business calculus [524, 2e ed.]
business calculus 2e
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english Pages [524] Year 2016
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Citation preview
lim f (x) = 1
x!1+
30, 00 f (x)
3,000 p
x
(0.5)(0.02) = 0.01
lim f (x) = 2
x!1+
p
x2 + 1
(0.4)(0.2) = 0.008
x2 + 9 + 4x2
k=0
50,000 22,903
x 6
xE
0
t
pE
pE , xE
- p
x x
x
C(x) 20
0
6 80 70 60 50 40 30 20 10
0
C(x) = 3.9x 0 x 20
f x
5
x
10
15
20
C(x)
' $ s
'
f
a
Rfs(a) Rs
s
$
f (b)
& %
&
b
%
0 20
[0, 20]
[0, 78] {1, 2, 3, 4, 5} {1, 2, 3, 4, 5} p
y = f (x) = x
2 x
y = g(x) =
3
2 y = f (x) =
x=3 x
{1, 4, 9, 16, 25}
y = g(x) = 2
p
x
x=3
3
3
x 6= 3
x x
2
x 6= 3
2
x x
x
2
x
2
2
[2, 1)
C(x) = 2.89x x
x C(8) = 2.89 ⇥ 8 = $23.12 p 3
f ( t)
f
f (x) = 2x2
3x
f (2) f (5a) f (x + h)
0
x
f (2) = 2(2)2
3(2) = 2(4)
f (5a) = 2(5a)2
3(5a) = 2(25)a2
f (x + h) = 2(x + h)2 f
6=8
⇣p ⌘ ⇣ p ⌘2 3 3 t =2 t
6 = 2. 15a = 50a2
p p 3 3 3 t = 2 t2
5x2
= 2x + 2h
f (x)
5x2
f (x + h)
5(x + h)2 = 2(x + h)
=2
10xh h 10x
10xh =
5x2
2x + 2h
2h
3x
3h.
f (x + h) h
f (x)
h 6= 0
x+h
f (x + h) = 2(x + h)
=
3(x + h) = 2x2 + 4xh + 2h2
p 3 3 t.
x
=
15a.
3(x + h) = 2 x2 + 2xh + h2
f (x) = 2x
f (x + h) h
x
5h2
5h2 ,
2x + 2h
10xh h =
5 x2 + 2xh + h2
5x2
5h2
h(2
10x h
10xh h
5h2
2x
5x2
2x + 5x2
5h)
h 6= 0.
5h,
f x
f
(x, y)
y = f (x)
(1, 1) (2, 4) (3, 9) (4, 16) (5, 25) s(x) = 1/x x
x=0
x=0 x=0
x 6= 0
y r
6 25 20 15 10
r
5
0
x
1
r
2
r 3
r
- x
4
5
y 46
s(x)
y = s(x)
3 2 1
- x
0 3
2
1
1
2
3
4
1 2 3
s(x) s(x)
y 6
y2 r
r 6
y1 r
r
6 r a
0
- x f (x)
x y1
a y2
G
G( 2) G( 1)
G(1)
y 4 3
s 4
3
2
c
1
2
6 s
1
0 1 2
c
1
2
3
4
x
y = G(x)
3 4
G
y x=
2
( 2, 2)
x=
1
( 1, 1) x=1
G(x) G( 2) = 2 G( 1) = 1
(1, 3)
G(1) = 3
p H( 4 a)
H(a2 )
f f (x) = 2x + 3
H(x + h)
f ( 1) f (0)
H(x + h) h
f (2)
f (t2 )
f (3a)
H(x)
h 6= 0
F
f (x + h) f (x + h) h
f (x)
h 6= 0
S S(x) = 4x
H
1
S( 1) S(0)
S(2)
G G(x) = 2/(x + 1)
S(t2 )
S(3a) S(x + h) S(x + h) h
S(x)
h 6= 0
G( 2) G( 1.5) G( 1.1) G( 0.9) G(0)
f
G(1)
S T 1)
T (x) = 1/(x F F (x) = 2x
2
2
T ( 1) T (0) T (0.5) T (0.9) T (1.1)
F ( 2) F ( 1) F (0) F (1) p F (5a) F ( a)
T (3)
F (2)
F (x + h) F (x + h) h
F (x)
f
h 6= 0
H 2
H(x) = x + 1
H( 2) H( 1) H(0) H(1)
H(2)
f ( 3)
f ( 2)
f ( 1)
f (0)
f (1)
f (3)
y
6
g( 2)
g( 1)
3
g(0)
g(1)
2
g(2)
g(3)
4
4
3
b
r
1
r
b
10 1
2
1
y 2
3
4
x
y = g(x) b
3 2
y = f (x)
2
6
4
@ r @1 0 2 1 2 @1 @ 2 @b @ @r 3 1
3 4
4
3
g
3
4
x
4
x = a
y 4
y
6
4
3
3
2
2
x=2
1 0 3
2
1
1
2
3
x=
- x
1
1 0
3
2
1
1
1
1
2
2
3
3
x=2 y=b
6
x=
1
2
3
- x
y
3
2
y
4
4
3
3
2
2
1
1
1
1
2
x
3
3
2
1
1
1
1
2
2
3
3
y=3
6
AA
2 1
y=
1 x 2
0 2
1
1
2
3
A A y = 2x A
- x
3
1 2 3
m (x1 , y1 )
m=
y2 x2
2
y
3
3
x
3
y=
y 4
2
2
4
6
3 2
A1 A0 1 A 1 1 A A 2 A A 3
2
3
- x
(x2 , y2 )
y1 . x1
x
y
y
6 #r #
#
#
6 b b
#
(x2 , y2 )#
#r #
(x1 , y1 # )
#
#
# #
(x2
#
(y1 (y2
b (x r 1 , y1 ) b b b y2 ) b
y1 )
b b (x r 2 , y2 ) b x1 ) b b b
(x2
x1 ) m=
y2 x2
y1 x1
>0
m=
- x
0
y2 x2
y1 x1
b
0 x1 =
x2 =
p b+ S = 2a
b
p 2a
S
=
( 3) + 2(2)
( 3) 2(2)
a=1 b=4
p
1
p
1
=
=
3+1 = 1; 4
3
1 4
=
1 . 2
S = b2
c=4
b 4 x= = = 2. 2a 2(1) a=1 b=
2
c=2
S = b2
4ac = ( 2)2
4(1)(2) = 4
S0
x=
1
f
1 g
1
P (x) = 0, R(x)
f (x) =
x2 + 2x 3 : x 1 f (x) = 0 f 1
f (x) =
x2 + 2x 3 = 0, x 1
x2 + 2x
3 = 0.
x2 + 2x
3 = (x
1)(x + 3). x=1
x=
3
f (x) = 0 x
f (x) =
x=1
x=1 x=
x 6= 1
1
3 x
x2 + 2x 3 (x 1)(x + 3) = = x + 3, x 1 x 1
1
x 6= 1. x=1
(1, 4)
y
6
5 4 3
x
f (x)
2
4
f (x)
q
q
a
f (x) =
x2 +2x 3 x 1
1
3
2
10 1
1
2
3
4
x
2
f (x) =
x2 + 2x 3 x 1
f (x) = xp
f (x) = x3
p p
f (x) = x
1
= 1/x p 1
n
f (x) = x 2 , n
f (x) = x m ,
m 1
f (x) = x 2 =
p
x.
x xn · xm = xn+m
(xn )m = xn·m
xn = xn xm
x
1
xn =
p n
x
m
n
m
=
xn =
y
n (xy)n = xn · y n ✓ ◆n x xn = n y y
1 xn p n
xm =
p n
x
m
f (x) =
f
[0, 1)
m
p n
p
xy =
x
p n
x·
p n
y
y 4
x
f (x)
6 f (x) =
p
7
8
x
3
r
2
r
1
0
f (x)
r
-
1
2
3
4
5
6
f (x) =
f (x) =
p
5x
6
r
p
x
9
x
x
f (x) = 0 f (x) 5x
6
0
6
5
x
6/5
6/5 [6/5, 1) f (x) = p
5x
p
p
5x
6
x=0
x
6 = x.
5x
6
2
= x2 ,
x2
5x + 6 = 0.
x2
5x + 6 = (x
5x
2)(x
6 = x2 .
3), x=2
f
f (2) = f (3) = 0
x=3
x=2 f (x) = 0
x=3
f (x) = |x| f f ( 2) f (0)
f (1) f x x
x
x
f
f (x) =
f (0) = 0
8
1
g
0 < 1 x 1
x g(3) = 2 g (1, 2)
x
y
6
4
x
g(x)
s
3
1
c
0
1
2
4
3
2
1
y = g(x)
2
3
4
x
1 2
g(x)
g(x) 8 < x+2 h(x) = : 2
h f (0) f (1)
f (3)
h(0) = (0) + 2 = 2 h(1) = 2
x 6= 1 x=1
f
h(1) = 2 h(3) = (3) + 2 = 5
x
h(x) = x + 2 x 6= 1
(1, 2) x 6= 1
y 4
x
3
h(x)
4
3
2
1
6 y = h(x)
b
1
r
0
1
2
h(x)
h
(1, 3)
2
3
1 2
h(x)
4
x
f (x) = 2x2
1
x2 + 1
f (x) =
f (x) = x2
f (x) =
x2 + 3x + 2 x+1
f (x) =
1
2x2 + 6x + 1
f (x) =
f (x) = x2 + 3x
4x + 4 2
1
f (x) =
p
x2
f (x) =
p
2x + 1
f (x) = f (x) =
4x = 0 4x3 + 4x2 = 0
f (x) =
x4 + x2 = 2 x
x
f (x) = 0
f (x) = x2 + 2x + 3
4
x2
2x
f (x) = 2x2 + 6x
x4
x2 4 x+2
x2 + 3x
f (x) =
x3
f (x) =
2
5x =
f (x) =
4
x2
4x =
3
x2
3x =
2
x2 x2
1 + 2x
9 4x + 3
x2 x2 x2
2x
4 x
6
3x + 2 x 2
x2 + 4x + 4 x+2
8 < x+2 f (x) = : 2x + 5
x1 x>1
f ( 1) f (0) f (1) f (2) x
2
4x = 2 2
3x + 1 = 0
2x2
3x + 2 = 0
2x
x2 4x2 x2
f (3) x1 x>1
5x + 1 = 0 4x + 1 = 0 6x + 9 = 0
f (x) =
p
f (x) =
p
f (x) =
x2 x
x
x+1 9 3
1
8 < 3 x g(x) = : 2x + 3
g( 1) g(0) g(1)
x0 x>0 g(2) x0
x>0
g( 1) g(0) g(1)
g(2) x0 x>0
F (x) = |x + 2|
F ( 3) F ( 2) F ( 1)
F (2) x< x
G(x) = |x
G( 1) G(0) G(1)
2 2
x1
f ( 1) f (0) f (1) f (2)
x 6= 1
1|
G(2)
x
8 < x+1 f (x) = : 3
4.9t2 + 19.6t + 98.
H(t) = f (3) x1
H(t)
x>1
120 24.5 t 8 < 3 x2 g(x) = : 2x + 3
H(t) = x0 x>0
4.9t2 + 24.5t + 120.
