not work. Do the two halves of the bar pull toward the ball with the same intensity? We can test this.

FIGURE 5.2

Bar and ball solution, Approach 1

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Integration and the fundamental theorem of calculus

Approach 2: Cut the bar into two pieces, treat each half as a point mass located at its centre of mass, then calculate the two forces separately (Figure 5.3). Since we got a different force than with Approach 1, the students realise that the initial approach is not accurate. But they also realise that they might get a more accurate answer if they cut the bar into more pieces. Approach 3: Try the same thing as Approach 2 but cut the bar into, say, six pieces (Figure 5.4). Approach 4: At this point the students are ready to imagine that you will get more accurate answers if you chop the bar into more pieces. A truly accurate answer would be achieved only by chopping the bar into infinitely many pieces, and then adding up infinitely many little forces. How could such a thing be done before the end of time? This is what integration is for. 5.2.1.2 Situation 2 – Revisiting the area bounded by a ‘generalised parabola’

Based on the way most textbooks introduce it, it is easy to think that the (defi­ nite) integral is the area bounded by a curve. Area is actually just one of many contexts for an integral, but it is the one that best allows many of the integral’s aspects to be visualised. We revisit the example from Section 2.1 of the area bounded by a generalised parabola to help illustrate the definitions of Riemann

FIGURE 5.3

Bar and ball solution, Approach 2

FIGURE 5.4

Bar and ball solution, Approach 3

Integration and the fundamental theorem of calculus

137

sums and integral and the front-door and side-door approaches for evaluating integrals (Figure 5.5). Returning to our discussion of Fermat and Roberval in Section 2.1, they had found an underestimate and an overestimate for the area A under the curve y = x k: 1 N k+1

[1k + 2k + . + ( N - 1)k J < A
1 and partition the domain at the points p, pr, pr2, pr3, etc. Find the Riemann sum, use a geometric series to write it in closed form. Then let r → 1.

A left-sided Riemann sum for such a partition is given by ( pr - p )

( pr

2

- pr

) ( pr1) + ( pr 2

3

- pr 2

)

1

( pr ) 2

2

+ ., which simplifies to

1

(

+ pr 2 - pr

) ( pr1) + ( pr

1 1 1 J r -1( 1 J r + 2 + ) = [ ) = . As r → 1, this approaches . r r p [ 1 - 1r ) p p Mathematter Can every antiderivative of r be seen as an accumulation function for r?

It is possible for F to be an antiderivative of f but yet for there to exist no a such xˆ

I

that F ( xˆ ) = f ( x ) dx. For instance, F ( x ) = a

3

p r -1( 1 1 J r -1( 1 J 1 + + 2 + ) = [ )= [ p r r p [ 1 - 1r ) 2

1 3x e + 500 is an antiderivative for, but 3

2

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Integration and the fundamental theorem of calculus

not an accumulation function for, f ( x ) = e 3x . The problem is that f only has way less than 500 area total under it between x = – ∞ and 0. This is one reason that the FTC Part 2 is not just an immediate corollary of Part 1, because it applies to all antiderivatives, not just to ones that are also accumulation functions.

Mathematter What values of a and b maximise the value of

b

J (6 + x - x ) dx? How many dif­ 2

a

ferent ways can you solve this, and how do those ways relate to each other?

One way is to graph the curve y = 6 + x - x 2 (Figure 5.23) and look for values of a and b that maximise and minimise the area bounded between the curve and the x-axis. If we require b > a, then the maximum net area will be achieved by letting a = –2 and b = 3, and there is no minimum.

FIGURE 5.23

Graph of y = 6 + x - x 2

Integration and the fundamental theorem of calculus

177

References Artigue, M. (1999). The teaching and learning of mathematics at the university level. Notices of the AMS, 46(11), 1377–1385. Bos, H. J. M. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90. Czarnocha, B., Dubinsky, E., Loch, S., Prabhu, V., & Vidakovic, D. (2001). Conceptions of area: In students and in history. College Mathematics Journal, 32(2), 99–109. Edwards, C. H. (1979). The historical development of the calculus. Springer-Verlag. Ely, R. (2012). Loss of dimension in the history of calculus and in student reasoning. The Mathematics Enthusiast, 9(3), 303–326. Ely, R. (2021). Teaching calculus with infinitesimals and differentials. ZDM Mathematics Education, 53, 91–604. Fisher, B., Samuels, J., & Wangberg, A. (2016). Student conceptions of definite integration and accumulation functions. In the online Proceedings of the nineteenth annual conference on research in undergraduate mathematics education. Pittsburgh, PA. http://sigmaa.maa.org/ rume/RUME19v3.pdf Grundmeier, T. A., Hansen, J., & Sousa, E. (2006). An exploration of definition and procedural fluency in integral calculus. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 16(2), 178–191. Jones, S. (2015b). The prevalence of area-under-a-curve and anti-derivative conceptions over Riemann-sum based conceptions in students’ explanations of definite integrals. International Journal of Mathematics Education in Science and Technology, 46(5), 721–736. Katz, V. (2009). A history of mathematics: An introduction (3rd ed.). Addison Wesley. Kouropatov, A. (2016). The integral concept in high school: Constructing knowledge about accumulation (Unpublished doctoral dissertation. Tel Aviv University, Israel). Newton, I. (1666/1967). The October 1666 tract on fluxions. In D. T. Whiteside (Ed.), The mathematical papers of Isaac Newton (Vol. 1, 1664–1666, pp. 400–448). Cambridge University Press. Parr, E., Ely, R., & Piez, C. (2021). Conceptualizing and representing distance on graphs in calculus: The case of Todd. In S. S. Karunakaran & A. Higgins (Eds.), Proceedings of the 24th annual conference on research in undergraduate mathematics education (pp. 214–222). Boston, MA. Simmons, C., & Oehrtman, M. (2019). Quantitatively based summation: A framework for student conception of definite integrals. In J. Monaghan, E. Nardi, & T. Dreyfus (Eds.), Calculus in upper secondary and beginning university mathematics – Conference proceedings (pp. 91–95). MatRIC. Retrieved December 21, 2019, from https://matric-calculus.sciencesconf.org/ Thomas, M. O. J., Yoon, C., & Dreyfus, T. (2009). Multimodal use of semiotic resources in the construction of antiderivative. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides (Proceedings of the 32nd annual conference of the mathematics education research group of Australasia, Vol. 2, pp. 539–546). MERGA. Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208. Thompson, P. W. (1994). Images of rate and operational understanding of the Fundamental Theorem of Calculus. Educational Studies in Mathematics, 26(2–3), 229–274. Thompson, P. W., & Ashbrook, M. (2019). Calculus: Newton, Leibniz, and Robinson meet technology. http://patthompson.net/ThompsonCalc/ Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), First compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics.

6 INTERLUDE The ordering of chapters 3, 4 and 5

This short chapter reflects on the ordering of the previous three chapters: limits, dif­ ferentiation and integration. This is a (the?) classic ordering of a first calculus course. Mathematically, in terms of the objects of mathematics, it is appropriate: limits are the most common objects used to develop the objects in differential and integral calculus, so introduce limits first. We note, however, that: (i) Infinitesimals are different mathematical objects that can be used to develop the objects of differential and integral calculus (ii) Integration can be developed prior to differentiation. Further to this there are extra-mathematical reasons why calculus teachers could develop alternative orderings: (iii) The order of the three mathematical topics in the history of mathematics is integration, differentiation and limits (iv) The mathematical limit notion is a particularly difficult one for students to grasp. This book is written for calculus teachers and education researchers interested in calculus as a focus of research. The question of whether or not the classic ordering is the optimal ordering for teaching is an important issue and we, the authors, feel that you, the reader, should be given food for thought on this issue – hence this chapter. We provide this food for thought by enlarging upon (iii), (iv), (i) and (ii) but first ask you to reflect on the ordering.

DOI: 10.4324/9781003204800-6

Interlude

179

Edumatter Do you default to teaching in the order limits, differentiation and integration? If so, then: • Why? (Habit? Principle?) • How (and how far) do you develop the limit notion before moving on to differentiation?

6.1 The order of the three mathematical topics in the history of mathematics is integration, differentiation and limits

The first part of Chapter 2 of this book presents a history of calculus where the order integration–differentiation–limits is clear. Further to this, Bressoud (2019) presents a respected historical account of calculus where the first two chapters are Accumulation and Ratios of Change; the reference to the classical ordering is explicit in the title of the book Calculus Reordered. But the historical ordering of these topics does not mean that we should adopt this as a didactical ordering of the topics unless one believes that the statement ontogeny reca­ pitulates phylogeny, that individual development mirrors the development of the species, applies to mathematical development and the history of mathematics. Further to this, there are practical reasons why calculus teaching should adopt the classical ordering:

• Many (most?) teachers will have been taught calculus in this order. If you’ve been taught in this way, then you have a model of teaching sequences that may require a lot of work to replace. • Many curricula and assessment tasks are written using this order; changing this is an educational task beyond that which most teachers can enact. 6.2 The mathematical limit notion is a particularly difficult one for students to grasp

We begin by recapping on matters raised in Chapter 3. It is undoubtedly the case that calculus teachers have long known of students’ difficulties with limits, but this knowledge was not in the public domain until the 1970s, initially through the writ­ ings of Rolf Schwarzenberger, David Tall and Bernard Cornu; see, for example, Schwarzenberger and Tall (1978) and Cornu (1991). Students’ problems with limits have many sources. Dynamic process versus static objects: 0.9, 0.99, . . . will ap­ proach 1 but it will never get to 1 (this problem is exacerbated by the fact that math­ 0 ematics is atemporal, we don’t consider ‘time to sum’ in evaluating 9 x 10 -i ).

E

i=1

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Interlude

Language: “I don’t really see how numbers can converge” (Monaghan, 1993). Pre­ calculus mathematics: decimal notation ‘teaches us’ that any number starting ‘0.’ is less than 1, so it makes sense that 0.9, there exists a δ>0 . . .’ is often internalised as ‘for every δ>0, there exists a ε>0 . . .’. Basically, limits are hard. ‘Hard’ in mathematics can be very satisfying but only when you eventually ‘get it’. Try asking a colleague or a student teacher if 0.9 0 C e e

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Calculus applications

The two solutions

y = Ae x , y > 0 y = - Ae x , y < 0 can be combined as y = Be x where B is an arbitrary constant [noting A ≠ 0]. Graphically, this gives us an infinite family of exponential curves, as we may have expected by looking at Figure 7.3. When B is positive, they lie above the x-axis; when B is negative, they lie below it. The particular solution y = 0 also satisfies the original d equation since ( 0 ) = 0. dx Alternatively, this DE may also be approached analytically using a result we know from differentiation (see Section 4.3) that: dy 1 = dx dx dy If we didn’t know this result, we could consider y = f ( x ) , where the inverse func−1 tion f exists.

FIGURE 7.3

A plot of the gradient at points ( x, y ) for the DE

dy =y dx

Calculus applications

191

dy dy dy dx = 1 , then 1 = = * , by the chain rule (see Secdy dy dx dy dx dy tion 4.3), then * = 1, and hence the result. dy dx So Using the fact that

dy 1 = = y dx dx dy and dx 1 = , dy y

y=0

It is good to note the condition that y ≠ 0 for this method, since at each stage of mathematical processes we need to ask under what conditions the statement is true. d (0 ) However, we also noted earlier that y = 0 is a solution since = 0 and we don’t dx want to miss it. 1 dy dy Of course, if we use differentials then the result is immediate since = 1.= . dx dx dx dy This illustrates well the advantages of sometimes using differentials. dx 1 In the equation = we are again able to use our antiderivative knowledge to dy y 1 ask what function of y when differentiated with respect to y gives as the answer. y As before we get x ( y ) = ln y + C, C E  , and proceed to solve it exactly as before. As a third possible method we could use the following: Integrate both sides of the original DE with respect to x. Since dy =y dx then 1 dy = 1, y dx

y=0

and integrating the functions on both sides with respect to x 1 dy

{ y dx dx = {1dx The left hand side is equivalent to a change of variable (see Chapter 5), giving 1 dy

1

{ y dx dx = { y dy

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and so 1

f y dy = x + K , K E  ln y = x + K and then proceed as before. We will return to analytical methods of solving DEs later, but first we highlight a fact that students need to be made aware of, namely that many DEs cannot be solved analytically and for others it may be too difficult to try to do so, although these DEs also have an infinite number of solutions. For example, consider the DE dy = x + y 2. However, there are methods that will enable us to find approximate dx solutions to such DEs numerically. One of these methods is called Euler’s method. This method is best illustrated graphically, as in the next section, but first we return to the plotting of the gradients that we introduced earlier. 7.2.1 Approximate solutions to DEs 7.2.1.1 Slope fields

The fact that each DE gives rise to an infinite family of solution functions can be seen by plotting what is called a slope (or direction) field. In this we plot a short line segment representing the gradient or derivative at points plotted using the values of the variables. We have already seen these in Figures 7.2 and 7.3. dy For example, consider the slope field for the DE = f ' ( x ) = x - 1. dx Notice that we could write down the solutions to this DE using direct antidif­ x2 ferentiation, which gives y = f ( x ) = - x + C. 2 dy We can calculate the value of the derivative at various values of x, as seen dx in Table 7.1, and note that from the use of differentials, dy = ( x - 1) dx. So, as in an earlier example, the y increment dy will increase as x increases. TABLE 7.1 A table of values for the DE

x –3 –2 –1 0 1 2 3 4

f ' (x ) = x - 1

f ' (x ) = x - 1 –4 –3 –2 –1 0 1 2 3

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193

Edumatter Digital technology is extremely useful for plotting slope fields. Here we will often use GeoGebra but there is other software available. To draw with GeoGebra we use the commands Slopefield[] and SolveODE[]. Are graphical facilities for plotting slope fields available in the digital technology that you and your students use?

The slope field plot we want in this example, produced by GeoGebra, is given in Figure 7.4, along with a solution function through the initial point (– 3, 6). For solutions through a particular point say, P = ( 2,1) , the command in GeoGebra is SolveODE[x – 1, P]. This gives the solution curve seen in Figure 7.5. By setting up a general point P it may be dragged on the screen to show the solution curve through other points, and this can be highly instructive. 7.2.1.1.1 Questions to ponder

First, is there a solution function through every point of the plane? Is any solution through a point in the plane unique? How can we tell? To answer questions like this

FIGURE 7.4

The slope field for the DE f ' ( x ) = x - 1 , along with the solution curve through the point (– 3, 6), using GeoGebra

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Calculus applications

FIGURE 7.5

The slope field for the DE f ' ( x ) = x - 1 , along with the solution curve through the point (2, 1), using GeoGebra

rigorously requires proof, and this involves results on continuous functions. This is called an existence theorem, and it guarantees that a solution exists, and hence we are not wasting our time trying to find it. However, even without this theorem we dy can reason that there will be a unique solution [to = f ( x,y ) ] through every dx point (x,y) as long as the equation specifies the required direction (i.e., f ( x,y ) is defined). Second, to understand the solution quantitatively we want students to ask questions such as the following and make conjectures about them. For functions of time, what happens in the long term? Are fixed points, such as the origin stable (attractors) rather than unstable repellers, are there periodic solutions? Slope fields and the phase plane (see later) can help us answer these kinds of questions. When we have two or more variables in the gradient function, such as for the DE dy y ' (t ) = = t + y 2 , we can draw the slope field by using a table of values of t and y dt dy where each number in the table is a value of the gradient function , obtained by dt 2 calculating t + y (see Table 7.2). A plot of these values, as shown in Figure 7.6, gives the slope field. We can find an approximate solution function by specifying a starting point for it (an initial

Calculus applications TABLE 7.2 The values of

195

t + y 2 for values of t and y y

t –3 –2 –1 0 1 2 3

–3 6 7 8 9 10 11 12

FIGURE 7.6

–2 1 2 3 4 5 6 7

–1 –2 –1 0 1 2 3 4

0 –3 –2 –1 0 1 2 3

1 –2 –1 0 1 2 3 4

The slope field and sample solution curves for the DE

2 1 2 3 4 5 6 7

3 6 7 8 9 10 11 12

dy = t + y2 dt

value – see more on this later) and using some technology to draw the result. As we see in Figure 7.7 simply moving the initial values slightly can result in quite a different solution curve/function. An alternative to GeoGebra, depending on what students have access to, is to use a calculator such as the TI-Nspire that will also plot DE solutions; in university, students will usually have programs such as Matlab, Maple or Mathematica available. In this chapter we have used the computer version of TI-Nspire. Set the graph TINspire drawing mode to differential equations by selecting: 3. Graph Entry/Edit and then 7. Diff Eq

196

Calculus applications

FIGURE 7.7

Slope fields and solution curves using TI-Nspire

The default setting is y1 as the dependent variable and x and the independent varidy able, so for the DE = x + y 2 we can enter x + y12 in the box provided. On dx pressing Enter or Return the slope (direction) field is drawn (see Figure 7.7), and the axes may be dragged to set the scales.

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Clicking on ‘Add initial conditions’ below the DE entry enables start points for the drawing of approximate solution curves using Euler’s method4 to be added. Two examples are shown in Figure 7.7, where we have the initial points (0.6, 2) and (2, 2), again demonstrating different kinds of solutions. 7.2.1.2 Euler’s method

One way to describe Euler’s method for constructing an approximate solution func­ tion of a DE is to use an algorithm, such as: (i) Select a starting point, or initial value, ( p, q ) on a set of ( x, y ) axes. (ii) Using substitution in the DE calculate the gradient of the tangent at ( p, q ) [or find the differential dy in terms of dx] (iii) Draw a short line segment of length x , called the step size (or use the differen­ tial dx), that is part of the tangent at ( p, q ) [or draw the third side of the dx, dy triangle) (ix) Repeat from step (i), using the end of the tangent line segment as the next point. Note that this approximate solution to the DE is obtained by a sequence of discrete values leading to a continuous function made up of tangent segments rather than be­ ing an exact solution function. In addition this function is not differentiable at points where the segments intersect unless the gradient does not change. Clearly, the size of x can be crucial and the smaller the value of x the more accurate the approximation will be. Tall (1991) explains this method in term of the concept of local straightness: First order differential equations tell us the gradient dy/dx=f(x,y) of a solution curve. Because the curve has a derivative (equal to f(x,y) at (x,y)), it must be locally straight. So an approximate solution may be constructed by building up curves out of tiny line segments having the direction specified by the differential equation. ( ibid, p. 12, bold in original) dy Example: Find an approximate solution to the DE = y - x with initial value dx (0, 0.5). dy 1 1 The gradient of the tangent at this initial point is given by = - 0 = . If we dx 2 2 1 choose a step size Lx = then the first tangent segment is part of the tangent line 2

y-

1 1

= (x - 0) 2 2

4 Or a more advanced version called a Runge-Kutta method (see later).

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or y=

1 ( x + 1) 2

1 1 ( 1 J J ( 1 3 J 0 + ,

+1 = , . Fig­ 2 2 2

2 4

ure  7.8a gives the values for finding successive segments, along with the values dy dy yI = of the gradient function. It is worth noting that the gradient function, , dx dx of the exact solution is a continuous function, but the gradient function of the ap­ proximate solution is strictly a discrete function, with the gradient only found at each x step, formed into a continuous function by the addition of line segments joining the points. The end of the line segment has coordinates

x 0 0.5 1 1.5 2

y 0.5 0.75 0.875 0.8125 0.46875

(

y' 0.5 0.25 -0.125 -0.6875 -1.53125

( , ) points on segments and gradients

FIGURE 7.8

Use of Euler’s method to find an approximate solution to

dy = y-x dx

Calculus applications

199

dy = y - x are given by the functions dx x f ( x ) = ke + x + 1, k E  (see solution later). Both an exact (with k = -0.5 ) and approximate solution function, using a step size x of 0.5, are shown in Figure 7.8b. With this relatively large step size, we can see that the error in the approximation grows from around 0.07436 (11.0%) at x = 0.5 , to 0.23414 (36.5%) at x = 1.0, and 0.55334 at x = 1.5. We can increase the accuracy of the approximation by decreasing the x step size. One way to see this would be to program GeoGebra using a slider for the step size. Once again thinking in terms of differentials could be useful. So dy = ( y - x ) dx and from the table in Figure 7.8a, with dx = 0.5, dy = 0.5 ( y - x ) and hence the y increments are easily found. The exact analytic solutions to

Mathematter There are more accurate numerical approximations to solutions of DEs based on Euler’s method that are commonly used. Among these are those called Runga-Kutte methods, and the classical second order Runge-Kutta method uses a step of length ∆x based on the midpoint of an interval. A full and rigor­ ous discussion can be found in Butcher (2008).

The idea of using Euler’s method on a computer to draw approximate, numeri­ cal solutions to DEs is not new and was employed a number of years ago in Tall’s (1990a, 1990b, 1991) Solution Sketcher. This used an enactive method of building a solution by pointing at any position in the plane and drawing a small line segment in the appropriate direction. Enactive here is said to mean “the user carries out the physical act of following a solution curve, while the computer calculates the gradi­ ent” (Tall, 1991, p. 113). This process was then able to be repeated from the end of the previous segment as seen in Figure 7.9, taken from Tall (1991).

Edumatter The importance of enactive and embodied thinking5 has been highlighted in many recent studies in mathematics education, often with different emphases. Examples for further reading where this is discussed include Lakoff and Núñez (2000), Arzarello and Robutti (2008) and Tall’s Three Worlds of mathematical thinking (Tall, 2008, 2014). It should be noted that these authors have different perspectives on the notion of embodiment.