D(p) = H(t)
D(p) =
21 p+3
S(p) = 3p
2
30 p+2
S(p) = 2p
5
D(p) = 20
p 2 p + 36 S(p) = p
D(p) = 10
p
p + 1 S(p) = 2p
2
3
t t 6% $1,000 ⇥ 6% = $1,000 ⇥ 0.06 = $60 $1,000 P (t) = 1,000 + 60t $1,000 ⇥ 0.06 = $60
t $60 ⇥ t
P (1)
P (1) = 1,000 + 60 = 1,000 + 1,000 ⇥ 0.06 = 1,000(1 + 0.06)
$1,060 ⇥ 0.06 = $63.60
$1,060 + $63.60 = $1,123.60
P (2) = P (1) + P (1) ⇥ 0.06 = P (1)(1 + 0.06) = 1,000(1 + 0.06)(1 + 0.06) = 1,000(1 + 0.06)2 P (3) = 1,000(1 + 0.06)3
t
t
P (t) = 1,000(1 + 0.06) = 1,000(1 + 0.06)t t t
f (x) = P0 ax a
1
P (t) = 1,000(1 + 0.06)t P0 = 1,000
P0
t
1.06
P0 r
P (t) = P0 (1 + r)t .
t
f (x) = 2x
g(x) =
✓ ◆x 1 2
h(x) = 5 · ⇡ x ( 1, 1) f (x)
g(x)
x
2x
( 12 )x
3
1 8
8
f !x""2x
2
1 4
4
15
15
1
1 2
2
10
10
0
1
1
5
5
g!x""!1 #2"x
y
y
1 2
1 2
4
1 4
3
8
1 8
!3 !2 !1
1
2
3
4
x !4 !3 !2 !1
a y = 2x
x
(0, 1)
a>1 y=
,
y = x2
y = 2x
x x
f (x) = x6
g(x) = 2x g(30)
f (2) = 26 = 64 6
1 x 2
a0 = 1
0 0)
r
T
T =
ln(2) r
r=
ln(2) . T ln(2) =
0.693147 · · · ⇡ 0.7 T =
ln(2) 0.7 70 ⇡ = . r r 100r 5%
70/5 = 14 10% $10,000
$40,000 $160,000
70/10 = 7 5% 10%
P (t) = P0 ert , r > 0,
P (t) = P0 e
rt
, r > 0.
t
t
P (t)
P (t) = P0 ert . P0 = 200 P (2) = 360,
200er(2) = 360.
r
200 er(2) =
360 = 1.8. 200
2r = ln(1.8),
r=
ln(1.8) . 2 t
P (t) = 200e(
ln(1.8) 2
P (5) = 200e(
)t .
ln(1.8) 2
)5 ⇡ 869
. r=
T =
ln 2 ln 2 = ⇡ 2.36 r (ln(1.8))/2
5,000 = 200e(
ln(1.8) 2
ln(1.8) 2
.
)t .
200 e(
ln(1.8) )t 2
✓
ln(1.8) 2
= ◆
5,000 = 25. 200
t = ln(25),
T (t) = (T0 r
C)e
rt
t=
ln(25) ⇡ 11 (ln(1.8))/2
.
+ C,
T0
C
130 70
1.5
125 T (t)
112
t
t T (t) T (t) = (T0
rt
C)e
C = 70 T (t) = 60e
+ C.
T0 = 130 rt
+ 70. r
T (1.5) = 125,
60e
r(1.5)
125
+ 70 = 125.
70 60e
r(1.5)
= 55. 60
e
r(1.5)
=
55 11 = . 60 12
r(1.5) = ln
✓
11 12
◆
,
r=
ln(11/12) ⇡ 0.058. 1.5 t
T (t) = 60e
0.058t
+ 70. 112
112 = 60e
0.058t
+ 70.
70 e
0.058t
=
60
42 = 0.7. 60
0.058t = ln(0.7),
t=
ln(0.7) ⇡ 6.15 0.058
.
6.15
112
103 = 1,000
271/3 = 3
ek = b
e
ah = J
10
2
=M 2
= 0.01
10b log3 (9) = 2
log16 (2) =
1 4
3
ln(10) = k
ln(h) =
3
ah
loga (H) = G
21/3
x
56x 2.5
10
loga (K) = h f (x) = 2x
g(x) = log2 (x) f (x) = 2x
g(x) = log2 (x)
f (x) = (3/2)x
log3 (K)
log5 (10)
log2 (7)
log6 (G)
loga (H)
loga (32)
p ln ( 5 e)
ln(e3 )
ln[(2e)
3
] + 3 ln(2)
ln[(3e)
4
] + 4 ln 3
g(x) = log3/2 (x) f (x) = (3/2)x
g(x) =
log3/2 (x)
p ln ( 3 ae)
1 3
ln(a)
p ln ( 4 ae)
1 4
ln(a)
$1,000 2.6%
t
$1,500 t
t
e = 10
e = 20
$8,000 5%
t
t
5 =9
120 = 10
$10,000
at = M
3
t
3t
e
0.05t
e
= 0.1
$1,000
= 0.1
7.6% $1,500
= 0.2 $3,000
3e4t = 9
100e
0.05t
6%
= 30 $5,500
60e
0.02t
+ 40 = 90
7e0.2t
10 = 25
$8,000 $10,000
e
$12,000
$14,000
$20,000 $80,000 5.6% 5.8% $30,000
3.8%
$75,000 5.8% $30,000 6%
5.5%
$15,000 $19,000
$30,000 $25,000 r%
$5,000 4.5% $25,000 $29,000
$20,000 $15,000
t r% $3,000 $2,900 $2,000
$8,000 $8,000
$1,000
5.1%
t
$4,000 $2,000
$3,800 $2,000
130 75 120 T (t) t
t 115 72 42 60 T (t) t
50 t 185 75 140 T (t) t
100
70 60 10 T (t) t
$12,000
' $ s
'
f
a
Rfs(a) Rs
s
$
f (b)
& %
&
b
%
f (x) = 2x2 x
f (2) = 2(2)2
3(2) = 2(4)
f (x + h) = 2(x + h)2
6=8
3x
6 = 2.
3(x + h) = 2(x2 + 2xh + h2 )
3(x + h) = 2x2 + 4xh + 2h2
3x
3h.
g 8 < x+2 g(x) = : 2 g(0)
x1 x>1 0 < 1
g(0) = (0) + 2 = 2
g(3)
3 > 1
g(3) = 2 y = f (x)
(x, y)
x
y = f (x)
f (a) a
y
f (x) = b x = a f (x) = mx + b
m
y
b
f (a)
y
y
y
6
6
6
r
y=b
b
-x
0
0 a
y = mx + b (x2 , y2 )
r
b (x1 , y1 )
- x
x=a
-x
0 y2 m= x2
m
m=
f (x) = mx + b (x1 , y1 )
y2 x2
(x2 , y2 )
y1 x1 f
f
f (x) = ax2 + bx + c a 6= 0
a>0
a0
f (x) = loga (x)
x = ay .
y = loga (x)
y = log10 (x) y = loge (x)
y = log(x)
y = ln(x)
f (x) = ax ax
g(x) = loga (x) x y loga (x) f (x) = ax
g(x) = ln(x)
g(x) = loga (x)
f (x) = ex
y = at
e
y = loga (x) at = e(ln(a))t
ax2 + bx + c = 0
loga (x) =
ln(x) . ln(a)
ax + b = 0 a 6= 0
x=
S = b2
S0 x1 =
p b+ S 2a
x2 =
b/a
b
p 2a
S
.
b 2a
4ac
p(x) = 0 x3 4x = 0 x3
x(x
4x = x(x2
4) = x(x
p
2)(x + 2),
2)(x + 2) = 0. x=0 x=2
x=
r(x) = p(x)/q(x) = 0 p(x) = 0
x
6
xE
s
0
pE
pE , xE
- p
2
f
(1, 2)
(3, 4) $45,000
f ( 3)
f ( 2)
f ( 1)
f (0)
f (1)
f (3) y b
4
3
$8,000 V t
6
V
t
4
r
3 2
r
1
10 b 1
2
2
b
1
2
3
4
x
y = f (x)
3 4
F F (x) = 2x2
2 x
F ( 2) F ( 1) F (0) F (1) p F (5a) F ( a)
F (2)
C(x) = 6,000 + 15x, 0 x 1,000.
F (x + h) F (x + h) h
F (x)
x
h 6= 0
R(x) x P (x) x
f f (x) =
1 3x
+6
P (x) = 0
3
( 1, 0)
P (x)
C
8 < x+2 f (x) = : x2 3
x1 x>1
f ( 1) f (0) f (1) f (2) x
f (3) x1
C x
x>1
$5,000 4.5%
f (x) = 2x2 + 6x f (x) =
1
2x2 + 6x + 1
x3
4x = 0
x4
5x2 =
$6,000 4.3% 4 $20,000 6%
x
2
x2
$20,000 5.2%
4x = 2 6x + 9 = 0
f (x) = 0
f (x) = f (x) =
x2
9 4x + 3
x2
4
x2 x2
x
$80,000
4.5% 6
x2 + 4x + 4 f (x) = x+2 p f (x) = x2 1 + 2x
6.8% 9.2%
$8,000 $10,000 $50,000 $175,000
$5,000 4.5%
5% 5.5% t
x = D(p) = 2,000 x = S(p) = 20p
10p 400
p
x
$8,000
$80,000 5.8% $30,000
$15,000 $19,000
$30,000 $25,000 r%
t et = 100
120 = 10t t
e
3t
= 0.01
60e
0.02t
+ 40 = 90
e 56x
21/3
ah
10
2.5
$1,000 3.6% $1,500
130 72 120
T (t) t 115
F (x) = x2
F F ( 1)
F (2)
F (5a)
p F ( a)
2x
F (x + h) F (x + h) h
F (x)
h 6= 0 2
(4, 0)
x2
x3
9x = 0
60e
0.02t
4x = 1
+ 40 = 90
C(x) = 5,000 + 20x, 0 x 1,000
x
R(x)
P (x)
x
x
P (x)
x = D(p) = 3,000 p
10p
x = S(p) = 20p x
600
$60,000
4.5%
6.8%
$2,000 4.6% $4,200
$6,000 $8,000
130 76 1.5
122 T (t)
t
3
110
$5,000 $5,000
=
100%.
y y = f (x)
6
(x + h1 , f (x + h1 ))
@
(x + h2 , f (x + h2 ))
@ (x + h3 , f (x + h3 ))
s @
(x, f (x))
@ @s
s
s
lim
h!0
f (x + h) h
- x
f (x)
= f 0 (x)
f (x) = 2x + 1
f (x)
x f (x)
x
x
2
2
x (x < 2)
f (x)
x
x (x > 2)
2 x
f 2
2
f (x)
x
2
2 f x
x=2 2 f (x)
f (x)
5 x
5 x
f (x)
2
lim f (x) = 5
x!2
y f (x) = 2x + 1
7 6 5 4 3 2 1 1
1
1
2
3
x
4
x
f (x) ! L
lim f (x) = L,
x!a
x x
L
2 f (x)
5
x ! a,
a
L a
f (x) a
lim f (x) = L
x!a
f (x)
L
x
a
x a
L
x
a
x
a
f
a
f
f
lim f (x) = L
x
a
a
f
a
lim f (x) = lim+ f (x) = L.
x!a
x!a
x!a
8 < x+2 g(x) = : 2x + 1
g
lim g(x);
lim g(x);
lim g(x);
x!2
x!2+
x!2
x 2)
g(x)
x
x!2
2
g(x)
g(x)
x
2
y 76 6 5 4 3
⌘ ⌘
⌘
⌘ 2 ⌘
⌘
⌘ ⌘
⌘ ⌘
s ◆
◆
◆
◆
◆◆ y = g(x)
c
1 0
1
1
-
2
3
4
- x
1
g
lim g(x) = 4
x!2
x=2
g(2) = 5 6= 4
lim g(x) = 3.
x!1
x ! 1 (x < 1)
g(x)
x ! 1+ (x > 1)
g(x)
x
1
g(x)
g(x)
x
1
y 76 6 5 4 3 2 ⌘
⌘
⌘ ⌘
⌘ ⌘ ⌘ q ⌘
s ◆
◆
◆
◆
◆◆ y = g(x)
c
⌘ ⌘ 1
0 1
-
1
2
3
x
4
1
g(x)
x=1
a=2 a=2 y lim g(x)
x!2
a=2 a=1
lim g(x)
x!2
y 66
◆
5 4 3
⌘ 2 ⌘
⌘
⌘
⌘ ⌘
◆
◆
◆ y = g(x)
⌘⌘
⌘ ⌘ 1
0 1
1
-
2
2
3
4
1
g(x)
f
lim f (x);
x!2
x
8 < x+2 f (x) = : 5 lim f (x);
x!2+
2 x 6= 2 x=2
lim f (x).
x!2
x
lim f (x) = 4,
lim f (x) = 4,
x ! 2 (x < 2)
lim f (x) = 4.
x!2
x!2+
x!2
x ! 2+ (x > 2)
f (x)
f (x)
x
f (x)
f (x)
y 76
y = f (x)
6 5 4 3
⌘ ⌘
⌘
⌘ 2 ⌘
⌘
⌘ ⌘
⌘ ⌘
s
⌘ c⌘
⌘
⌘ ⌘
⌘
⌘ ⌘
1 0
1
-
1
2
3
4
- x
1
f (x)
f (2) = 5 6= 4
lim f (x) = 4
x!2
h
h(x) =
lim h(x);
2
4 (x + 2)(x = 2 x 2
4 2
lim h(x);
lim h(x).
x!2
x x 6= 2
x=2 x2 x
x2 x
x!2+
x!2
h(x) =
2)
= x + 2,
lim h(x) = 4,
2
x 6= 2.
x 6= 2 h x!2
x=2
f
h (2, 5)
b)
lim h(x) = 4,
x!2+
lim h(x) = 4.
x!2
x
lim h(x) = 4
h
x!2
x2 x!2 x lim
4 22 = 2 2
4 0 = 2 0
x=2
does not exist . x=2 x=2
1 = big tiny
1 = tiny big 0
s lim s(x);
lim s(x) =
x!1
1 x
1
+2
lim s(x);
lim s(x).
x!1
x!1+
x!1
x
s(x) =
1 1 x
lim s(x) = 1
x!1+
lim s(x) =
x!1
1,
lim s(x) = 1,
x!1+
1
lim s(x)
lim s(x)
x!1
lim s(x).
x!1+
x!1
x ! 1 (x < 1)
s(x)
y 76
y = s(x)
6 5
x
1
4 3
x ! 1+ (x > 1)
2
s(x)
1 0 1
-
1
2
3
4
x
1
s(x)
x
x=1
1
f
x
a
a
x
f (x)
lim f (x) = 1,
x!a
lim f (x) =
x!a
1
1
1
1.
1
1 H
lim H(x);
x!1
( 1, 1) H(x) = lim H(x);
x!1+
1 (x
1)2 lim H(x).
x!1
x ! 1 (x < 1)
x x
x ! 1+ (x > 1)
H(x)
H(x)
1
x
1
1
lim H(x) = 1,
lim H(x) = 1,
lim H(x) = 1.
x!1
x!1+
x!1
y = f (x)
x L
f (x)
x
L
lim f (x) = L.
x!1
s
1
s(x) =
x
1
+2
lim s(x)
x!1
y 76
y = s(x)
6
x!1
s(x)
5 4 3 2 1
x!1
0 1
1
2
3
4
1
s(x)
lim s(x) = 2.
x!1
- x -
s(x) =
1 x
1
+2 1
G
lim G(x)
x! 1
y
6
3
4
3
2
r b
lim G(x)
x! 1
r
4
2
lim G(x)
x!0+
lim G(x)
x!1 1
b
10 1
1
2
3
4
lim G(x)
x
x!1
lim H(x)
x! 2
y = G(x)
2
lim H(x)
x! 2
3
lim H(x)
4
x!2+
H
lim H(x)
x!1
lim H(x)
y
x!1
lim G(x)
x! 1+
lim G(x)
x!0
lim G(x)
x!0
lim G(x)
x!1+
lim G(x)
x!3
lim H(x)
x! 2+
lim H(x)
x!2
lim H(x)
x!2
lim H(x)
x!1+
lim H(x)
x!3
6
T
4 3
y = H(x) b
2 1 4
3
2
10 1 2 3 4
1
b
2
r
lim T (x)
3
4
x
x! 3
lim T (x)
x! 3
lim T (x)
x!0+
lim T (x)
x! 1
lim T (x)
x! 1
lim T (x)
x! 3+
lim T (x)
x!0
lim T (x)
x!0
lim T (x)
x! 1+
lim T (x)
x!1
y
6
lim H(x)
x! 1
4
lim H(x)
y = T (x)
3
lim H(x)
x! 1+
x! 1
2 1 4
3
10 1
2
1
2
3
4
x f (x) = 2x2
2 3
lim f (x)
f (x) = |x|
4
lim f (x)
x!0
x!2
lim f (x)
lim f (x)
x! 1
x2 4 (x + 2)2 lim g(x)
x!0
g(x) =
lim g(x)
x! 2
f (x) = 2x + 3
lim f (x)
x! 2
2
f (x) = x + 1 g(x) =
lim f (x)
x! 1
1 x
lim g(x)
2
s(x) =
(x
lim g(x)
1)2
lim s(x)
lim s(x)
x!1
lim F (x)
x!1
lim G(x)
x!2
8 < 3x S(x) = : 2x
lim S(x)
H(x) =
x!1
: 3
lim s(x)
lim s(x)
x!1
x!0
s(x) = ln x lim s(x)
lim s(x)
x!1
x!0+
8 < 4 F (x) = : 2x
lim G(x)
lim+ F (x)
lim F (x)
x!1
x!1
8 < 4x S(x) = : x
lim S(x)
x2
x!0
x0
x! 1
x>2 lim+ G(x)
lim G(x)
x!2
x!2
lim g(x)
x!1
x
x!2
x>1
x = 1, x 6= 1 lim F (x)
lim F (x)
x!1
x!1+
x=2 x 6= 2 lim G(x)
lim G(x)
x!2
x!2+
1
x0 lim S(x)
lim S(x)
x!0
x!0+
lim H(x)
x! 1
x
1
x>
1
lim H(x)
x! 1+
lim H(x)
lim S(x)
lim S(x)
x!0
x!0+
x+2
lim g(x)
x!0
8 < 5 G(x) = : x+1
x 1,
8 < 2x + 3 G(x) = : x 1
s(x) = 2e
1 1
lim F (x)
lim s(x)
x! 3
8 < x+5 F (x) = : x 6
x2 x
g(x) =
x!1
1 s(x) = 5 x+3 lim s(x)
8
1
f (x) =
p
x
lim f (x)
x!4
lim f (x)
x!1
f (x) =
p 3
x
lim f (x)
x2 + x 6 g(x) = x 2
1 (x
lim g(x)
lim G(x)
lim g(x)
x! 1
x!0
lim S(x)
x!1
x!1
x! 1
8 < x2 F (x) = : x+2
x! 1
lim H(x)
x>2
x! 1
lim (x2 + 3x
lim x = 2
5)
lim 5 = 5.
x!2
x!2
lim 3x = 3 · 2 = 6
x!2
lim x2 = 22 = 4.
x!2
lim (x2 + 3x
5) = 4 + 6
lim (x2 + 3x
5) = (2)2 + 3(2)
x>0 lim G(x)
5 = 5.
5=4+6
5 = 5.
lim G(x)
x!0
x1 lim S(x)
lim S(x)
x!1
x!1+
lim H(x)
x2
x!2
x0
8 < 1/(x + 1) H(x) = : ln(x + 1)
lim s(x)
lim F (x)
x!2
x!0+
8 < 3x 3 S(x) = : ln x
x!1
1 s(x) = x+1 lim s(x)
lim F (x)
x!2+
8 < ex G(x) = : x3 + 1
lim s(x)
x!1
x!2
x!2
1
1)2
lim s(x)
x!2
lim F (x)
x!0
x! 2
x2 + 3x + 2 g(x) = x+1 s(x) =
lim f (x)
x! 1
x
1
lim H(x)
x! 1+
lim
x!1
lim
x!1
p
x2
p x2
2x + 5 =
2x + 5
p (1)2
2(1) + 5 =
p
4 = 2.
c lim c = c x!a
lim x = a
x!a
lim f (x) = L
x!a
c lim cf (x) = cL x!a
n lim [f (x)]n = [ lim f (x)]n = Ln x!a
n lim
x!a
n
x!a
p n
f (x) =
q n
lim f (x) =
x!a
lim g(x) = M
x!a
lim [f (x)±g(x)] = L±M
x!a
lim [f (x) · g(x)] = [ lim f (x)] · [ lim g(x)] = L · M
x!a
x!a
lim f (x) f (x) L = x!a = x!a g(x) lim g(x) M
M 6= 0
lim
x!a
0 = lim 0 = lim x!0
✓
1 x!0 x lim
◆
x!a
x!0
✓
✓
1 x
1 x!0 x lim
◆
1 x
,
◆
.
p n
L
0
L
y6
y6
x
0
x
0
y6
y6 c
c
s x
0
x
0
f (x)
x
a
lim f (x)
x!a
f (a)
f f (a)
f
x=a x=a
lim f (x)
x
x!a
a
lim f (x) = f (a)
x!a
I
I
f (x) = x2 + 3x f (2) = (2)2 + 3(2)
5
x=2
5 = 5,
f (2) lim f (x) = 5,
x!2
lim f (x) = f (2).
x!2
f
x=2 x=a
f (a)
f
x=a
lim f (x)
x
x!a
f (a)
lim f (x) 6= f (a)
lim f (x)
x!a
x!a
f (x) =
x2 x
x=2
4 2
x=2
f (2)
f
x=2 8 < x+2 f (x) = : 2 lim f (x) = 3
x!1
x1 x>1 x=1
lim f (x) = 2.
x!1+
lim f (x) 6= lim+ f (x)
x!1
lim f (x)
x!1
x!1
x=1
y6 4 3 2 1
4
3
2
1
0
s c 1
2
3
1 2
f (x)
4
x
a
8 < x+2 f (x) = : 2
x 6= 1 x=1 x=1 y 4 3 2 1
4
3
2
1
0
6 b r 1
2
3
4
x
1 2
f (x) f (1) = 2 lim f (x) = 3.
x!1
lim f (x) 6= f (1)
x!1
x=1
f lim f (x) = f (a).
x!a
6e2x + x2 lim p . x!0 2x + 9 6e2x + x2 f (x) = p 2x + 9
x=0
6e2x + x2 6e2(0) + (0)2 6 lim p = p = = 2. x!0 3 2x + 9 2(0) + 9 lim (6x2
4xh + h2 ).
h!0
x
x
h=0 lim (6x2
h!0
4xh + h2 ) = 6x2
x+2
x
4x(0) + 02 = 6x2 .
0 (
x=4
x+2 x0 x 4 x > 0, x
lim
x!
x!
4
x=0 3x 4
x2 1 . 1 x+1
x x 6=
lim
x+2 x 4
4
1 1
x2 1 (x 1)(x + 1) = lim = lim (x 1 x+1 x! 1 x! 1 x+1
lim
h!0
2 3+h
h
h h 6= 0 2 3+h
2 2 3 = · 3 3+h 3
2 3
1) = ( 1)
1=
2.
. 0
2 3+h 6 (6 + 2h) 6 6 2h 2h · = = = . 3 3+h 3(3 + h) 3(3 + h) 3(3 + h)
1 h
h 2 3+h
2 3
h
lim
2 3+h
2h 1 2 · = , for h 6= 0. 3(3 + h) h 3(3 + h)
=
2 3
= lim
h
h!0
h!0
2 = 3(3 + h)
(x + h)2 h!0 h
2 = 3(3 + 0) x2
lim
2 . 9
. x
(x + h)2 h!0 h
x2
lim
h h 6= 0
x2 + 2xh + h2 h!0 h h(2x + h) = lim h!0 h = lim (2x + h)
0
x2
= lim
h!0
= 2x.
lim
x! 1
x 1 x+1 x
1
2 x
x 1 lim x! 1 x + 1 x!
1
(x
x!
f (x)
f (x)
x!