5 See Section 1.3.

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Calculus applications

FIGURE 7.9

Tall’s Solution Sketcher

7.2.2 Analytical methods for solving DEs 7.2.2.1 Variables separable

Variables separable is perhaps a common starting point for introducing analytical methods for solving DEs. It is used when it is possible to write first-order linear DEs in the form: x =

dx = f ( x ) g (t ) dt

We call this kind of linear DE variables separable, since f is separated from g. For exam­ ple, f ( x ) g (t ) = xe x + 1 .cost is of this type, whereas h ( x, t ) = x cos t - e t sin x + 1 cannot be written as a product of a function of x and a function of t. To solve DEs with variables separable we note that we can always rearrange them in the form:

(

)

1 dx = g (t ) f ( x ) dt for f ( x ) e 0, and so integrating the functions on both sides with respect to t (as­ suming the functions f and g are integrable) we get 1

dx

| f ( x ) dt dt = | g (t ) dt

Calculus applications

201

and so, using the substitution formula (see Chapter 5) 1

{ f ( x ) dx = { g (t ) dt Writing the DE in terms of differentials, we can change

dx = f ( x ) g (t ) to dt

dx = f ( x ) g (t ) dt dx = g (t ) dt f (x ) and summing the arbitrarily small increments on each side we get dx

{ f ( x ) = { g (t ) dt as before. Common examples in real-world contexts where this kind of DE is found involve growth and decay (e.g. population growth and radioactive decay), concentration of drugs in the blood stream or spread of a virus in a population. For example: Let x be the number of people infected with a virus after t days. Given a 10% growth rate in numbers per day and 10,000 people initially infected, find the number infected after 7 days. The 10% growth rate may be written as: dx = 0.1x dt dx = f ( x ) g (t ) , with f ( x ) = x and g (t ) = 0.1 [or equivalent, dt such as f ( x ) = 0.1x and g (t ) = 1]. We will see shortly that DEs of this form, where the right-hand side is independent of t, the independent variable, are called autonomous linear systems. Usually a linear system has more than one DE (which gives rise to the name system). First, we should note that the function x (t ) = 0 is a solution of the DE, since d (0 ) = 0.1 X 0 = 0. Then, separating the variables and integrating with respect to dx t, gives

This is of the form

1 dx

{ x dt dt = 0.1{dt, 1

{ x dx = 0.1t + C

x=0

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dx [or using differentials, = dx 0= .1xdt , 0.1dt and summing both sides] x

ln x = 0. 1t + C since the number of people x > 0. From our initial values x ( 0 ) = 10000 so C = ln10000 and ln x = 0.1t + ln 10000 ln

x = 0.1t 10000

x = e 0.1t 10000 x = 10000e 0.1t 0.7 So when t = 7 days = x 10000e = 20,137.5 or 20137 people. We note that the solution for x (t ) has the form x (t ) = x ( 0 ) e 0.1t and it can be shown, as seen earlier, that the general solution of growth and decay DEs of the form

dx = Ax dt where λ may be positive (growth) or negative (decay) is x ( t ) = x ( 0 ) e -t . EXERCISE

A more realistic DE for the spread of a virus includes recovery rates (there are of course even more sophisticated DE models for virus spread, as used in the Covid-19 pandemic and these may use DEs with versions of susceptible, infected and recov­ ered individuals6). The logistic equation models this. This is: dx = ( A - Bx ) x dt 6 See, for example, Harjule, P., Tiwari, V., & Kumar, A. (2021) Mathematical models to predict COVID-19 outbreak: An interim review. Journal of Interdisciplinary Mathematics, 24(2), 259–284. https://doi.org/10.1080/09720502.2020.1848316.

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203

where A is the constant rate of increase of spread and Bx the recovery rate. Solve this initial value problem DE when x ( 0 ) = 50 , with A=1 and B=0.5. What happens long term when B is close in value to A? 7.2.2.2 First-order linear DEs

A second class of first-order DEs is called linear and they have the general form: dy + f ( x ) .y ( x ) = g ( x ) dx We note that if g ( x ) = 0 then we call it a homogeneous linear DE and these reduce to a form that can be solved by separating the variables. To solve linear DEs we make use of an integrating factor. When f is integrable on a suitable interval, if we multiply each term by e ) ef

f ( x )dx

.

f ( x )dx

then the following happens.

f ( x )dx f ( x )dx dy + ef . f ( x ) .y ( x ) _ e f .g ( x ) , dx

and we note (this may not be obvious at first) that the left hand side can be written e|

f ( x )dx

.

dy + e| dx

f ( x )dx

| | f ( x )dx .y ( x ) | d |e| | . f ( x ) .y ( x ) = | dx

which was the reason for the multiplying factor! Hence, the DE becomes l i f ( x )dx d le) .y ( x ) j dx

= e)

f ( x )dx

.g ( x )

and integrating with respect to x, or thinking in terms of differentials and summing, gives ef

f ( x )dx

.y ( x ) = e f

f

f ( x )dx

.g ( x ) dx + C

and hence y ( x ) , providing the integral on the right can be found. NOTES

Since e ) factor.

f ( x )dx

> 0 for all x the DE is not changed by multiplying through by this

We are assuming that f ( x ) is integrable.

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Calculus applications f ( x )dx

We are assuming that e { . g ( x ) is integrable. Some real-world contexts where we find linear first-order DEs include linear motion with air resistance, virus growth in a population with recovery, population growth with migration and concentrations of solutions in fluids where there is both an inflow and an outflow. dy The DE we considered earlier in the Euler method, = y - x , is linear first-ordx der with f ( x ) = -1 and g ( x ) = -x. The slope field for this is given in Figure 7.10. To solve the DE analytically we have e{

f ( x )dx

= e{

-1dx

= e -x +c = e c e -x

We can choose the constant c = 0 to simplify the function needed, since e 0 = 1 , and multiplying through by e −x (>0) we get e -x

(

dy - e - x y = -x.e -x dx

d e -x y dx

) = -x.e

FIGURE 7.10

-x

The slope field for

dy = y-x dx

Calculus applications

205

and integrating with respect to x (right-hand side by parts – see Chapter 5) e -x y = e -x ( x + 1) + K or, multiplying through by e x y ( x ) = x + 1 + Ke x for real K, and this is the solution we used previously. Figure 7.12 shows two of the solution curves corresponding to K = 3 and K = -0.5. We notice that as x - o _ Ke x - 0, and the solution functions y ( x ) - x + 1, as seen on the right­ hand side of Figure 7.11. dy = y - x could be to It is worth noting that an alternative approach to solving dx use the fact that the sum of any two solutions to a linear homogeneous DE is also dy a solution. Thus we could solve the homogeneous DE - y = 0 (solved earlier) dx x to obtain y = Ke and then add the particular solution y = x + 1 inferred from the d ( x + 1) dy = x + 1 - x = 1 and slope field (and it is easily verified that = 1, making dx dx it a solution). 7.2.2.3 An initial value example

A tank contains 10 litres of water. A solution with a chemical concentration of 2 g of chemical per litre is added to the tank at a rate of 1 litre per minute. The solution, which is kept well mixed, is drained from the tank at a rate of 1 litre per minute. If C (t ) is the amount of the chemical (in g/litre) in the solution after t minutes, rep­ resent the initial value problem7 by a differential equation. What is the concentration of the chemical after 5 minutes? To solve this initial value problem we use the rate of change of chemical in the solution, C ' (t ) =

dC = rate of salt in - rate of salt out dt

where: the rate in = 2 x 1 = 2 g/minute C the rate out = x 1 g/minute, 10 + t

7 An initial value problem comprises a DE along with a set of initial values and specifies a single solution among the infinite number of solution functions.

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Calculus applications

FIGURE 7.11

Solution curves for

dy = y - x , and as x -> 8 - y ( x ) -> x + 1 dx

Calculus applications

207

C since 10 + 1 x t is the volume of solution after t minutes and the proportion 10 +t of chemical in the solution after t minutes. So dC C = 2_ dt 10 + t is the differential equation representing the situation. Rewriting this as dC C + =2 dt 10 + t we can see that it has the form of a linear first-order DE with f (t ) =

1 10 + t

and g (t ) = 2. However, before we try to solve this analytically it would be useful to consider what the solution to the initial value problem might look like using the slope field. We can draw this using GeoGebra again (see Figure 7.12). Note that C (t ) is only

FIGURE 7.12

dC C + = 2 and a solution curve for the initial dt 10 + t

value problem with C ( 0 ) = 0

The slope field for

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Calculus applications

defined for t ≥ 0 , and this gives the region we are interested in looking at for solu­ tions. In particular we are solving the initial value problem where C ( 0 ) = 0 , and this approximate solution function is shown on the slope field. For the analytical solution, we first find ef

f (t )dt

1

=e

f 10 +t dt

ln 10+t +k = e ( ) = e k (10 + t ) , k E 

Note 10 + t > 0 so the natural logarithm is well defined.

Choosing k = 0 and multiplying through by this term we get

dC C = 2 (10 + t ) + (10 + t ) dt 10 + t

dC (10 + t ) + C = 2 (10 + t ) dt

(10 + t )

d (C (10 + t ) ) dt

= 2 (10 + t )

and integrating both sides with respect to t [ t2 ] C (10 + t ) = 2 10t + + m = t ( 20 + t ) + m 2 C (t ) =

t ( 20 + t ) + m 10 + t

Initially, C ( 0 ) = 0, so 0 = C (t ) =

mE

0+m and m = 0, giving 10

t ( 20 + t ) 10 + t

and C ( 5) =

5 ( 20 + 5 ) 10 + 5

=

25 1 = 8 g/litre 3 3

That this solution fits the slope field can be seen in Figure 7.13, where the point B 1

is approximately ( 5, 8 } . 3

Calculus applications

FIGURE 7.13

Verifying graphically the analytical solution for

C (0) = 0

209

dC C + = 2 , with dt 10 + t

EXERCISE

A closed electrical circuit contains a resistance R and an inductance L. When a volt­ age V is applied the current i(t) at time t is given by L

di (t ) dt

+ R.i (t ) = V

Solve the initial value problem i(0)=0 to find I in terms of V, R and L. What is the ‘steady’ current (i.e. the limit of the current as t - o) ? 7.3 Second order DEs

A second order DE in two variables contains a second derivative of the dependent variable with respect to the independent variable. Examples are: d 2S = -kS, k constant dt 2 5

d 2z dr

2

-nz

dz = r 3

dr

210

Calculus applications

7.3.1 Second order linear DEs with constant coefficients

A common class of second order DEs that can be solved analytically is those with constant coefficients. The first DE mentioned previously is of that type, the second is not, due to the z in the second term. Mathematter The derivative is an example of a broader mathematical concept of a linear operator, where, for example, a linear transformation, with linear operator L, is defined as,

(

) (

) (

L f (x ) + g (x ) = L f (x ) + L g (x )

)

or L (f + g ) = Lf + Lg and

(

)

(

L af ( x ) = aL f ( x )

)

or L ( af ) = aL (f ) where f and g are functions and a, b scalars or real constants (Note that the integral function can also be viewed as a linear operator). We will comment on d as a linear operathis in the method later, where we can think of using D = n dx d n tor, with D = n .

dx dn Any combination of derivatives Dn = n will comprise a linear operator, in dx a polynomial form. Why do you think an operator like L is called linear? How would you introduce this idea to your students?

A general linear second order DE has the form d 2x dt

2

+ m (t )

dx + n (t ) .x = r (t ) dt

or x "" (t ) + m (t ) x " (t ) + n (t ) .x = r (t ) In the case where r (t ) = 0 we call the DE homogeneous, as before.