1)2 1
x
x!1
(x < 1)
x ! 1+ (x > 1)
f (x)
f (x)
x
f (x)
1
1 f
f (x) =
(
3x2 + 2 2 4x
x3 . x>3
lim f (x)
lim f (x)
x!2
x!3
f (x) = 3x2 + 2
x
3x2 + 2
2
lim f (x) = lim 3x2 + 2 = 3(2)2 + 2 = 14.
x!2
x!2
x=3
lim f (x) = lim (3x2 + 2) = 3(3)2 + 2 = 29;
x!
x!3
lim f (x) = lim (2
x!3+
4x) = 2
x!3+
4(3) =
lim f (x) 6= lim+ f (x)
x!3
10.
lim f (x)
x!3
x!3
f f (x) =
(
x2 4x
x2 x>2
3x 10
lim f (x)
f
x!2
x=2
x=2 lim f (x) = lim (x2
3x) = (2)2
3(2) =
lim f (x) = lim+ (4x
10) = 4(2)
10 =
x!2
x!2
x!2+
x!2
lim f (x) = lim+ f (x) =
x!2
f (2) = f
x!2
2
2
lim f (x) =
x!2
x=2
f (x)
2; 2.
lim f (x) =
x!2
2
2
x
y
6
4
lim (2x
5)
x!2
lim (5x2
lim (6x + 3)
lim (x2
3x + 1)
x!1
3
x! 1
4
x2 lim x!1 x
4 2
x lim x!2 x2
lim (e3x + ln x)
lim (2e
x!1
x! 1
x
lim
h!0
1
b
3
r
10 1
2
5 1 + 5x2 )
1 h!0 x(x + h)
lim (3x2 + 3xh + h2 )
r
9)
x! 3
2
y = s(x)
1
2
3
4
x
f ( 1) lim f (x)
lim f (x)
x! 1+
x! 1
lim f (x)
x! 1
f
x=
1
f ( 2) ( 4, 4)
lim f (x)
lim f (x)
x! 2+
x! 2
lim f (x)
x! 2
f
y
4
3
2
r b
2
6
1
2
3
4
x
b
10 1
1
2
3
4
x=3
lim g(x)
x
x=
lim g(x)
6
x!3
2
lim g(x)
lim g(x)
x!3
x=3 s( 2) lim s(x)
1 10 b 1
1
x!3+
g
2
2
lim g(x)
x! 1
g(3)
3
3
lim g(x)
x! 1+
x! 1
g
4
4
lim h(x)
x!3
g( 1)
y = h(x)
2
y
lim h(x)
x!3+
h r
lim h(x)
x!1
x=1
x!3
6
1 2
h
lim h(x)
2
3
lim h(x)
x!1+
h(3)
3
4
lim h(x)
x!1
y = f (x)
2
2
h(1)
1
10 1
y
x=
1
2
3
y = g(x)
4
x
x! 2
s
lim s(x)
x! 2+
x=
lim s(x)
x! 2
2
s(0) lim s(x)
x!0
lim s(x)
x!0+
lim s(x)
x!0
s
x=0
8 < x+2 f (x) = : x2 3
x2 x!2 x
lim f (x)
x2 + x 2 x!1 x 1 lim f (x)
x!1+
lim f (x)
x!1
lim
x!1
x=1
x1 x>1
lim
x2 + x 6 3 x+3
lim f (x)
x!1+
f
lim f (x)
x!1
lim
x!2
(1 + h)3 h!0 h
1
(2 + h)3 h!0 h
(x + h)2 h!0 h
x2
3(x + h)2 h!0 h
lim
1 1+h
lim
x>1
(1 + h)2 h!0 h lim
lim
lim
1
1 x
3 x+h
lim
h
h
h!0
f (1)
lim f (x)
x!1
lim f (x)
x!1+
f
lim f (x)
x!1
x=1 lim
x!2
8 < x+5 f (x) = : 4x + 2
x1
lim
x>1
f (1)
lim f (x)
x!1
f
lim f (x)
x!1+
x=1
lim f (x)
x2 x
2 3 2)2
lim
x
lim
x+1 (x 3)2
(x
lim
2 x
x!0
lim
x+1 x2
x!0
x!0
4
x!1
x!2
x!3
lim
x+3 x
x!1
x!0
lim
x
2 x2
3 4
h
h!0
1 x+h
h!0
3 4+h
lim
h
h!0
6
2 4
4
lim
x1
x x2
(2 + h)2 h!0 h
x=1
4
x!
1 1
lim
f (1)
8 < x2 f (x) = : 3x
x x2
lim
8 < x 2 f (x) = : x2 x!1
x2 4 2 x+2
x!
lim
f
lim f (x)
lim
x>1
f (1) x!1
4 2
lim
x1
1
3 x
1
8
3x2
s 6
(6, 90)
80 (5.5, 70)
60
(5, 40)
40
s
s
s
20
0
40 5
0 4
70 5.5
40 5
s
-t
4
=
5
40 1
=
6
= 40
30 0.5
.
= 60
.
y = f (x) y
x x1 y1 = f (x1 ) y2 x2
x2 y2 = f (x2 )
y1 f (x2 ) = x1 x2
f (x1 ) x1
y
x
x2 6= x1 . x1 , f (x1 )
x2 , f (x2 )
y y = f (x)
6
f (x2 ) f (x1 ) x2 x1
(x2 , f (x2 ))
@ s
s @
(x1 , f (x1 ))
-
y 10 6
f (x) = 2x2
s
8
6
s
4
s
2
0
1
y = f (x) = 2x2 x
s
2
f (x) = 2x2 f (x2 ) x2
- x
f (x1 ) x1
x x
x1 = 1 f (2) 2
x2 = 2
f (1) f (2) f (1) 2(2)2 2(1)2 8 2 = = = = 6. 1 1 1 1
x
x1 = 1
x2 = 1.5
f (1) f (1.5) f (1) 2(1.5)2 2(1)2 4.5 2 = = = = 5. 1 0.5 0.5 0.5
f (1.5) 1.5
x=1
x2 = 1.1
f (1) f (1.1) f (1) 2(1.1)2 2(1)2 2.42 2 = = = = 4.2. 1 0.1 0.1 0.1
f (1.1) 1.1
x2 x1
x
x2
x+h
x1 h
x2
h
f (x) x2 6= x1
y2 x2
y1 f (x2 ) = x1 x2
h 6= 0
f (x1 ) f (x + h) f (x) f (x + h) = = x1 (x + h) x h
f (x)
.
(x, f (x)) y
6
y = f (x)
f (x+h) f (x) h
(x + h, f (x + h))
@ s
s @
(x, f (x))
y = f (x) = 2x2 x x x
x1
f (x + h)
- x
f (x + h) h
f (x)
(x + h, f (x + h))
x=1 f (1 + 1) 1
f (1)
h=2 =
x=1 f (1 + 0.5) 0.5
f (1)
x=1 f (1 + 0.1) 0.1
1=1
f (2)
f (1) 1
h = 1.5 =
=
2(2)2
2(1)2 1
=
8
2 1
= 6.
1 = 0.5
f (1.5) f (1) 2(1.5)2 2(1)2 4.5 2 = = = 5. 0.5 0.5 0.5
h = 1.1 f (1)
=
1 = 0.1 f (1.1) f (1) 2(1.1)2 2(1)2 2.42 2 = = = 4.2. 0.1 0.1 0.1
y = f (x) = 2x2 f (x + h) h
f (x) x=1
h=
1
0.5
0.1 f (x) = 2x2
f (x + h) = 2(x + h)2 = 2 x2 + 2xh + h2 = 2x2 + 4xh + 2h2 .
f (x + h)
f (x) = (2x2 + 4xh + 2h2 )
f (x + h) h
f (x)
=
h(4x + 2h) = 4x + 2h, h x=1
f (x + h) h
2x2 = 4xh + 2h2 = h(4x + 2h).
f (x)
h 6= 0.
h=1
= 4x + 2h = 4(1) + 2(1) = 4 + 2 = 6.
x=1
h = 0.5
4(1) + 2(0.5) = 4 + 1 = 5. x=1
h = 0.1
4(1) + 2(0.1) = 4 + 0.2 = 4.2.
h h f (x + h) h
y = f (x) = x3 f (x) = x3 f (x + h) = (x + h)3 = x3 + 3x2 h + 3xh2 + h3 .
f (x + h)
f (x) = (x3 + 3x2 h + 3xh2 + h3 )
x3
= 3x2 h + 3xh2 + h3 = h(3x2 + 3xh + h2 ).
f (x + h) h
f (x)
=
h(3x2 + 3xh + h2 ) = 3x2 + 3xh + h2 , h
h 6= 0.
f (x)
f (x + h) h
y = f (x) = 4x + 3
f (x)
f (x) = 4x + 3 f (x + h) = 4(x + h) + 3 = 4x + 4h + 3.
f (x + h)
f (x) = (4x + 4h + 3)
f (x + h) h
f (x)
=
4h = 4, h
y = f (x) = 1 x
f (x) = f (x + h)
f (x)
(4x + 3) = 4x + 4h + 3
=
f (x + h) h
1 x
f (x)
=
1 1 1 x 1 x+h = · · x+h x x+h x x x+h x (x + h) x x h h = = . x(x + h) x(x + h) x(x + h) 1 h
h 1 1 · = , x(x + h) h x(x + h)
h 6= 0.
t=0 t s(2)
s(0)
s(2) 2
s(0) 0 s(2)
s(2) 2
s(0) = 23,914
23,822 = 92
s(0) 92 = = 46 0 2 $3,000 t
A(t) = 3,000(1 + 0.05/4)4t = 3,000(1.0125)4t . A(2)
A(0)
A(2) 2
A(0) 0
A(4)
A(2)
f (x)
1 . x+h
h f (x + h) h
3 = 4h.
h 6= 0.
f (x + h) =
=
4x
s(0) = 23,822
t=2 s(2) = 23,914
A(4) 4
3,313.46
A(2) 2
A(2) 3,000 = $313.46 A(2) 2
3,659.67
A(0) = 3,000(1.0125)4(2)
3,000(1.0125)4(2) = 3,000(1.0125)16
3,000(1.0125)8 ⇡
A(2) 346.21 ⇡ = 173.105 2 2
t 16t2 0 t 3.
s(t) = 144
s(2)
s(2)
s(0)
s(2) 2
s(0) 0
s(0) = 80 s(2) 2
3,000 ⇡
A(0) 313.46 ⇡ = 156.73 0 2
A(4) A(2) = 3,000(1.0125)4(4) 3,313.46 = $346.21 A(4) 4
3,000(1.0125)4(0) = 3,000(1.0125)8
s(0) = 144 144 = 64 s(0) = 0
64 = 2
16(0)2 = 144
32
0 = 144
s(2) = 144
16(2)2 = 80
x2 x = 2
f (x) = 4 h=
0.5
0.1
3
x=2
f (x) =
f (x+h) f (x) h
h=
x 1
0.01
0.2
0.001
0.1
0.01
0.1
0.01
f (x) = 2x3 x = 1 x
h
f (x) = h=
1
0.1
0.01
1
0.2
0.2
h=
2x + 3 x = 5
1
0.2
2
f (x) = x + 3x
2x2 x = 2
f (x) = x2 h=
0.2
1
f (x) = x2
f (x) = 3x2 x = 3 h=
h=
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
h=
1
0.1
0.01
1 x=3
0.2
0.1
0.01
x x=1
1
0.2
f (x) = 2x2 + 3x x = 1 h=
1
0.2
f (x) = 5x + 2 x = 3 h=
1
f (x) = 2x h=
0.2
t s(t) = 256
16t2 0 t 4.
s(3)
s(1)
s(3) 3
s(1) 1
5 x=1
1
0.2
3 x=1 x h= 0.5 0.1
f (x) =
5 x=2 x h= 0.5 0.1
0.01
0.001 t
f (x) =
0.01
0.001
2 f (x) = 2 x = 2 x h= 0.5 0.1
0.01
0.001
1 x=2 x2 0.5 0.1
0.01
0.001
f (x) = h=
s(t) = 196
16t2 0 t 3.5.
s(2)
s(1)
s(2) 2
s(1) 1
t
f (x) = 4 x = 2 h=
1
f (x) = h=
0.1
0.01
0.1
0.01
2 x=1 1
f (x) = 2 h=
0.2
0.5
0.2 2x2 x = 1 0.1
0.01
0.001
s(t) =
16t2 + 80t 0 t 5.
s(2)
s(0)
s(2) 2
s(0) 0
$2,000 t
A(3)
A(2)
A(3) 3
A(2) 2
A(t) = 2,000(1.00375)12t . A(1)
A(0)
A(1) 1
A(0) 0
A(2)
A(1)
A(2) 2
A(1) 1
g(x) =
g(x)
t A(t) = 4,000e0.04t . A(0)
A(1) 1
A(0) 0
A(2)
A(0)
A(2) 2
A(0) 0
0 x 10.
x
$4,000
A(1)
2x2 + 60x
g(2)
g(0)
g(2) 2
g(0) 0
g(10)
g(8)
g(10) 10
g(8) 8
S(x) =
x2 + 60x
0 x 30.
x S(x)
$2,000
S(10)
S(0)
S(10) 10
S(0) 0
S(30)
g(20)
S(20) 20
S(10) 10
t A(t) = 2,000(1.04)4t A(3)
A(0)
A(3) 3
A(0) 0
0 t 3.
x
C(x) =
x 0.02x2 +40x+200 0 x 250.