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211

If we consider just those DEs where m (t ) and n (t ) are constant functions, then we can find analytic solutions. In other cases we reduce the DE to an autonomous linear system, using the method described next, and find approximate solutions. Here, considering the homogeneous DE with constant coefficients d 2x dt

2

+a

dx + bx = 0

dt

based on experience, and noting that its derivative is closely related to the function itself, if we try substituting x (t ) = ke mt , then x ' (t ) = mke mt x '' (t ) = m 2ke mt substituting gives m 2ke mt + amke mt + bke mt = 0 factorising

(

)

ke mt m 2 + am + b = 0 and since e mt > 0, Vm,t, m is a root of what is called the auxiliary equation m 2 + am + b = 0 giving the values m1 and m2 of m for the solutions k1e m1t and k2e m2t and the general

solution takes the form of the complementary function x (t ) = k1e m1t + k2e m2t , for real k1 , k2 , m1 , m2 . The case where real roots m1 and m2 do not exist is considered later. We note that the auxiliary equation corresponds directly with the idea of the lin­ d ear operator D introduced earlier, where using D = would give D 2 + aD + b = 0, dx and we can also think of D 2 + aD + b itself as a polynomial linear operator. EXERCISES

(i) Show that D 2 + aD + b satisfies the conditions for a (polynomial) linear opera­ tor previously given, namely L ( f + g ) = Lf + Lg and L ( af ) = aL ( f ) . (ii) Substitute x (t ) = k1e m1t + k2e m2t into the left hand side of the DE to verify that it gives 0 and hence is a solution.

212

Calculus applications

Mathematter In general, if f ( x ) and g ( x ) are two linearly independent8 solutions of a linear homogeneous equation then a linear combination of them, af ( x ) + bg ( x ) , a,b constants, is also a solution. How would you check this? The set of linearly in­ dependent solutions of a linear homogeneous DE span a vector space.

If the roots of the auxiliary equation are real and equal, then the solution becomes x (t ) = k1e mt + k2 xe mt = ( k1 + k2 x ) e mt d 2x dx Check again by substituting this into the left hand side of the DE 2 + a + bx = 0 dt dt to verify that it gives 0 and hence is a solution. We may wonder what happens if the values of m from the auxiliary equation are complex rather than real. What then? We note that if m1 = p + iq,

m2 = p - iq

p, q real, are the complex roots of the auxiliary equation then y = e pt ( k1 cos qt + k2 sin qt ) is the general solution of the DE. EXAMPLES

(i) Find the general solution of the homogeneous DE x "" (t ) + 3x " (t ) - 4x = 0 The auxiliary equation is m 2 + 3m - 4 = 0 with

( m + 4 )( m - 1) = 0

m = -4 or 1

8 So one is not a constant multiple of the other, on the same interval.

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213

and the general solution is x (t ) = k1e -4t + k2e t Figure 7.14 shows the solution curves for some of these functions. (ii) Simple harmonic motion Simple harmonic motion (SHM) occurs when a mass moves back and forth about an equilibrium position under the influence of a restoring force proportional to its displacement and it has a number of applications in physics, such as a shock absorber and a pendulum. If, for example, a mass M attached to a spring is in vertical motion under gravity then if S is the displacement of the mass from the equilibrium position, we can obtain the DE for the motion as S '' (t ) = -

Y S (t ) M

where λ is a constant for the spring. Solving this linear homogeneous DE in the form S '' (t ) +

FIGURE 7.14

Y S (t ) = 0 M

Graphs of some analytical solution functions for x'' (t ) + 3x' (t ) - 4x (t ) = 0

214

Calculus applications

the auxiliary equation is m2 +

A =0 M

with complex roots m1 =

f f .i and m2 = .i M M

and the general solution, from the general solution given earlier, is | y y | S (t ) = e 0xt | k1 cos .t + k2 sin .t | | M M || | and so S (t ) = k1 cos

y y .t + k2 sin .t. M M

EXAMPLE

A mass M stretches a spring by 0.05 m at the equilibrium point and then it is pulled down 0.12 m below that equilibrium point and released. What is the general equa­ tion of its motion? At equilibrium the weight is balanced by the force in the spring which is given by F = Al , from Hooke’ law, where again λ is a constant for the spring. So at this point

Al = Mg and in this case 0.05A = 9.8 M using g = 9.8 ms -2

A = 196M

A = 196 M Substituting in our general solution S (t ) = k1 cos 196.t + k2 sin 196.t S (t ) = k1 cos14t + k2 sin14t

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215

To find k1 and k2 we use the initial conditions that S ( 0 ) = -0.12 (below the equi­ librium point but measuring travel upwards) and S ' ( 0 ) = 0 (i.e. it starts at rest). So S ( 0 ) = -0.12 = k1 cos 0 = k1

k1 = -0. 12 and S ' ( 0 ) = -14k1 sin 14.0 + 14k2 cos 14.0 = 14k2 14k = 0,= k2 0 2 The general solution is then S (t ) = -0.12 cos 14t This enables us to answer questions such a (i) Where is the mass after 5 seconds? (ii) What is the velocity of the mass after 2 seconds? ANSWERS

(i) S ( 5 ) = -0.12 cos 14 X 5 = -0.12 cos 70 : -0.076 m i.e. 0.076 m below the equi­ librium point (ii) S ' ( 2 ) = 0.12 x14 sin 14 x 2 = 1.68 sin 28 ~ 0.455 ms−1, that is upwards EXERCISE

When does the mass first come to rest? 7.3.2 Autonomous linear systems

One class of important second order linear DEs are those with independent variable t that can be written as an autonomous linear system, that is one where the righthand sides of the equations do not explicitly contain the independent variable, t. One advantage of such a system is that we can consider the behaviour of the DE without having to find explicit analytical solutions. While higher-order DEs may be converted to linear systems by introducing new variables for the derivatives, we will only consider a basic second order example here, where the system is autonomous. For example, taking d 2x dt

2

+a

dx + bx = 0

dt

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Calculus applications

and letting y =

dx , then dt

dy + ay + bx = 0 dt giving the system of 2 DEs: dx =y dt dy = -ay - bx dt which is an autonomous linear system, since the right-hand sides, and hence both dx dy and , are independent of t. This system can be used to describe damped har­ dt dt monic motion, such as arises from damped spring oscillations. For such systems ( x (t ) , y (t ) ) is the position in the x-y plane at time t. Since dx y (t ) = , the velocity in the x direction, this plot of x against y, its velocity, is often dt called the phase space or phase plane. As t varies the point ( x (t ) , y (t ) ) moves along dx dy a curve in the phase space. We note that at ( 0, 0 ) , = = 0, indicating that the dt dt origin of this phase space is an equilibrium or rest point. dx The other points where = 0 are called x nullclines and give a gradient paral­ dt dy lel to the y-axis, since the x velocity is zero, and those where = 0 are called y dt nullclines and give a gradient parallel to the x-axis, since the y velocity is zero. The fixed points of the system are where both the x velocity and y velocity are zero and dx dy = 0 and = 0. Classifying the nature of so are found to be points where both dt dt the fixed points is one of the goals of investigating the system. This is because these are the equilibrium points of the system and may be stable (attractor) points or unstable (repellor) points. For practical applications knowing the stable points of a system can be very useful. An example of a plot of the slope field and a solution for the system earlier with ' y (t ) = 0.8y - 1.6x are shown in Figure 7.15. In this case the x nullclines are given dx = 0 - y = 0 , the x-axis, where we see the vertical gradients. dt dy The y nullclines are found from = 0 - y = 2x , and we can see the horizontal dt gradients along this line. They divide the plane into regions based on the sign of the x velocity and y velocity. dx dy The fixed points of the system occur when both = 0 and = 0 so y = 0 dt dt and y = 2x giving (0,0) as the only fixed point. by

Calculus applications

FIGURE 7.15

217

The phase space slope field and a solution curve for x' (t ) = y,

y ' (t ) = 0.8y - 1.6x

( dx dy ) At the point (1,2) on the y nullcline y = 2x we have | , | = ( 2, 0 ) , so the ( dt dt ) direction of the spiral is towards positive x. Similarly, the velocity at the point ( -3, 0 ) is ( 0, 4.8 ), which is towards positive y. This information tells us that the spiral solution curves move away from the unstable fixed point at the origin (see Figure 7.15). It can be very instructive to allow students to investigate systems such as dx =y dt dy = -ay - bx dt using a program such as GeoGebra, with sliders for the parameters a and b. They can consider the nullclines, the fixed points, whether they are classified as nodes, spirals or saddles, and if they are stable or unstable (the formal classification is made using a vector/matrix approach and eigenvalues). Examples of the solution curves are shown in Figure 7.16. Each has x nullcline y = 0 and only the origin (0,0) as a fixed point. How would we identify its stability?

218

Calculus applications

( a, b ) = ( -0.3, -0.6 )

( a, b ) = ( -0.5, 5) FIGURE 7.16

Nullcline y = -2x

Nullcline y = 10x

( a, b ) = (1.4, -0.9 )

( a, b ) = ( -2, 3.6 )

Nullcline y =

9 x 14

Nullcline y = 1.8x

Saddles and spirals for autonomous linear systems

Mathematter and Edumatter Ideas about autonomous linear systems are used extensively in the theory of dynamical systems, the study of which is beyond the intended scope of this book. However, simple systems include examples such as the first-order DEs for growth and decay over time have been considered. This important area of study builds strong links between vectors and matrices and differential calculus. Further information can be found in books such as Campbell and Haberman (2008).

7.4

What we know about differential equations from research

There has been relatively little research into the learning of differential equations. This may be partly due to the fact that in many parts of the world they are only

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219

studied at the university level, and research on undergraduate mathematics has only recently been considered to any extent (see Bosch et al., 2021). One of the early studies by Artigue (1992) began a research theme of looking at the value of visualisation by looking at geometrical approaches to the understanding of DEs that has continued (e.g. Fardinpour & Gooya, 2018). This trend has been a response to the difficulties experienced by students in learning about DEs by tradi­ tional approaches, which have emphasised finding exact solutions to DEs in closed form (Kwon, 2002), although, in practice, solutions can rarely be expressed in such a form (Rasmussen et al., 2006). As one of the pioneers of research into DEs, Ras­ mussen (1997, p. 4) observed: Until recently, numerical and qualitative methods for analyzing differential equa­ tions were not part of most introductory courses in differential equations, but rather were reserved for graduate or advanced study. This can be attributed in part to the fact that numerical and qualitative methods require extensive comput­ ing, hand-sketching of graphs (in particular, direction fields, isoclines, time series, and phase portraits), and careful analysis of the differential equations themselves. In a later paper (Rasmussen, 2001) he identified several key difficulties that students had. For example, they did not see solutions to DEs as functions, and that these can be presented graphically, understand equilibrium solutions as constant functions that satisfy the DE and appreciate that a numerical approximation and an exact solution are not the same. Others have found that even students who could find algebraic solutions for DEs “did not fully understand the related concepts, and they had seri­ ous difficulties in relation to these concepts” (Arslan, 2010, p. 873). In order to address these problems an inquiry-oriented DEs curriculum has been employed focused on a student-centred approach and aiming to be more visual and technology rich. Originally developed by Rasmussen (2002), and building on his prior research, a realistic mathematics education (RME)-inspired inquiry-oriented approach has been used in a number of research studies. These inquiry-oriented ma­ terials draw on a dynamical systems point of view that “treats differential equations as mechanisms that describe how functions evolve and change over time” (p. 86). These materials have been described as a rare example of advanced mathematics “re­ form-minded, research-based principles of instruction. The materials include several series of problems, activities, and accompanying Java applets designed to guide stu­ dents through the discovery of the core concepts of a dynamical systems approach to differential equations” (Wagner et al., 2007, p. 252). The aim has been to use a con­ nected, long-term series of problems where students create and elaborate symbolic models of their informal mathematical activity (Rasmussen, 2001). This modelling is supported by both individual activity and whole-class discussions to promote stu­ dent engagement in progressive mathematisation (Kwon, 2002) as they engage in guided reinvention. One example, seen in Rasmussen and King (2000) was where students were guided toward reinvention of Euler’s method using autonomous linear

220

Calculus applications

systems of DEs. It was concluded that such a program was a promising means for promoting conceptual reasoning about rate (ibid). Others agree (Arslan, 2010) that courses need to include activities aimed at en­ hancing student understanding of basic concepts, including “what a DE and a solu­ tion to a DE are, the relationship between a DE and its solutions, the geometric interpretations of DE and solution concepts, and modelling via DEs” (p. 887). These studies have informed us that students tend to overgeneralise, for example, by thinking there are equilibrium solutions whenever a differential equation is zero, even if the equation is not autonomous and in their use of approximate solutions when using numerical methods (Rasmussen, 1997; Zandieh & McDonald, 1999). We also know that, according to Trigueros (2004), students are often unable to dis­ tinguish between straight line solutions and nullclines. In a study based on autonomous systems of DEs, Stephan and Rasmussen (2002) identified the emergence of six classroom mathematical practices that included aspects such as reasoning about the way slopes change over time (including invariance), creat­ ing and organising collections of solution functions, realising that graphs of solution functions do not touch each other but graphs are horizontal shifts of each other. They also noted that it was surprising that the idea that for autonomous DEs slopes are invari­ ant horizontally and that solution functions are horizontal shifts did not emerge in a linear manner for students. The reform-based RME study of Kwon (2002) concluded that engaging students in mathematising activity that supports reinventing conventional representations can cause slope fields and graphs of solution functions to emerge. Habre (2012) introduced a writing component into introductory differential equations course classes since this dovetailed well with the reformed teaching ap­ proach adopted. An example of the writing tasks was Your task is to introduce to your audience in the simplest way the concept of a (ordinary) differential equation. Elaborate in a short paragraph how you would complete this task. Support your ideas with examples and describe/explain the various approaches to solve such equations. (p. 157) There was some evidence that such writing improved understanding of DEs. In summary, there has recently been good progress in research on DEs, and the research has focused on employing reform approaches, including an emphasis on geometrical considerations, to improve difficulties students face while trying to un­ derstand solutions to DEs (Fardinpour & Gooya, 2018). However, many challenges still lie ahead for the teaching and learning of DEs (Kwon, 2020). 7.5 Applications of integration

A basic principle, established in Section 5.3, is that when we know how fast some­ thing is changing, i.e. the rate at which it is changing, we can accumulate small

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221

increments in the value of the variable using Riemann sums and then when we take the limit of these sums over a range of the variable, we obtain a definite integral. Alternatively we can think in terms of summing of differentials, such as xdx, to obtain the integral. We will approach all of the different examples of applications of integration here using this basic method. 7.5.1 Work done by a force

The work done by a constant force F moving a distance d is equal to F.d. Thus when a force F ( s ) varies over distance s we can use this formula to write the work done in moving a small distance δ s (in time period δ t ) as

8 W = F .8 s where δ s is the distance travelled in time δ t. The accumulation of these small amounts δ W of work over distance s = a to s = b can be summed as i =n

W =

LF ( s J .8 s i

i=1

and taking the limits as 8 s - 0 we get i =n

W = lim

8 s-0

b

L

J

F ( si J .8 s = F ( s J ds

i=1

a

Once again, we can think of this as summing the arbitrarily small differentials F .ds b

f

as s changes from a to b to obtain F ( s ) ds . a

If F is a function of t it requires a change of variable to find the work done over t. If i =n

W =

i =n

ss

EF (t ) .s s = EF (t ) . s t .s t i

i

i=1

i=1

and taking the limit as 8 t - 0 we get i =n

t2

t2

t1

t1

ds 8s W = lim F (t i ) . .8 t = F (t ) . dt = F (t ) .v (t ) dt 8 t H0 dt 8t i=1

"

J

J

222

Calculus applications

EXAMPLE

According to Hooke’s law the tension force T in a spring is proportional to the ex­ tension from its natural length (but within the elastic limit). So T ( s ) = ks where k is a constant for a given spring. So the work done in stretching a spring horizontally 0.5m from its natural length is equal and opposite to the tension and 0.5

0.5

0

0

f

W = T ( s ) ds =

f ksds =

[ k 2 ] 0.5 k s = joules 0 2 8

(k can be found by considering the extension when the spring is stretched vertically by a known mass and hangs in equilibrium.) Mathematter When the force is not acting in a straight line the questions are best done using vectors. Then we can use b

f

W = F.ds a

and t2

|

W = F.v dt t1

where the dot is the scalar product.

7.5.2 Volume of revolution by accumulating disks

Consider the graph of the function y = h ( x ) (see Figure 7.17) from x = a to x = b. This portion of the graph is rotated through 2π radians or 360˚ about the x-axis. The problem is to find the volume of revolution thus generated. If we take the volume to be made up of very small disks, each of width δ x , then each small disk is approximately cylindrical in shape and so its volume is given by Volume = area of cross section x width 2

sV = t ||h ( xi ) || x s x

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223

y=h(x) h(xi

x

FIGURE 7.17

Discrete examples of the slices of width δ x accumulated in the volume of revolution of the graph of the function y = h ( x ) from x = a to x = b

224

Calculus applications

Accumulating the volume by summing n of these small disks in the interval [a, b ], we get i =n

V =t

E ||h ( x )|| i

2

xsx

i=1

b-a , and the actual volume is the limit of this sum as 8 x - 0 (i.e. n n - o, see Section 5.3). where 8 x =

i =n

V = p lim

s x-0

E i=1

b

|

2

2

||h ( xi ) || x s x = p ||h ( x ) || dx a

Alternatively we can again think of summing differentials, avoiding the need for limits. EXAMPLE

Airplanes often use a parabolic shaped nose cone because it creates less drag due to good airflow around it. If the 2.8 m long nose cone of a Boeing airplane (see Fig­ ure 7.18) can be modelled by rotating the graph of f ( x ) = 0.5 2.8x , 0 : x : 2.8, through 360˚ around the x-axis, what volume does it contain? b

f

2

Using the volume of revolution formula V = n [h ( x ) J dx , we have a

2.8

V =1

f [0.5

2

2.8x r dx

0

: V = 4

2.8

2.8

0

0

' 2.8xdx = 0.7: ' xdx = 0.7:

V = 0.7n

[ 2.82 ] 2

[ x2 [ 2

2.8 0

= 0.7 x 3.92n : 8.62m3

7.5.3 Accumulating washers

Each small washer shape in Figure 7.19 can be seen as the difference between two approximately cylindrical shapes and so its volume is given by Volume = outer cylinder – inner cylinder 2

2

{

2

2

}

sV = t |\ g ( xi ) || x s x - t \|h ( xi ) || x s x = t \| g ( xi ) || - \|h ( xi ) || s x

Calculus applications

FIGURE 7.18

Modelling the nose cone of a modern Boeing airplane

225

226

Calculus applications

FIGURE 7.19

Discrete examples of the shell slices accumulated as the graphs of the functions y = g ( x ) and y = h ( x ) from x = a to x = b are rotated about the x-axis

Accumulating the volume by summing n of these small washers in the interval [a, b ] , we once again get

E{[[ g ( x )[] i =n

V =#

i

i=1

2

2

}

- [[h ( xi ) [] 8 x

b-a , and the actual volume is the limit of this sum as 8 x -> 0 [ie n n -> 8], giving

where 8 x =

E{ i =n

V = # lim

8 x ->0

i=1

2

2

}

[[ g ( xi ) ]] - [[h ( xi ) ]] 8 x = #

{ {[[ g ( x )]] b

a

2

2

}

- [[h ( x ) ]] dx

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227

Alternatively, again, we can think of this in terms of differentials and summation, so

{

2

2

}

dV = t || g ( xi ) || - ||h ( xi ) || dx and

fdV = n f {[ g ( x )J

2

i

2

}

- [h ( xi ) J dx = n

f {[ g ( x )J b

a

2

2

}

- [h ( x ) J dxx

EXAMPLE

The conical end (or frustum) of a 60˚ conical brewing tank (see Figure 7.20) is made of metal 3 mm thick. It is 1 m high and is 20 cm wide at the drainage hole end. What volume of metal is in the conical end? We can model the upper inside part of the conical end (on its side) with the function f ( x ) = 10 +

3 x cm, 0 < x < 100 3

and the outside with g ( x ) = 10 + 0.2 3 +

3 x cm, 0 : x : 100 3

Hence the volume of metal in the conical part of the tank is given by

| {|| g ( x )|| b

V =t

2

2

}

- || f ( x ) || dx = t

a

100

V =n

J (0.12 + 4

100 |

| 0

2| 2 3 | | 3 | | || x | - | 100 + x | | dx ||| 10 + 0.2 3 + 3 || || 3 || | ||| |

)

3 + 0.4x dx

0

100 | | 0.4x 2 | || | V = t | |0.12x + 4 3 x + | | 2

| 0 | || | |

(

V = 7 12 + 400 3 + 2000

)

V ≈ 8497cm 3

So the volume of metal in the conical end of the brewing tank is approximately 8497 cm3.

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Calculus applications

FIGURE 7.20

A brewing tank with a truncated conical end modelled graphically, with close up; (Brewing tank image made available by Paul Brennan under CC0 Public Domain license)

Calculus applications

7.5.4

229

Accumulating lengths

In order to find the length of a curve9 (see Figure 7.21) we can accumulate small lengths along it. A small length δ l is approximately given by

( 8 x ) 2 + ( 8 y )2

8l =

8l = 8x

( 8 x )2 ( 8 y )2 + ( 8 x )2 ( 8 x )2

( 8y ) = 8x 1+ | | (8x )

2

since 8 x > 0. Accumulating the arc length by summing n of these small lengths over the interval x e [a,b ] , we get i =n

L=

E i=1

2

(8y ) 1+ | | 8x (8x )

b-a , and the actual length is given by the limit of this sum as 8 x -> 0 n (i.e. n → a ), giving where 8 x =

b

L=

{ a

2

( dy ) 1 + | | dx = ( dx )

FIGURE 7.21

b

{

1 + y ' ( x ) dx 2

a

Accumulating the length of the graph of a function

9 Also considered in Section 5.3.5.

230

Calculus applications

EXERCISE

Rewrite the earlier information using the differentials, dx, dy, dl.