C(101) 101
C(100) 100
0.1x2 + 100x 0 x 200.
R(101) 101
R(100) 100
x
x C(x) = 2x2 + 15x + 1500 0 x 200. C(101) 101
R(x) =
C(100) 100
R(x) =
0.02x2 + 100x 0 x 100.
R(51) 51
R(50) 50
y = f (x) f (x + h) h
f (x)
. h f 0 (x)
x
f
x
x
y = f (x) f 0 (x) = lim
h!0
x
f (x + h) h
f (x)
f 0 (x)
f 0 (x)
, f
x
f0
y = f (x)
f (x + h) h
f (x) 0
h
0 h
0
y = f (x) = 2x2 f 0 (x) f 0 (1) f 0 ( 2)
4x + 2h f (x + h) h
f (x)
f (x + h) h
f (x)
= =
=
2(x + h)2 h
2x2
2x2 + 4xh + 2h2 2x2 h 2 4xh + 2h h(4x + 2h) = h h 4x + 2h, h 6= 0. =
h f 0 (x) = lim
h!0
f (x + h) h
f (x)
0
= lim (4x + 2h) = 4x. h!0
0
f (x) = 4x x=1 f 0 ( 2) = 4( 2) =
f 0 (x) = 4x
f 0 (1) = 4(1) = 4
8
y = f (x) = x3 f 0 (x) f 0 ( 0.5) f 0 (0.2) 3x2 + 3xh + h2 h f 0 (x) = lim
h!0
f (x + h) h
f 0 (x) = 3x2
f (x)
= lim 3x2 + 3xh + h2 = 3x2 . h!0
0
x=
f 0 (x) = 3x2
0.5
f 0 ( 0.5) = 3( 0.5)2 = 3(0.25) = 0.75
f 0 (0.2) = 3(0.2)2 = 3(0.04) = 0.12 f 0 (x)
y = f (x) = 4x + 3
f 0 (5) 4
f (x + h) h!0 h
f (x)
f 0 (x) = lim 0
= lim 4 = 4. h!0
0
f (x) = 4
f (5) = 4
y = f (x) =
1 x f (x + h) h
f 0 (x)
lim
h!0
f 0 (a)
f (x + h) h
f (x + h) h a=
f (x)
=
f (x)
f (x)
3
1
2
1 . x(x + h) h
f 0 (x) = lim
h!0
f (x + h) h
f 0 (x) =
= lim
h!0
1 1 1 = = 2 . x(x + h) x·x x
1 x2 x=3
f 0 (3) =
f (x)
0
1 1 = = (3)2 9
f 0 (x) =
1 x2
1 . 9
f 0 ( 1) =
1 1 = = ( 1)2 1
f 0 ( 2) =
1 1 = = ( 2)2 4
f (x) (x, f (x))
1 1 4
(x + h, f (x + h))
h
f 0 (x) (x, f (x)) y y = f (x)
6
(x + h1 , f (x + h1 ))
@ s
(x + h2 , f (x + h2 ))
@ (x + h3 , f (x + h3 ))
s @
@ @s
s
lim
h!0
(x, f (x))
f (x + h) h
f (x)
= f 0 (x)
- x
f (x) = 2x2 y
x=1
f (x) = 2x2
86
6
4
2
0
y = 4x
2
s 1
2
- x
f (x) = 2x2 4 y
2 = 4(x
x=1
f 0 (1) = 4 (1, 2)
x=1
1). y = 4x h
2
f (1) = 2
s v(t) = s0 (t) t 16t2 0 t 3.
s(t) = 144 v(t) v(2)
s(t + h) h
s(t)
s(t + h) h
s(t)
= =
= =
2
144
16(t + h) h
16 t2 + 2th + h2 ] h
[144
16t2
[144
=
16h2 ]
32th
[144
16t2 ]
[144
16t2 ]
h 16h2 144 + 16t2 h 32th 16h2 h( 32t 16h) = h h 32t 16h, h 6= 0. 16t2
144
=
v(t) = s0 (t) 144 16t2
32th
h s(t + h) h!0 h
s0 (t) = lim
v(t) = s0 (t) = v(2) =
s(t)
= lim ( 32t h!0
16h) =
0
32t.
32t
64
f
x
f x=a x=a f 0 (a) = lim
h!0
f (a + h) h
f (a)
. f
x=a h
lim [f (a + h)
h!0
f (a)] = 0
lim f (a + h) = f (a).
h!0
f 0 (a)
f 0 (x)
f
x=a x=a x=a
x=a x=a f (x) = |x|
f (x) = |x|
x=0
y6 4 3 2 1
4
3
2
1
0
1
2
3
4
1
y = |x| f (x) = |x|
lim
h!0
f (0 + h) h
f (x) =
8
h > : =1 h
h
h!0
h!0+
=
8 > > > >
0
x
x=0
f
x=0 f 0 (a)
x=
a x=a f
x=a
x=a x=a x=a x=a x=a x=a a
y 6
q
y
y
y
6
6
6
- x
- x
- x
- x
f (x) = 2x2 + 3x a=
2
0
0.5
0
3
1
0
3
1
1
3
1
1
5
1
f (x) = 5x + 2 f (x+h) f (x) h
a=
lim f (x+h)h
f 0 (x)
f (x) = 2x
f (x)
h!0
a=
f 0 (a)
a
f (x) =
1
1
f (x) = x2 a=
0
1
f (x) =
2
a=
2x2
a=
f (x) =
0
1
0
0.5
5
3 x
a=
f (x) = 3x2 a=
1
5 x
2
x 2
1
f 0 (x)
lim f (x+h)h
h!0
f (x)
y
f 0 (a)
a
(a, f (a))
f (x) =
a q
2 x
a= f (x) =
6
q a
q
q
q
q
x4
x5
a
q
x6
x7
q
q
q
1
2
1
2
x 0 x1
x2 x 3
x8
x9 x10
1 x
a= f (x) =
x
a=
2
2
0
0
2
0
2
0
1
0
1
0
1
16t2 0 t 4.
v(t) v(3)
2x2
f (x) = 2 1
x2
f (x) = 4 a=
2
f (x) =
t
1
s(t) = 256
1 f (x) = x2 2 a= 1 a=
x
a=
t s(t) = 196
3
1
16t2 0 t 3.5.
v(t) v(2)
f (x) = 2x3 a=
1
f (x) = x a=
2
2x + 3 2
0
f (x) = x2 + 3x a=
1
t
1 1
0
s(t) = 1
16t2 + 80t 0 t 5.
v(t) v(3) t=3
y
6
q
a q
q
q
q a
q
x3 x 4
x5
q
q q
q
t
q
s(t) =
x 1 x2
x6
x7
x8 x 9
16t2 + 120t 0 t 7.5.
v(t) v(3) -
x0
q
x10
x
t=3
x
f 0 (x) y
x
f 0 (x)
y = f (x)
dy . dx y dy
x
dx
y = x2 dy = 2x. dx
f 0 (x) =
d d 2 f (x) = x = 2x. dx dx x=3 dy dx
f 0 (3)
. x=3
dy = 2x dx dy dx
x=3
= 2x x = 3 = 2 · 3 = 6. f 0 (x)
y 0 = 2x. Dx y
dy dx
y0
y
y = x2
y 0 , f 0 (x),
dy d df , f (x), , Dx f, dx dx dx
Dx y
dy f (x) dx
d 2 x = 2x dx
dy 2 x 6= 2x. dx
y = f (x) = C
C
f (x + h) = C
f (x + h) h 6= 0
h f 0 (x) = lim
h!0
f (x + h) h
f (x)
= lim
h!0
C
C h
= lim
h!0
0 = lim 0 = 0. h h!0
d C=0 dx x y f (x+h) f (x) h
6 s
f (x) = C
=
c c h
=0
s
f (x + h) = C
- x
f (x) = C
C=0
d C = 0, dx y = f (x) = C y 0 = f 0 (x) =
y=
dy = 0. dx
4;
y=
1 ; 3
y=
d ( 4) = 0 dx
d dx
p
5
⇡;
y = e3 .
✓ ◆ 1 =0 3
d ⇣ 5⌘ e3 = 0 dx
d p ⇡ =0 dx
y = f (x) = xk k
x3
3x2
x2
2x1 = 2x
x = x1
1 · x0 = 1
1 =x x 1 =x x2
1
2
1·x
2
2·x
3
= =
1 x2 2 x3
k d k x = k · xk dx
1
,
y = f (x) = xk y 0 = f 0 (x) =
dy = k · xk dx
1
.
xk
k k
x
y = x4 ;
y=x
d 4 x = 4 · x4 dx d x dx
3
=
1
d ⇡ x = ⇡ · x⇡ dx
y = x1/2 ;
;
= 4x3 .
3 1
3·x
1 d 1 1 x2 = · x2 dx 2
3
k
1
1
=
=
1 x 2
3x
1 2
4
=
3 . x4
.
.
d 1 ; dx x4
d p 3 x5 . dx
d 1 p ; dx x
xk d 1 d = x 4 dx x dx d 1 d p = x dx x dx
4
=
1 2
=
4·x
4 1
1 ·x 2
5 d p d 5 5 3 x5 = x3 = · x3 dx dx 3
1
=
4x
1 2
1
=
5 2 x3 , 3
=
5
4
,
1 x 2
x5 3 2
, 5p 3 x2 . 3
.
1 p . 2 x3
y = x⇡ .
y = ex y = f (x) = ex
lim
eh
h!0
1 h
h!0
= 1.
eh
(h < 0)
1
h ! 0+ (h > 0)
h
eh 1 h
eh
1 h
eh 1 h
h!0
f (x) = ex e f (x + h) h
f (x)
= =
x+h
x
=e ·e
ex+h ex ex · eh ex = h h h ex (eh 1) 1 x e =e · , h h
h
h 6= 0.
f 0 (x) lim
h!0
f 0 (x)
= =
h eh
1 h
=1
f (x + h) f (x) eh 1 = lim ex · h!0 h!0 h h h e 1 ex lim = ex · 1 = ex . h!0 h lim
d x e = ex dx
ex d x e = ex , dx y = f (x) = ex y 0 = f 0 (x) =
dy = ex . dx
ex
0
h ! 0+
F (x) = k f˙(x) F (x) F (x + h) h
F (x)
f 0 (x) F (x + h) = k · f (x + h) = =
k · f (x + h) h f (x + h) k· h
k · f (x) f (x)
,
=
k · [f (x + h) h
f (x)]
h 6= 0.
F 0 (x)
h f (x)
F 0 (x)
= =
F (x + h) F (x) f (x + h) = lim k h!0 h h f (x + h) f (x) k lim = k · f 0 (x). h!0 h lim
f (x)
h!0
d d [k · f (x)] = k · f (x). dx dx
y = F (x) = k · f (x) y 0 = F 0 (x) = k · f 0 (x), d d [k · f (x)] = k · f (x). dx dx
d 5x3 ; dx 2
d 3 ; dx x
d 7ex ; dx
d 1 p . dx 4 3 x
d 5x3 5 d 3 5 15 2 = x = · 3 · x2 = x . dx 2 2 dx 2 2 d 3 d =3 x dx x dx
1
=3·
1·x
1 1
=
3x
2
,
3 x2
.
0
d d 7ex = 7 ex = 7ex . dx dx d 1 d p = dx 4 3 x dx
✓
1 x 4
1 3
F 0 (x)
= = = =
F (x)
=
1 d x 4 dx
f 0 (x) F (x)
F (x) = f (x) + g(x) F (x + h) h
◆
1 3
=
1 · 4
1 ·x 3
1 3
1
4 3
,
1 p . 3 12 x4
[f (x + h) + g(x + h)] [f (x) + g(x)] h [f (x + h) f (x)] + [g(x + h) g(x)] = h f (x + h) f (x) g(x + h) g(x) = + , h 6= 0. h h F 0 (x) f (x) g(x) =
F (x + h) F (x) h!0 h f (x + h) f (x) g(x + h) g(x) lim + h!0 h h f (x + h) f (x) g(x + h) g(x) lim + lim h!0 h!0 h h f 0 (x) + g 0 (x). lim
g(x)] =
d f (x) dx
d g(x). dx
y = F (x) = f (x) + g(x) y 0 = F 0 (x) = f 0 (x) + g 0 (x), d d d [f (x) + g(x)] = f (x) + g(x). dx dx dx y = F (x) = f (x)
1 x 12
g 0 (x)
d d d [f (x) + g(x)] = f (x) + g(x). dx dx dx d [f (x) dx
=
g(x) y 0 = F 0 (x) = f 0 (x) d [f (x) dx
g(x)] =
g 0 (x),
d f (x) dx
d g(x). dx
h
0
d 3x2 dx
5x + 8e
d 3x2 dx
5x + 8ex
=3
d 2 x dx
d dx =3·
✓
5
3 x2
p 5
x3
2 1
◆
7 ;
7 =
d = dx
4·
✓
3 3 · x5 5
✓
d dx
d 3x2 dx
d d x + 8 ex dx dx
4
2·x
x
1
◆
=
4
p 5
x3
5 · 1 + 8ex
.
3
12 x 5
2 5
=
d 7 dx
0 = 6x
⌘ d ⇣ p d 5 4 x3 = 3 x dx dx 6x
◆
d d (5x) + (8ex ) dx dx
d 7 = 3 · 2x dx 3 x2
3 x2
6 x3
2
4
5 + 8ex .
d 3 x5 dx
12 p . 5 5 x2
y = f (x) f 0 (x0 )
(x0 , f (x0 )) x0 f (x) = x3
3x2 + 1
x=1
0
f (x) d x3 dx =
3x2 + 1 =
d 3 x dx
d d 3x2 + 1 dx dx
d 3 d d x 3 x2 + 1 = 3 · x3 dx dx dx f 0 (1) = 3(1)2 6(1) = 3 x=1 (1, 1)
y
y=
( 1) =
3(x
1
3 · 2 · x2
1
f (1) =
1
+ 0 = 3x2
6x. m=
1).
3x + 2. (1, 1)
f (x) = x3 (a, f (a)) f 0 (x) = 3x2 3x2
6x = 0.
6x
3x2 + 1 f 0 (a) = 0
3
3x(x
2)
=
0
3x = 0
x
x=0
2=0
x = 2. x=0
f (0) = (0)3
3(0)2 + 1 = 1.
f (2) = (2)3
3(2)2 + 1 = 8
12 + 1 =
3. (0, 1)
y
(2, 3)
f !x""x3 !3 x2 #1
4
2
1
!1
2
3
x
!2
!4
f (x) = x3
3x2 + 1
y 0 = f 0 (x) = 4x3
y = f (x) = x4 f (x) y 00 = f 00 (x) =
d 2y d2 = f (x). dx2 dx2
y = f (x) = x4 d 2y d2 4 d y = f (x) = = x = dx dx dx 00
00
y 000 = f 000 (x) =
d 4 x dx
◆
=
d 4x3 = 12x2 . dx
d 3y d3 = 3 x4 = 24x. 3 dx dx n
y n = f (n) (x) =
✓
dn y dn = f (x). dxn dxn
n
4
x=2
d3 dx3
3x2
5x + 8ex
d2 dx2
7 ;
d 3x2 dx d2 dx2
3x2
5x + 8ex
7
= = =
d2 dx2
3x2
✓
3 x2
◆ p 5 4 x3 .
3 x2
5x + 8ex
7 = 6x
d (6x 5 + 8ex ) dx d d d 6 x 5 + 8 ex dx dx dx 6 · 1 0 + 8ex
6 + 8ex , d 5x + 8ex 7 = (6 + 8ex ) dx = 0 + 8ex = 8ex . ✓ ◆ p d 3 12 5 3 4 x = 6x 3 x 2 dx x 5 ◆ ✓ ◆ p d 12 2 5 4 x3 = 6x 3 x 5 dx 5 2 d 12 d = 6 x 3 x 5 dx 5 dx 2 12 2 3 1 = 6· 3·x · ·x 5 1 5 5 24 7 18 24 4 = 18x + x 5 = + p . 5 4 25 x 25 x7 =
d3 dx3
✓
2 5
s v(t) = s0 (t)
s
a(t) = v 0 (t) = s00 (t).
t s(t) = 144 v(t)
16t2 0 t 3. a(t)
a(2) v(t) = s0 (t) = a(t) = v 0 (t) = a(2) =
32. 32
2
32t
5 + 8ex
f 0 (x) f (x) = 2x7
4x5
f (x) =
p
p 3
f (x) = 5 x
f (x) =
p 5 3 x7
p f (x) = 2 x5
f (x) =
p 5 3 x3
p 3 f (x) = 2 x2
f (x) = 3 x
2 f (x) = 2 x
f (x) =
2 f (x) = p x dy dx y=
y=
3 x4
3 f (x) = p 3 x2
5x 7
y=
7ex 2
y=
3 y= p 24x
3x 2
2
2ex 3
2 y= p 35x
y=
x2 3 + 2 3 x
y = 5ex
xe
d 8x2 dx d 3x2 dx
2.4x
0.1
y=
y = 2x3
10x2 + 6ex
y = 5x4
3x
1.7x0.4
5x + 10
8x2
6x + 4
d2 dx2
3x2
5x + 10 ◆ 3 p x ◆ 7 p 3 x
d dx
✓ ✓
p 5 x p 23x
d5 7 x dx5
d4 dx4
d7 (3ex ) dx7
p y=23x
p
x3
2 x2
y=
f 0 (4) f 0 (2)
3
f 00 (4) f 00 (2)
f (x) = 2x3 3x2 +x 1
7ex + 8 1;
x = 0;
2 3e + x
x = 1.
x
5ex +
f (x) = x2
5 x3
x=
1;
x = 1;
y0 y=
3x4
d8 (4ex ) dx8
f (x) =
x= p y=3 x
y = 3xe + 2ex
d2 dx2
d dx
x3 2 + 3 2 x
6x + 4
f (x) = y=
y=
p 5x +3 x 7 p 3x + 5 x3 4
5 p 3 x2 f (x) = 2 p 3 x
f (x) =
1 2 x 2
1
2x2 + 3
1 x x = 2.
f (x) = 13 x3
2
5x3 + 1
f (x) =
f (x) = 10x
x2
f (x) = 6x
3x2
f (x) = 0.3x
2
t s(t) = 196 v(t)
1.2x + 3
f (x) = 0.5x2 + 2x f (x) = 2x3 + 3x2
a(t)
s(2) v(2)
a(2)
1 12x + 1
f (x) = 2x3
7x2 + 4x + 3
f (x) = 13 x3
2x2
f (x) = 2x3
3x2 + 5
t s(t) = v(t)
f (x) = x3
12x + 3
f (x) = x3
3x
16t2 + 80t 0 t 5. a(t)
2
f (x) = 7
f (x) = 3x
16t2 0 t 3.5.
f (x) =
1
5
f (x) = 4x + 5 t s(t) =
t
v(t)
s(t) = 256 v(t)
16t2 0 t 4.
a(t)
s(3) v(3)
y = x2 ex
y=
x2 + 1 ex
a(3)
16t2 + 120t 0 t 7.5. a(t)
f (x)
g(x)
d d d [f (x) + g(x)] = f (x) + g(x) dx dx dx
c d d [c · f (x)] = c · f (x), dx dx
f (x) = x2
f (x) · g(x)
g(x) = x3
f (x) · g(x) = x2 · x3 = x5 , d d 5 [f (x) · g(x)] = x = 5x4 . dx dx f 0 (x) = 2x f (x) · g(x)
f (x)
g 0 (x) = 3x2 f 0 (x) · g 0 (x) = 2x · 3x2 = 6x3
y = F (x) = f (x) · g(x)
g(x)
y 0 = F 0 (x) = f (x) · g 0 (x) + f 0 (x) · g(x), d d d [f (x) · g(x)] = f (x) · g(x) + f (x) · g(x). dx dx dx
f (x)
g(x) @
@
0
g 0 (x)
f (x) f 0 (x) · g(x)
+
f (x) · g 0 (x)
f (x) = x2 f (x)
g(x) @
@
f 0 (x) f 0 (x) · g(x)
g 0 (x)
+
f (x) · g 0 (x)
g(x) = x3
5x4
x2
x3 @
@
=
3x2
2x 2x · x3
x2 · 3x2 = 2x4 + 3x4 = 5x4 .
+
d (x2 · x3 ) = 5x4 dx
y = x3
2x2 + 1
x4
2 ;
y=x
3 x
e .
f (x) = x3
f (x)
2x2 + 1
g(x) @
@
0
g 0 (x)
f (x) f 0 (x) · g(x)
f (x) · g 0 (x)
+ x3
2x2 + 1
x4
2
@
@
=
3x2 3x2 d (x3 dx
4x3
4x
4x · x4
2
2x2 + 1)(x4
+ 2) = (3x2
x3
2x2 + 1 · 4x3 .
4x) · (x4
2) + (x3
f (x) = x
f (x)
g(x) @
@
f 0 (x) f 0 (x) · g(x)
g 0 (x)
+
f (x) · g 0 (x)
2x2 + 1) · 4x3 .
3
g(x) = ex
g(x) = x4
2
3
x
ex @
@
=
4
3x
3x d x dx
4
ex
· ex
3 x
e =
+ 3x
4
x
3
· ex + x
· ex . 3
· ex = (x
3)x
4 x
e
3)ex x4
f (x) = x2
g(x) = x3
F (x) =
1
F 0 (x) =
f (x) x2 1 = 3 = g(x) x x
x2 f 0 (x) · g(x) f (x) · g 0 (x) 2x · x3 x2 · 3x2 = [g(x)]2 [x3 ]2
F 0 (x) =
=
(x
2x4
3x4
=
x4 1 = 2. x6 x
x6 ✓ 2◆ d x 1 = 2 dx x3 x
f (x)
g(x)
y = F (x) = y 0 = F 0 (x) =
d dx
f (x) = g(x)
⇥
d dx
f (x) g(x)
f 0 (x) · g(x) f (x) · g 0 (x) , [g(x)]2 ⇤ d f (x)] · g(x) f (x) · [ dx g(x) . [g(x)]2
g(x)
y=
x3
2x2 + 1 ; x4 2
y=
ex . x3 f (x) = x3
2x2 + 1
g(x) = x4
2
dy dx
= = = =
f 0 (x) · g(x) f (x) · g 0 (x) [g(x)]2 2 (3x 4x) · (x4 2) (x3 2x2 + 1) · 4x3 [x4 2]2 6 5 2 (3x 4x 6x + 8x) (4x6 8x5 + 4x3 ) [x4 2]2 6 5 3 x + 4x 4x 6x2 + 8x . [x4 2]2
d x3 2x2 + 1 = dx x4 2
dy dx
= = =
x6 + 4x5 4x3 6x2 + 8x (x4 2)2 f (x) = ex g(x) = x3
f 0 (x) · g(x) f (x) · g 0 (x) [g(x)]2 x 3 e ·x ex · 3x2 (x3 )2 x 2 e x (x 3) ex (x 3) = . x6 x4
d ex ex (x 3) = 3 dx x x4 y=
y=
x3
2x2 + 1 ; x2
ex x3
y=
y=x
xex . x 1 f (x) = x3
y=
x3
dy =1 dx
2x2 + 1 x3 = x2 x2
3
2x
=1
2x2 1 + 2 =x 2 x x
2+x
f (x)
.