Mathematter Generally a chain will hang in the form of a catenary10 (from the Latin catena for x ( x - J l [ a a + e e x ( J chain), whose equation is of the form y = acosh [ l = a [ l . However, 2 /a/ l [ l [ / / /x/ as we see in Figure 7.22, which shows the catenary y = -1.5 + 2 cosh / / and /2/ the parabola y = 0.3x 2 + 0.5, the difference over even short lengths between a catenary and a parabola can be small and hence the difference in accumulated length will be small too, as we will see in the example later.

A catenary also has the property that the area underneath the curve is equal to a times the length of the curve over a given interval. d

a xL = a

| c

2

| | x || 1+ \ sinh | | | dx = a | a || |

d

| c

d

|x| |x| cosh2 | | dx = a cosh | | dx = A |a| |a|

| c

EXAMPLE

Consider, for example, if we want to know the length of chain that will be needed to cross a ravine in order to hold a light walkway. We will assume that the end of the chain will be fixed to supports 2 m above the ground at each side of the ra­ vine, the distance between the supports on either side will be 50 m and the chain should not sink more than 0.5 m below the ground level. First, we need to model the curve. In this case take the origin as the midpoint of the ravine and the lowest point of the chain, we need the graph to pass through ( 50, 2.5 ) and ( -50, 2.5 ). This can be

10 The formula for a catenary in terms of ex was first found in the 1690s by Huygens, Newton, Johann Bernouilli and Leibniz, with Huygens the first to use the term catenary. It was the function satisfying 2

the DE T0 .z ' ( x ) = w 1 + [/z ( x ) ]/ , where z = y ' ( x ), w the weight per unit length of the metal, and T0 the constant horizontal tension.

Calculus applications

FIGURE 7.22

231

A parabola and a catenary

( x ) modelled approximately by the function C ( x ) = 500 cosh | | - 500, x e [ -50, 50 ] ( 500 ) and then the length of the chain, using symmetry, is 50

L=2

{ 0

50

L=2

{ 0

50

1 + C ' ( x ) dx = 2 2

{ 0

( x ) 1 + sinh 2 | | dx ( 500 ) 50

( x ) ( x ) cosh | dx = 2 cos h | | | dx ( 500 ) ( 500 ) 0 2

{

1 1 [ ] ( ) L = 1000 |sinh - sinh 0 | = 1000 | sinh - 0 | ~ 100.16675002 m. 10 10 [ ] ( ) If instead we were to model the supporting cable using a parabola, we can use the 1 2 x x e [ -50,50 ] function C ( x ) = 1000

232

Calculus applications

and 50

L=2

{ 0

( x2 1+ | | 5002 (

) 1 || dx = 250 )

50

{

5002 + x 2 dx

0

This integral may be calculated using a combination of a substitution ( x = tan u ) followed by integration by parts (using sec u. sec2 u to integrate sec3 u). We won’t put the details here, but it gives L ≈ 100.1664175552082 m. Both of the graphs are shown on the same axes in Figure 7.23. At this scale it is not possible to distinguish one from the other. In fact the difference in the calculated lengths is approximately 100.16675002 - 100.16641756 = 0.00033246 that is only around 0.332 mm! 7.5.5

Accumulating surface areas

In Section 7.4.2, we considered the volume generated when the graph of the function y = h ( x ) from x = a to x = b is rotated through 2π radians or 360˚ about the x-axis. The problem here is to find the surface area of the revolution thus generated. Each small disclike volume of revolution of the curve (see Figure 7.24) may be approximated by a frustum of a cone of width δ x in shape by joining each successive

FIGURE 7.23

x ) The catenary function C ( x ) = 500 cosh (| | - 500, x e [[-50, 50]] ( 500 )

Calculus applications

233

y=h(x)

8x FIGURE 7.24

Discrete example of slices accumulated to form the surface area of revolution of the graph of the function y = h ( x ) from x = a to x = b

point on the curve at x and x + 8 x with a straight line.11 So an approximation to the curved surface area (CSA) of the volume generated is given by accumulating these, where CSA = mean circular circumference x slant length between and since the slant length lies along the curve it is of length δ l , giving ( h ( xi ) + h ( xi +1 ) ) 8 A = 2# | x8l | || 2 ( ) and using the formula for the arc length 2 ( h ( xi ) + h ( xi +1 ) ) (8y ) 8 A = 2# | x 1+ | | | 8x | | 2 (8x ) ( )

Using differentials again we can get ( h ( xi ) + h ( xi+1 ) ) dA = 2# | || x dl | 2 ( ) 11 Note an approximation using small cylindrical slices does not work in this case.

234

Calculus applications

and summing these b

2

( dy ) CSA = 2n y 1 + dx. dx

b a

Alternatively, accumulating the area of these small strips, to get the total area i =n -1

CSA = 2n

n

[ h ( xi ) + h ( xi+1 ) ] 2

i=1

2

x 1+

[8y ] 8x 8x

and the actual volume is the limit of this sum as 8 x - 0 (i.e. n - o). i =n -1

CSA = 2t lim

s x-0

E i=1

b

2 \ h ( xi ) + h ( xi+1 ) \ \sy \ \\ \\ x 1 + \ \ sx 2 \sx \ \ \ 2

b

CSA = 2n y 1 + a

( dy ) dx dx

in the case where y = h ( x ) is positive on [a, b] and has a continuous derivative. Note that as 8 x - 0 then h ( xi+1 ) - h ( xi ), and hence the y [= h ( x ) J in the formula. EXAMPLE

A parabolic mirror, where all incoming parallel rays are all reflected to the same focal point, is approximately 100 cm across and 20 cm deep at the centre needs recoating. What area of mirror will the coating have to cover? The mirror can be modelled using the function (see Figure 7.25) f ( x ) = 5 5x cm, 0 < x < 20 . Therefore the CSA to be covered is given by 20

f

CSA = 2n 5 5x 1 0

5 5x 2

1 2

2

20

dx = 10n 5

f

125 dx 4x

x 1

0

20

= 5 55

f

4x + 125dx

0

CSA = 5 5n

[2 3

3

x

3

3 ] 20 1 5 5n [ 205 2 - 125 2 ~ 90001.2cm 2 ( 4x + 125) 2 = 0 4 6 [

The area of mirror to be coated is approximately 9001cm 2 or 0.9001 m 2.

Calculus applications

FIGURE 7.25

7.6

235

The functions f ( x ) = 5 5x and g ( x ) = -5 5x ; Note: only f is used for the curved surface area calculation

Approximating integrals

In practice a great many functions cannot be expressed in a closed analytic form. In this case the methods for exact integration using antiderivatives described earlier in this book (see Section 5.3.6) are not applicable, and we need other means of evaluating definite integrals. These involve numerically approximating the integral rather than trying to find an exact value. There are a number of different approximation methods that have been used, and we will now examine two of these. In order to do this we present an example of a problem where these techniques have been used. Such problems will usually involve a set of discrete values of two related variables. PROBLEM – A SHIP’S STABILITY

When a ship floats at a particular draft (or depth in the water), any trimming moment acting on the ship acts about the centroid of the area of the water plane, which

236

Calculus applications

is called the centre of flotation. In order to find its position it is necessary to calculate the area of the water plane. Since the shape of the ship’s hull is not constructed to a closed analytic mathematical formula, we need an approximate method to find the area of the water plane. What would you use? This problem will be addressed next. 7.6.1 Trapezium method12

If we can measure ordinates, such as AC and BD in Figure 7.26a, then we can es­ timate the area under that part of the graph of the function between A and B by drawing the line segment AB and finding the area of the trapezium ABCD. Since 1 the area of this trapezium is ( AC + AD ) .CD , then this is equivalent to taking a 2 rectangle of height equal to the mean of AC and BD, as seen in Figure 7.26b. These two figures suggest that, in this case, the approximation for the area may not be that accurate. However, we can improve the accuracy by reducing the size of CD. In Figure 7.27 we see that the error from each trapezium appears to be much smaller. If we use this method to estimate the area from x = a to x = b with n + 1 b-a equally spaced ordinate values, then we have n trapeziums, each of width h = . n 1 Adding up the areas using the equivalent of ( AC + AD ) .CD for each but with 2 y1 , y2 ,y3 .,yn+1 for the ordinates we get a total area of 1 1 1 1 A = h( (y1 + y2 ) + ( y2 + y3 ) + ( y3 + y4 ) + ... + ( yn + yn +1 )) 2 2 2 2 h A = ((y1 + y2 ) + ( y2 + y3 ) + ( y3 + y4 ) + ... + ( yn + yn+1 )) 2 h A = ((y1 + yn+1 ) + 2 ( y2 + y3 + y4 + ... + yn )) 2 which is the trapezium rule formula.

Edumatter It is not uncommon to find questions that say something like: Use the trape­ zium rule to estimate the area under a function such as y = x 3 + 1 from x = 1 to x = 4. Why may this kind of task be less motivating than one such as the ques­ tion above about the ship? How would you reason with a student who ques­ tions why we would bother to find an estimate for this area when we can use a definite integral to find the exact answer?

12 The trapezium method and Simpson’s rule are also briefly considered in §5.2.5.

Calculus applications

FIGURE 7.26

237

The trapezium method for approximating the area under the graph of a function

238

Calculus applications

FIGURE 7.27

7.6.2

Reducing the width of the trapezium increases the accuracy of the approximation

Simpson’s rule

To improve on the accuracy of the trapezium rule estimate of area instead of using a straight line segment to join the ordinate values of the shape Simpson’s rule uses a curve. There are a number of versions of Simpson’s rule that use different curves to join the points. There are at least two ways to arrive at Simpson’s rule, and we will employ a version using a quadratic function, as seen in Figure 7.28. This gives rise to what is called the one third rule, as we will see. In the general case we take the middle of 3 x-values (abscissa) as x1 and considering the values an equal distance either side as x1 − h and x1 + h , with g ( x1 - h ) = y1 , g ( x1 ) = y2 , and g ( x1 + h ) = y3, as seen in Figure 7.29. Then there is a unique quadratic function whose graph passes through the three points with ordinates y1, y2 , and y3 . If we translate this graph so that the point F is at the origin, the area is invariant. So now the integral for the area under the quadratic graph becomes +h

A=

{(

-h

h [ ax 3 bx 2 ] ax + bx + c dx = | + + cx | 2 |[ 3 ]| -h 2

)

h

h

Calculus applications

5

239

f

4

g 3

y1

2

y3 y2

1

F

0 0

FIGURE 7.28

=

1

2

3

x1 x1 + h

x1 – h

4

Simpson’s rule using a quadratic function

ah3 bh 2 ah3 bh 2 2ah3 h + + ch - [ + - ch l = + 2ch = 2ah 2 + 6c [ 3 l 3 2 2 3 3

(

)

Now, since y1 = a ( -h ) + b ( -h ) + c, y2 = c, y2 = a ( h ) + b ( h ) + c ,

2

(

2

)

(

y1 + 4y2 + y3 = ah 2 - bh + c + 4c + ah 2 + bh + c

)

= 2ah 2 + 6c +h

(

f(

)

h ax 2 + bx + c dx = ( y1 + 4y Hence the area under the quadratic may be written as A = 3 h -h ax 2 + bx + c dx = (y1 + 4y2 + y3 ). 3 This approximation for the area under the graph of the function f using a quad­ ratic function g is called Simpson’s rule (or the one third rule, named from the h ). 3 In practice we will use more ordinates than just three, taking equally spaced or­ dinates 3 at a time, which gives 2 strips, and applying the rule to each group of 3 ordinates, using a different quadratic function for each group. When we do this with n ordinates (n odd to give an even number of strips) we get:

)

A=

=

h h h ( y1 + 4y2 + y3 ) + (y3 + 4y4 + y5 ) + (y5 + 4y6 + y7 ) + . 3 3 3 h + (yn-2 + 4yn-1 + yn ) 3 h ( y1 + 2 ( y3 + y5 + . + yn-2 ) + yn + 4 ( y2 + y4 + y6 + . + yn -1 ) ) 3

240

Calculus applications

h × (the sum of the end ordinates + 2 × the sum of the odd numbered ordinates 3 + 4 x the sum of the even numbered ordinates). Note that if the integral of f is from x = a to x = b then with n equal strips we b-a . get h = n

Or

7.6.2.1 Simpson’s 3/8 rule

If we were to use a cubic function to approximate the area under f rather than a quadratic function and taking 3 strips at a time (4 ordinates) we could obtain, in a similar manner but using n equal strips with n a multiple of 3, Simpson’s 3/8 rule where A=

3h [ y1 + yn + 3 ( y1 + y4 + + y7 + . + yn-2 ) + 3 ( y2 + y5 + +y8 + . + yn-1 ) ] 8 +2 ( y3 + y6 + y9 + . + yn-3 )

Note that the middle two brackets here could be combined, but one would lose the pattern somewhat. Mathematter Since the area estimate will depend on the function used, how many differ­ ent possible quadratic functions can be drawn through any three points (see Figure 7.29)? How would you justify your answer? Since a quadratic function has three coefficients, or unknowns, we can form a linear system with three equations by substituting the coordinates of the three points in turn into a gen­ eral equation y = ax 2 + bx + c . This gives a linear system with three equations in three unknowns. What are the conditions for this system to have: a unique solution? No solution? An infinite number of solutions? How many different cubic graphs can be drawn through four points? These questions are best an­ swered using the concepts and methods of linear algebra (see for example, Lay et al., 2021).

7.6.3 Applications of the Trapezium and Simpson’s rules

Let’s return to the problem with the application to a ship’s stability. Recall from ear­ lier that when a ship floats at a particular draft (or depth in the water), any trimming moment acting on the ship acts about the centroid of the area of the water plane, which is called the centre of the flotation. In order to find its position it is necessary to calculate the area of the water plane. Since the shape of the ship’s hull is usually not constructed to a closed analytic mathematical formula, we need an approximate method to find the area of the water plane.

Calculus applications

241

2 1 0

0

1

FIGURE 7.29

2

3

4

5

6

7

8

9

10

8

9

The half-breadths at a waterline of a 35 m boat

TABLE 7.3 The half-breadth values at a waterline of a 35 m boat in metres

Half-Breadth Position (x)

0

1

2

3

4

5

6

7

10

Ordinate length (m) at a 0.62 1.38 1.75 1.94 2.0 2.0 1.88 1.5 1.19 0.62 0 waterline ( yi )

What data will we need to apply our two approximations that we have? We need the value of the ordinates. Books on stability, such as that by Barrass and Derrett (2006) employ the half-breadth measurements as the ordinates, where the x-axis is a line of symmetry from the bow to the stern. So we calculate the area of one half and double the answer. Consider the data describing a boat’s shape and size given in Figure 7.29 and Table 7.3. Using the trapezium rule formula from earlier, A= with = h

h ((y1 + yn+1 ) + 2 ( y2 + y3 + y4 + ... + yn )) 2 35 = 3.5 to find the area at this waterline gives 10

A = 1.75{( 0.62 + 0 ) + 2 (1.38 + 1.75 + 1.94 + 2 + 2 + 1.88 + 1.5 + 1.19 + 0.62 )} A = 1.75 ( 0.62 + 2 (14.25 ) ) = 1.75 ( 29.12 ) = 50.96 m 2

This gives a total area at this waterline of approximately 101.92 m2. In practice the captain will use Simpson’s rule since it gives a much better esti­ mate, and we use Simpson’s 1/3 rule here. It is left as an exercise to compare the result from the 3/8 rule. Using Simpson’s 1/3 rule, of A=

h ( y1 + 2 ( y3 + y5 + ... + yn-2 ) + yn + 4 ( y2 + y4 + y6 + ... + yn-1 ) ) , 3

242

Calculus applications

we obtain, with h = 3.5 again, A=

3.5 {( 0.62 + 0 ) + 2 (1.75 + 2 + 1.88 + 1.19 ) + 4 (1.38 + 1.94 + 2 + 1.5 + 0.62 )} 3

3.5 {0.62 + 2 ( 6.82 ) + 4 ( 7.44 )} = 33.5 {0.62 + 13.64 + 29.76} 3 3.5 = x 444.02 ~ 51.36 m 2 3

A=

This gives a total area at this waterline of approximately 102.72 m2. This is used to calculate the x-value, x, of the centre of flotation of the boat or ship using the formula

x=

L

\

\

L

\ xydx \\= \ xydx \\ 2 x \ ydx \ \ ydx \ \ \

2 x

0 L

0 L

0

0

L

f

where L is the length of the vessel and 2 x ydx the area we approximated previ­ L

0

∫

ously. To estimate xydx , we use the values in Table 7.4. Hence 0 L

f

xydx = 2 x

0

=

3.5 1( 0 + 0 ) + 2 ( 3.5 + 8 + 11.28 + 9.52 ) } 3 +4 (1.38 + 5.82 + 10 + 10.5 + 5.58 )

7 {2 ( 32.3) + 4 ( 33.28 )} = 37 ( 64.6 + 133.12 ) = 37 (197.72) ~ 461.35m 2 3

Then for the centre of flotation of the boat, x=

461.35 ~ 4.49 m 102.72

from the stern, or aft, of the boat. This working can be somewhat simplified in practice and a worked example can be seen in Barrass and Derrett (2006, p. 87). Captains will often have available a table TABLE 7.4 Calculating moments from the half-breadths

Half-Breadth Position (x) 0 Ordinate length (m) at waterline ( yi )

xyi

1

2

3

4

5

0.62 1.38 1.75 1.94 2.0 2.0 0

1.38 3.5

6

7

8

9

10

1.88

1.5

1.19 0.62 0

5.82 8.0 10.0 11.28 10.5 9.52 5.58 0

Calculus applications

FIGURE 7.30

243

A table of the USNA Yard Patrol craft half-breadths (reproduced with permission from USNA)

of half-breadth values at different waterlines, such as that seen in Figure 7.30, which are those for the United States Naval Academy (USNA) Yard Patrol craft. PROBLEM – VENTRICULAR HEART EJECTION FRACTION VOLUMES

A problem for radiologists has been to calculate the volume of blood leaving the heart’s left ventricle, known as the left ventricular ejection fraction (LVEF). During a radiography scan they can see an image of the ventricle and can take measurements on-screen. Using (half of) these as ordinates and Simpson’s rule applied to volumes V =

d ( A1 + 2 ( A3 + A5 + ... + An-2 ) + An + 4 ( A2 + A4 + A6 + ... + An-1 ) ) 3

enables one (using a computer program) to calculate the required volume (see the example in Figure 7.31). In practice, in many cases, what is called a modified Simpson’s rule is used. In this case the left ventricle is modelled using a cylinder attached to a truncated cone (or frustum), attached to a cone, as seen in Figure 7.32. The volume is then estimated by using just the three ordinates and the formula V =

L L \ Am + Ap Am + \\ 3 3\ 2

\ 1 L \\ + . Ap \ 3 3

244

Calculus applications

FIGURE 7.31

Biplane Simpson Method using the end diastolic and end systolic apical 4and 2- chamber views for estimation of LV volume and calculation of the ejection fraction; contributed by Ateet Kosaraju, and taken from Kosaraju et al. (2021) 13

3

FIGURE 7.32

3

3

Diagram for the modified Simpson’s rule, based on Folland et al. (1979)

13 Copyright © 2022, StatPearls Publishing LLC. This book is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license, and any changes made are indicated.

Calculus applications

245

Here the circular areas are found from the measurements taken and, for example, Ap = t y 2p . Other practical applications of Simpson’s rule to volume calculations include estimating the volume of the hull of a ship, the volume of earthwork to be exca­ vated using contour maps (see the worked example at www.howtoexcel.info/Civil/ earthwork-volume/) and subsurface reservoir (e.g. of gas or oil) volume calculations using anticlines or contours (see e.g. Slavinić & Cvetković, 2016). In this chapter we have considered the importance of applications of mathemat­ ics. The examples used have considered how DEs and integral calculus may be ap­ plied to the solution of some real-world problems. We feel it is important to open up such applications to our students so that they may begin to see calculus as relevant and applicable to the world they live in. We hope that you will agree. References Arslan, S. (2010). Do students really understand what an ordinary differential equation is? International Journal of Mathematical Education in Science and Technology, 41(7), 873–888. Artigue, M. (1992). Functions from an algebraic and graphic point of view: Cognitive dif­ ficulties and teaching practices. In G. Harel & E. Dubinsky (Eds.), The concept of func­ tion: Aspects of epistemology and pedagogy (pp. 109–132). The Mathematical Association of America. Arzarello, F., & Robutti, O. (2008). Framing the embodied mind approach within a multimodal paradigm. In L. English (Ed.), Handbook of international research in mathematics educa­ tion (2nd ed., pp. 720–749). Routledge. Ball, D. L., Thames, M. H., & Phelps, G. C. (2008). Content knowledge for teaching: What makes it special? Journal for Teacher Education, 59(5), 389–407. Barrass, C. B., & Derrett, D. R. (2006). Ship stability for masters and mates. ButterworthHeinemann. Bosch, M., Hochmuth, R., Kwon, O-N., Loch, B., Rasmussen, C., Thomas, M. O. J., & Trigueros, M. (2021). Survey on research in university mathematics education at ICME 14. European Mathematical Society Magazine, 122, 57–59. https://doi.org/10.4171/MAG-42 Butcher, J. C. (2008). Numerical methods for ordinary differential equations (2nd ed.). John Wiley & Sons. https://doi.org/10.1002/9780470753767 Campbell, S. L., & Haberman, R. (2008). An introduction to differential equations with dynamical systems. Princeton University Press. Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathemat­ ical thinking. Basic issues for learning. Proceedings of the 23rd conference of the international group for the psychology of mathematics education, 1 (pp. 3–26). Haifa, Israel. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. https://doi.org/10.1007/ s10649-006-0400-z Ely, R. (2017). Reasoning with definite integrals using infinitesimals. Journal of Mathematical Behavior, 48, 158–167. Ely, R. (2021). Teaching calculus with infinitesimals and differentials. ZDM Mathematics Edu­ cation, 53, 591–604. https://doi.org/10.1007/s11858-020-01194-2