2 . x3 f (x) = xex
0
2
2x2 + 1
g(x) = x
1
g(x) = x2
3 x
e
dy dx
f 0 (x) · g(x) f (x) · g 0 (x) [g(x)]2 (ex + xex ) · (x 1) xex · 1 (x 1)2 xex + x2 ex ex xex xex ex (x2 = 2 (x 1) (x
= = =
d xex ex (x2 = dx x 1 (x
x 1) 1)2
f 0 (x)
f (x) = x7 · x2
f (x) = x5 · x3
f (x) = 3x3 2x2 + x
f (x) =
p
x·
p 3
f (x) = x 5x4
x
f (x) =
p 3
x·
3x2
p 4
x
f (x) =
2x 1 x3 + 5
f (x) =
3x2 + 5 2x 1
f (x) =
2ex x2
f (x) =
3ex x4
f (x) =
2x ex
f (x) =
f (x) x6 f (x) = 2 x
f (x) =
2x8
f (x) =
x5 x2 1 x+1
f (x) =
f (x) = 3x5
f (x) =
✓
3
2x4 + 6
p 8 3 x+ x
◆
p f (x) = 2ex + 5 3 x
7x2
x3 + 3x x2 x4 4 x2 + 2
y=
y = 2x3
10x2 + 6ex
3 ex
y = 5x4
3x
6 +3 x2
p 5 y = 2 3 xex + 3 x
y0
2
e x + x2 + 4
3x4 +
2x4 + 7xex
7ex + 8 ex
p 2 y = 3 xex + x
6x + 1
5x3 + 2x2
✓
3x2 5 2
y = 2x + x3 ex
p 5x + 3 x 2 (x + 3) +3 ⇣ p ⌘ 3x y= 3 + 5 x3 + 3 (2x 1) 4x 5 y=
8
y=
7ex 2ex y= 2x + 1 3x 4 (x2 3x + 2) x3 2x2 + 1 p p y= y = 24x+3 35x+5
f 0 (x) f (x) = 2x7 + 5x4
5x + 3 7
y=
x7 f (x) = 4 x
x2
3x ex
dy dx y=
0
f (x) =
x 1) . 1)2
◆
y=
7x2
x2 3ex + 2 + 3ex + 2 x2
y=
d dx
x3 2ex + 3 + +3 x3
f (x) = (3x + 2)ex
2ex ✓
3xex x2 + 4
◆
d dx
d2 (3x + 1) x2 dx2
✓
◆
f (x) = (2x3
5x)ex
x=0
5x + 2
d2 (4x + 3) 2x2 + 7x dx2 d2 x2 ex dx2 d2 3 dx2 3x + 1
x2 ex x+3
f (x) =
4 x2 +2
1 x=
1;
x = 0;
d2 (3xex ) dx2 d2 1 dx2 x2 + 4
f (x) = x=
4%
1;
x = 0;
x = 1.
x2
x +4 x = 1.
t
S(t) = 500e0.04t . S(t) y = 1 + x3
2
,
y = 1 + 2x3 + x6 ,
dy = 6x2 + 6x5 . dx p=2
2(1 + x3 )
y f (x) = 1 + x2 S(t) = 500e
0.04t
4
y = 1 + x3
2
y y = f (u) = u2
u = g(x) = 1 + x3 . 2(1 + x3 )
2
dy dx
dy = 6x2 + 6x5 = 2(1 + x3 ) · 3x2 . dx du dx
u = g(x) = 1 + x3 y = [g(x)]k g(x) dy = k[g(x)]k dx
1
·
= 3x2
g 0 (x)
d g(x). dx
g(x) y 0 = F 0 (x) = k[g(x)]k d [g(x)]k = k[g(x)]k dx
y = x3
4x2 + 5
5
;
y=
1 (2x +
1
1
·
3
4x2 + 5
5
= 5 x3
g(x) = 2x + e
x
4x2 + 5
3; ex )
p
1 + x3 .
g(x) = x3
4
· 3x2
8x .
y = (2x + ex ) g (x) = 2 + ex 3
= ( 3) (2x + ex ) y = 1 + x3
g(x) = 1 + x
3
3
0
d 1 d = (2x + ex ) dx (2x + ex )3 dx k = 1/2
y=
d g(x). dx
8x
d x3 dx k=
y = F (x) = [g(x)]k
· g 0 (x),
k = 5 g 0 (x) = 3x2
k
0
g (x) = 3x
4
· (2 + ex ) =
1/2
2
d p d 1 1 + x3 = (1 + x3 )1/2 = (1 + x3 ) dx dx 2
1/2
3x2 · 3x2 = p . 2 1 + x3
3 (2 + ex ) (2x + ex )
4
.
4x2 + 5
y = eg(x) g(x)
y = F (x) = eg(x)
g(x)
y 0 = F 0 (x) = eg(x) · g 0 (x), d g(x) d e = eg(x) · g(x). dx dx
y=e
0.04x
y = ex
;
3
+2x
;
y=e
p
x
. g(x) =
0.04x
g 0 (x) =
0.04 d e dx
0.04x
=e
0.04x
· ( 0.04) =
0.04e
0.04x
. g(x) = x3 + 2x
3 d x3 +2x e = ex +2x · 3x2 + 2 , dx
3x2 + 2 ex
3
+2x
.
g(x) = p
p d px 1 e x e =e x· p = p . dx 2 x 2 x
y = (x3
2
2x2 + 1)3 e3x .
y f (x) = (x3 f 0 (x) = 3 x3 2x2 + 1 2 g 0 (x) = 6xe3x
2
f (x)
2x2 + 1)3 3x2 4x
g(x) @
@
f 0 (x) f 0 (x) · g(x)
g 0 (x)
+
f (x) · g 0 (x)
g(x) = e3x
2
g 0 (x) = 3x2 + 2
p
x
1 g 0 (x) = p 2 x
(x3
2x2 + 1)3
e3x
2
@
@
=
3 x3 3 x3
2x2 + 1 2
2x2 + 1
3x2
d x3 dx
2x2 + 1
3 3x2
= 3 x3
2x2 + 1
2
= [3 3x2
2
1
e5x
4x3 + 3x2
1
=
2
2
x3
+
2x2 + 1
3
2
· 6xe3x .
4x · e3x + x3
2x2 + 1 ] x3
2x
x3
2x2 + 1
3
2x2 + 1 2x2 + 1
2 3x2
e
· 6xe3x
2
2 3x2
e
.
x2 1 e5x y =
= x2
d x2 1 = dx e5x =
4x · e3x
2
3x2
y= x2
6xe3x
4x
e
4x + 6x x3
= 3 2x4
3x2
2x · e5x
1 e
5x
d dx
x2
· e5x
1
x2
2 (e5x )
x2 (e5x )
1 · 5e5x
2
1 ·
d dx
e5x
5x2 + 2x + 5 e5x
=
(e5x )
2
5x2 + 2x + 5 . e5x f (x) = x2 f (x)
1
g(x) = e
g(x) @
@
f 0 (x)
g 0 (x)
f 0 (x) · g(x) x2
f (x) · g 0 (x)
+
1
e
5x
@
@
=
5e5x
2x
2x · e
5x
+
x2
1 · ( 5)e
5x
.
5x
d x2 1 d = x2 5x dx e dx 5x
= 2x · e
+ x2
= 2x + x2
1 · ( 5)e 5x
5x
5x
1 ( 5) e
5x2 + 2x + 5 e
=
5x
1 e
.
dy dx
f 0 (x) f (x) = 2x3 + 5x2
3
f (x) = 3x5 2x4 + 6 p f (x) = 3 2x3 + 2ex p f (x) = 5 6x3 + 2ex f (x) = 5e0.06x f (x) = 2e3x
2
7 5
f (x) = 7e0.2x
+x
f (x) = 3e4x
3
5x
y=
✓
y=
✓
y=
r
2x + 3 3x 5
y=
r
y=
7 (2x + 1)3
y=
5x + 3 7x 2
◆3
y = 3x2 + ex
5
0.02x
f (x) = 5e
7
2
y = 4x3
2)
f (x) = (2x + 3)5 · (x
2)4
2
3
y0
y= y=
f (x)
f (x) =
x2
2x ex
f (x) =
x2 3x 2x + 2
f (x) =
x3 + 3x ex
p 3 p 3
3
x5
3x3 + 2
x4
2x3
x2 + 1 3ex + 2
4
x3 + 2 y= 2ex + 3
2
y=
0
x3 + 2x x2 + 1
2
y = 5x4 3x + 8 p y = 3 2x + 1 · ex p y = 2 3x + 5 · ex
f (x) = 2 x2 + 1 · ex p f (x) = x · x2 + 1 p f (x) = x · 3x2 + 5
f (x) =
2x4 + 7xe3x
y = 10x2 + 6x
2
f (x) = (x + 1) · (3x f (x) = 4x5 · e3x
y=
x2 2
3
2
· e2x+5
· e3x+1
3
⌘ d ⇣ x2 3e + (2x + 1)5 dx ⌘ d ⇣ x3 5e + (3x 2)4 dx
◆4
5x + 1 7x 3
(3x
y = 2x2 + x3 e4x+1 f (x) = 2e
3x2 5 2x + 1
4)5 2ex
4
d2 x 2 e dx2
f (x) = 2e2x
2
+3x
x=0
d2 2x3 e dx2 d2 3 x2 + 1 dx2 d2 2 x2 + 4 dx2
f (x) =
4
x=
1;
4 (x2 +1)2
x = 0;
x = 1.
3
f (x) = ex
2
f (x) = +2x
x=
f (x) = 1 + x2
4
f (x) = 3e
1;
0.4x
f (x) = ln x
x = 0;
3 (x2
+ 4)
2
x = 1.
4%
t
S(t) = 500e0.04t . f (u) = 500eu
u = g(t) = 0.04t S(t) = f (g(t)),
S = f (u)
S(t)
u = g(t).