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Fardinpour, Y. K., & Gooya, Z. (2018). Comparing three methods of geometrical approach in visualizing differential equations. International Journal for Research in Undergraduate Math­ ematics Education, 4, 286–304. Folland, E. D., Parisi, A. F., Moynihan, P. F., Jones, D. R., Feldman, C. L., & Tow, D. E. (1979). Assessment of left ventricular ejection fraction and volumes by real-time, twodimensional echocardiography. A comparison of cineangiographic and radionuclide tech­ niques. Circulation, 60(4), 760–766. Habre, S. (2012). Improving understanding in ordinary differential equations through writing in a dynamical environment. Teaching Mathematics and Its Applications, 31, 153–166. Harjule, P., Tiwari, V., & Kumar, A. (2021). Mathematical models to predict COVID-19 out­ break: An interim review. Journal of Interdisciplinary Mathematics, 24(2), 259–284. https:// doi.org/10.1080/09720502.2020.1848316 Kosaraju, A., Goyal, A., Grigorova, Y., & Makaryus, A. N. (2021). Left ventricular ejection fraction [Updated 2021 May 3]. In StatPearls [Internet]. Treasure Island, FL: StatPearls Publishing. www.ncbi.nlm.nih.gov/books/NBK459131/ Kwon, O. N. (2002). Conceptualizing the realistic mathematics education approach in the teaching and learning of ordinary differential equations. In D. Hallett & C. Tzanakis (Eds.), Proceedings of the international conference on the teaching of mathematics (at the Under­ graduate Level) (pp. 60–69). Hersonissos, Crete. Kwon, O. N. (2020). Differential equations teaching and learning. In S. Lerman (Ed.), Ency­ clopedia of mathematics education (pp. 220–223). Springer. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. Lay, D. C., McDonald, J. J., & Lay, S. R. (2021). Linear Algebra and its applications (6th ed.). Pearson Education Limited. Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 205–236). Sense. Rasmussen, C. L. (1997). Qualitative and numerical methods for analyzing differential equations: A case study of students’ understandings and difficulties (Unpublished doctoral dissertation, Uni­ versity of Maryland, College Park). Rasmussen, C. L. (2001). New directions in differential equations: A framework for interpret­ ing students’ understandings and difficulties. Journal of Mathematical Behavior, 20, 55–87. Rasmussen, C. L. (2002). Instructional materials for a first course in differential equations (Unpub­ lished document, Purdue Calumet University). Rasmussen, C. L., & Blumenfeld, H. (2007). Reinventing solutions to systems of linear dif­ ferential equations: A case of emergent models involving analytic expressions. Journal of Mathematical Behavior, 26, 195–210. Rasmussen, C. L., & King, K. D. (2000). Locating starting points in differential equations: A realistic mathematics education approach. International Journal of Mathematical Education in Science and Technology, 31(2), 161–172. Rasmussen, C. L., Kwon, O. N., Allen, K., Marrongelle, K., & Burtch, M. (2006). Capital­ izing on advances in mathematics and K-12 mathematics education in undergraduate mathematics: An inquiry-oriented approach to differential equations. Asia Pacific Education Review, 7(1), 85–93. Schwarzenberger, R. L. E. (1969). Elementary differential equations. London: Chapman and Hall. Slavinić, P., & Cvetković, M. (2016). Volume calculation of subsurface structures and traps in hydrocarbon exploration – a comparison between numerical integration and cell based models. Open Geosciences, 8(1), 14–21. https://doi.org/10.1515/geo-2016-0003

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Stephan, M., & Rasmussen, C. L. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459–490. Tall, D. O. (1986). Lies, damn lies . . . and differential equations. Mathematics Teaching, 114, 54–57. Tall, D. O. (1990a). Real functions & graphs: SMP 16–19, Rivendell software (for BBC compatible computers), prior to publication by Cambridge University Press. Tall, D. O. (1990b). A versatile approach to calculus and numerical methods. Teaching Math­ ematics and its Applications, 9(3), 124–131. Tall, D. O. (1991). Intuition and rigour: The role of visualization in the calculus. In W. Zim­ mermann & S. Cunningham (Eds.), Visualization in mathematics, M.A.A. Notes No. 19 (pp. 105–119). Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24. Tall, D. O. (2014). How humans learn to think mathematically: Exploring the three worlds of math­ ematics. Cambridge University Press. Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Jour­ nal for Research in Mathematics Education, 7(2), 67–87. Trigueros, M. (2004). Understanding the meaning and representation of straight line solu­ tions of systems of differential equations. In D. E. McDougall & J. A. Ross (Eds.), Proceed­ ings of PME-NA (pp. 127–134). PME-NA. Wagner, J. F., Speer, N. M., & Rossa, B. (2007). Beyond mathematical content knowledge: A mathematician’s knowledge needed for teaching an inquiry-oriented differential equations course. Journal of Mathematical Behavior, 26, 247–266. Zandieh, M., & McDonald, M. (1999). Student understandings of equilibrium solution in differential equations. Proceedings of the twenty-first PME conference. Cuernavaca, Mexico.

8 BEYOND ELEMENTARY CALCULUS

Our book on elementary calculus effectively finished in the last chapter. In this chapter we consider some possible directions that a first course in calculus may take a student. This includes calculus immediately beyond what was covered in earlier chapters, the transition to university analysis, calculus in disciplines other than math­ ematics and the preparation of future school calculus teachers. While it comprises a rather cursory look at some of the topics that may comprise a second course, and even those beyond, the aim is to provide a suitable context and a means to answer the question ‘Where does calculus lead next after an elementary course’? In Section 1.3 we introduced, among other theoretical ideas, the concept of mathematical knowledge for teaching (MKT) (Ball & Bass, 2002; Ball et al., 2005). In a later paper (Hill et al., 2008) they introduce what they call an ‘egg’ to represent the six dimensions of pedagogical content knowledge and subject matter knowledge (see Figure 8.1).

Common Content Knowledge (CCK) Knowledge at the mathematical horizon

FIGURE 8.1

Knowledge of Content and Students (KCS) Specialised Content Knowledge (SCK)

Knowledge of Curriculum Knowledge of Content and Teaching (KCT)

The pedagogy and content knowledge ‘egg’ based on Hill et al. (2008, p. 377)

DOI: 10.4324/9781003204800-8

Beyond elementary calculus

249

As noted by Zazkis and Mamalo (2011), initially, in the Hill et al. (2008) paper where the diagram appeared, there was no explanation of what was meant by the mathematical horizon in the subject matter knowledge half. However, in other papers the construct was further refined, such as in the statement that “Horizon knowledge is an awareness of how mathematical topics are related over the span of mathematics included in the curriculum” appearing in Ball et al. (2008, p. 403). This was echoed in Ball and Bass (2009), which defines horizon knowledge as an aware­ ness of the mathematical landscape that lies ahead rather than a deep curriculum knowledge of it. Expanding on this clarification Ball and Bass (2009, p. 6) went on to describe four constituent elements of horizon knowledge as: (i) A sense of the mathematical environment surrounding the current “location” in instruction (ii) Major disciplinary ideas and structures (iii) Key mathematical practices (ix) Core mathematical values and sensibilities. They also suggest that paying “Attention to the mathematical horizon is thus important in orienting instruction to embody both pedagogical foresight and mathematical integ­ rity” (ibid., p. 6). It is with this background in mind that we have included this chapter in our book. We hope that what is presented will provide teachers of first courses in calculus with a sufficient sense of the mathematical environment surrounding it and an awareness of the landscape that lies ahead in order to deepen and strengthen both the pedagogical soundness and mathematical integrity of their classroom activity. 8.1 The landscape that lies ahead

It is important to remember, as we have mentioned earlier in this book, that the con­ tent, year of teaching and rigour of a first course in calculus will vary considerably depending on the country where it is taught. Some will meet it first in Year 12 (age 17 years) of school (or occasionally even earlier), while others will not study it until their first year in university. So we have chosen here some topics that we think will often not appear in many first courses. Since this is intended to be a fairly broad sweep of these in a short space, we have not been able to include a number of topics that the reader might feel we should have. In addition, the intent is not to go too deeply into each topic but hopefully some readers will feel motivated to engage in further reading on some of the topics. In the sections that follow, we consider what it means for a sequence, series or a function to converge and look at introductory multivariable calculus. 8.1.1 Convergence of sequences and series

While many students may have met both sequences and series before calculus, they will often not have explicitly considered the topic of convergence. Students may

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wonder why the topic of convergence is covered in a calculus context. What is the connection? Historically, Newton (see Section 2.2) worked out how to represent a number of rational and transcendental functions using power series1 (see later, where Taylor and Maclaurin series are discussed). Later, Euler also represented some func­ tions in this way. For example, they knew that arctan ( x ) = tan -1 ( x ) = x -

x3 x5 x7 + +. 3 5 7

Why was this useful? It enabled the functions to be integrated or differentiated term by term, and hence the link to calculus. However, before students start applying calculus procedures to the terms in a series, we would like them to be aware of potential pitfalls. One of these is whether or not the series converges, and if so, for what values of the independent variable do they do so? For instance, it can be shown that the previous arctan series only converges for -1 : x : 1. How would you set about showing this? Let’s look at the topic of convergence to see how we can tell.

8.1.1.1 Sequences

First, it is important to distinguish sequences and series. We can think of a sequence as an ordered list of infinitely many numbers, or a function f : N → R. For exam­ ple, we may write, a1 , a2 , a3 , a4 … for an arbitrary sequence. Some specific examples could be 2, 3, 5, 7,11… and 1 1 1 1, , , ,… 2 4 8 n 2 So while the nth term of a sequence will often have a formula, such as an = ( -1f , n it may not (e.g. the sequence of prime numbers). One very well-known sequence is the Fibonacci sequence, which can be written as

a1 = 1, a2 = 1 an = an-1 + an-2

n > 3.

This is an example of a sequence defined by a recurrence relation, and leads to the terms 1, 1, 2, 3, 5, 8, 13, . . . The topic of convergence of these sequences may be considered in first calculus courses in some countries, but not in others.2 We say that a sequence converges to 1 A power series is an infinite series whose terms are polynomial functions.

2 See Viirman et al. (2022) for further information on limits in the curricula of England, France and

Sweden.

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a limit L if the terms in the sequence eventually become arbitrarily close to L. But what does arbitrarily close mean? The convergence of sequences is an important topic in analysis (see further), and we don’t have the space here to do it justice. We will just consider a few examples. Not all limits are easy to find. Consider for example, the sequence (previ­ n 1 J ( , which converges to ously mentioned in Section 3.4) given by an = 1 + n e = 2.718281828., although this is not obvious. Two ideas we think worthy of mention here on the topic of convergence are the squeeze theorem (introduced in Section 3.4 for functions) and Cauchy sequences. The squeeze theorem for sequences, which is an aid to determining convergence, asserts that, if ( an ) , ( bn ) , ( cn ) are three sequences such that an ≤ bn ≤ cn for all (large) n, and (an ) converges to L, that is lim an = L, some L E 

n-o

and lim cn = L

n-0

then ( bn ) is convergent and also converges to the limit L. We can see that ( bn ) is getting squeezed between the values of the other two sequences. For example, does the sequence

We know that

0