S
f
g
u 500eu e
p
0.04t
x2 +1
u p v
eu p
f (x) =
g(x) = x3
x
f g(x) ;
v x2 + 1
x
g f (x) ;
f f (x) ;
g g(x) .
x
f
g f g(x) = f x3
p
x =
x3
x.
p
u x3
u
x
g f (x) = g
p
x =
p
x
x
g 3
p
x =
p ⇣p 2 x x
⌘ p 1 = x(x
f
1).
u
u3
p
u
x
f f (x) = f
p
x =
q p
f
x=
p 4
u
p
u
x
= x9
x = x3
3x7 + 3x5
x3
f
x
p
g g(x) = g x3
x
x
g x
3
x3
x3 x = x9
g
x 3x7 + 3x5
2x3 + x.
u u3
f (x) y = f (u)
x3
u
x
y = f g(x) y = f g(x)
g(x) u = g(x).
u f (u)
g(x) y = [g(x)]k
y = f (u) = uk
u = g(x),
u uk
g(x)
dy dx
g y = uk
dy dx
dy dx
dy dy du = · . dx du dx
u = g(x),
u g(x)
uk kuk
•
1
g 0 (x) y = eg(x)
y = f (u) = eu
u = g(x).
dy dx
g
y = eu
dy dx
dy dy du = · . dx du dx
u = g(x),
u eu eu
g(x) •
g 0 (x) y = f (g(x)) = (f
f
f (x)
g(x)
y = F (x) = f g(x)
y 0 = F 0 (x) = f 0 g(x) · g 0 (x), y = f (u) dy dy du = · . dx du dx
u = g(x)
g)(x)
dy dx
u f (u)
g(x) •
f 0 (u)
g 0 (x) dy dy du = · dx du dx
y = f g(x) dy = f 0 (u) du
y = f (u)
u = g(x)
du = g 0 (x) dx
dy dx
u g(x) dy dy du = · dx du dx
y=
p
y = ex
ex + x2 ;
dy d 1 1 = u2 = u du du 2
3
+2x
. y=
1 2
p
u
1 = p , 2 u
du d x = e + x2 = ex + 2x. dx dx dy dy du 1 ex + 2x = · = p · (ex + 2x) = p . dx du dx 2 u 2 e x + x2
u e x + x2
u1/2 1 1/2 2u
• y = eu
dy d u = e = eu , du du du d = x3 + 2x = 3x2 + 2. dx dx
(ex + 2x) u = x3 + 2x
u = ex + x2
3 dy dy du = · = eu · 3x2 + 2 = ex +2x · 3x2 + 2 dx du dx
3x2 + 2 ex
3
+2x
.
u eu
x3 + 2x •
eu
y = 3u2 dy ; du
u = 7x5 + 2ex du ; dx
3x2 + 2
2
dy . dx
dy d = 3u2 = 6u du du ⌘ 2 2 2 du d ⇣ 5 = 7x + 2ex = 35x4 + 2ex · 2x = 35x4 + 4xex . dx dx ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ 2 2 2 dy dy du = · = 6u · 35x4 + 4xex = 6 7x5 + 2ex · 35x4 + 4xex . dx du dx
y = ln x f (x) = ln x x = ey .
y = ln x y
f (x)
ef (x) = x.
d f (x) d e = x. dx dx d dx
d f (x) e = ef (x) · f 0 (x). dx ef (x) = x d f (x) e = ef (x) · f 0 (x) = x · f 0 (x). dx
x=1
f (x) = ln x
x · f 0 (x) = 1,
1 . x
f 0 (x) =
ln x
1 x
x
d 1 ln x = , dx x y = f (x) = ln x y 0 = f 0 (x) =
y=
p
dy 1 = . dx x
y = (ln x)2 .
x ln x; y
f (x) =
g(x) = ln x f (x)
p
g(x) @
x @
@
f 0 (x) 0
f (x) · g(x)
=
g 0 (x)
+
ln x
0
f (x) · g (x)
@
1 p 2 x
1 p · ln x 2 x
p 1 d p 1 x ln x = p · ln x + x · dx x 2 x ln x 1 ln x + 2 p = p +p = . 2 x x 2 x g(x) = ln x d 1 2 ln x (ln x)2 = 2(ln x) · = . dx x x
y = ln g(x)
+
1 x p
x·
1 . x
p
x
d g 0 (x) ln g(x) = , dx g(x) y = ln u
u = g(x) dy 1 du = . dx u dx
y = ln ex + x2 ;
y = ln ln x ;
y = ln 3x . g(x) = ex + x2
d ex + 2x ln ex + x2 = x . dx e + x2 g(x) = ln x d 1/x 1 ln(ln x) = = . dx ln x x ln x g(x) = 3x d 3 1 ln(3x) = = . dx 3x x
y = ln
✓
x3 + 5 2x + 1 ln
y = ln x3 + 5
d ln dx
✓
◆
;
M = ln M N
y = ln
3
⌘ x4 .
ln N
ln(2x + 1).
x3 + 5 2x + 1
◆
✓ d = ln x3 + 5 dx
ln(2x + 1)
ln(a)k = k ln a y = ln x4/3 =
⇣p
4 ln x. 3
⇣p ⌘ d d 4 4 1 4 3 ln x4 = ln x = · = . dx dx 3 3 x 3x
◆
=
3x2 x3 + 5
2 . 2x + 1
y = ax
y = loga x
a = eln a
a>0 y = eln a
x
y = ax
= e(ln a)x .
dy = (ln a)e(ln a)x = (ln a)ax . dx
d x a = (ln a)ax . dx y = loga x x = ay = e(ln a)y , (ln a)y = ln x
y=
ln x ln a
d dy 1 loga x = = . dx dx (ln a)x
f (g(x))
g(f (x))
y = (2x + 3)5
y=
f (x) = x2 g(x) = 3x + 1 f (x) = x3 g(x) = 2x + 4 f (x) =
p
f (x) =
p 3
x g(x) = x2
p
2
f (x) = 3x2 + 2x + 1 g(x) = ex
u = x2 + 2
u
y = ln u y = f (u)
dy dx
p
y = u2
u = 3x2 + 4 2u
u=
p
x
u = x4 + 1
u = g(x) y = u2
1 5x + 4
y = ln 3x2 + 1
u = 3ex + 2
y = eu
5x g(x) = 2ex g(x)
+x
y = u3 y=
f (u)
y=p
dy du du dx
x g(x) = x2 + 3
f (x) = x2
ex + 2
y = e2x
1
y = 4x2 + 5x + 1
u = ln x
4
d (ln x)3 + ln 4 dx ⌘ d ⇣p (ln x) + ln 5 dx
f 0 (x) f (x) = 2x + 3 ln x + ex f (x) = x3
4 ln x + 3ex
d2 ln(3x + 1) dx2
f (x) = 5 ln 2x2 + 3 f (x) = 4 ln 3x4 + x2 + 5 dy dx y = ln
✓
5x + 3 4x + 1
◆
d2 ln(2x dx2
y = ln
✓
2x + 1 3x 2
y = ln(4x)
y = ln(5x)
p y = ln ( 4 x)
y = ln
y = x ln x
y = x2 ln x
⇣p 5
x3
◆
d2 dx2
x2 ln x
d2 (3x ln x) dx2 f (x) = (3x + 2) ln x
⌘
f (x) = ex ln x2 + 2 x=0
y0 y = (ln x + 4ex
3x2 )6
f (x) = ln(x + 4)
y = (5x4 4 ln x + 3)5 p y = 3 x ln 6x p y = 2 3 x ln 7x
x=
x x
a
lim f (x) = L,
x = 0;
1;
x = 1.
x = 0;
L a
x!a
1;
f (x) = ln(2x + 5) x=
L
2)
x = 1.
f (x)
a
f (x) ! L as x ! a.
f (x) x = a
f
f (a) f
x = a
g 8 < x+2 g(x) = : 2x + 1
x 2)
⌘ ⌘
g(x)
⌘
3
⌘ 2 ⌘
⌘
⌘ ⌘
⌘ ⌘
s ◆
◆
◆
0 1
-
2
g(x)
H(x) =
1 (x
lim H(x)
x!1
1)2
.
y = g(x)
c
1
H
◆◆
1
1
g(x) x ! 2+
◆
3
4
- x
x ! 1 (x < 1)
x!1
H(x) x
x ! 1+ (x > 1)
H(x)
H(x)
H(x)
x ! 1+
1
lim H(x) = 1.
x!1
f (x)
f (a)
lim f (x) = f (a)
f
x!a
x=a
lim f (x) = f (a).
x!a
x!a
x = a
(x + h)2 h!0 h x lim
x2
.
h h 6= 0
(x + h)2 h
=
x2
x2 + 2xh + h2 h
=
h!0
x2
h(2x + h) = (2x + h), (h 6= 0). h
(x + h)2 h!0 h lim
f (x + h) h
⇥
x2
= lim (2x + h) = 2x. h!0
(x + h)2 f (x)
.
⇤ x2 /h
f (x) = x2
0
x
y y = f (x)
6
(x + h1 , f (x + h1 ))
@ s
(x + h2 , f (x + h2 ))
@ (x + h3 , f (x + h3 ))
@ @ s
s @
s
lim
h!0
(x, f (x))
(x, f (x))
-
(x + h, f (x + h)) x
f (x + h) h
f (x)
= f 0 (x)
x
x+h
h x (x, f (x))
f
f 0 (x) = lim
h!0
f (x + h) h
f (x)
. f 0 (x)
y = f (x) = 2x2 x f (x) = 2x2 x x=1
f (x + h) h
f (x)
=
⇥
=
4xh + 2h2 h
=
⇥
2(x + h)2
2x2 + 4xh + 2h2
x h
h
=
h
x
⇤
(x + h) h ⇥
h(4x + 2h h
2x2 1)
⇤
⇥
⇤
=
x
f 0 (1)
2x2
x
⇤
2x2 + 4xh + 2h2
x h
= 4x + 2h
1,
h 6= 0. h
f 0 (x) = lim
h!0
f (x + h) h
f 0 (x) = 4x
f (x)
= lim 4x + 2h h!0
1 = 4x
f 0 (x) = 4x
1
f 0 (1)
1),
1.
f 0 (1) = 4(1)
1=3 f (x) = 2x2
(1, f (1)) = (1, 1) 1 = 3(x
0
1 x=1
y
2x2 + x
h
y = 3x
2.
x
x=1
y
f (x) = 2x2
46
x
3
2
y = 3x
2
s
1
0
1
- x
2
y = f (x)
f 0 (x)
y0
dy dx
d f (x). dx
x
f 00 (x)
y 00
d 2y dx2
d c=0 dx
• d 3=0 dx •
d2 f (x). dx2
d k x = kxk dx
d e dx
0.2
=0
1
d 3 d 1 d x = 3x2 = x 3 dx dx x dx d p d 1/3 1 1 3 x= x = x 2/3 = p 3 dx dx 3 3 x2 •
d x e = ex dx
3
=
3x
4
=
3 x4
d 1 ln x = dx x
•
d [k · f (x)] = k · f 0 (x) dx
• d 2ex dx
3 ln x + x2
5 = 2ex
d [f (x) ± g(x)] = f 0 (x) ± g 0 (x) dx
3 + 2x x
d [f (x) · g(x)] = f 0 (x) · g(x) + f (x) · g 0 (x) dx
•
d x4 ex = 4x3 ex + x4 ex = (4 + x)x3 ex dx d f (x) f 0 (x) · g(x) f (x) · g 0 (x) = dx g(x) [g(x)]2
•
d x2 2xex x2 ex (2 x)xex (2 x)x = = = . x x 2 x 2 dx e (e ) (e ) ex d f g(x) = f 0 g(x) · g 0 (x) dx
•
y = 3eu + u4
dy dy du = · dx du dx
u = 5x3 + 4x2
dy dx
du = 15x2 + 8x dx dy dy du = · = 3eu + 4u3 dx du dx
15x2 + 8x
h i 3 2 3 = 3e5x +4x + 4 5x3 + 4x2 15x2 + 8x .
•
d [g(x)]k = k[g(x)]k dx
1
· g 0 (x)
d 9 9 (2 + 3ex )10 = 10 (2 + 3ex ) (3ex ) = 30ex (2 + 3ex ) dx d g(x) e = g 0 (x) · eg(x) dx
• 2 d 3x2 e = 6xe3x dx
•
d g 0 (x) ln[g(x)] = dx g(x)
dy = 3eu + 4u3 du
d 15x4 + 2ex ln 3x5 + 2ex = dx 3x5 + 2ex
t s(t) = 144
16t2 0 t 3. v(t)
v(2)
v(t) = s0 (t) = v(2) =
a(t)
a(2)
a(t) = v 0 (t) = s00 (t) =
32t, 32(2) =
64
a(2) =
32
32. 2
$1,000 5%
k = 0.05 P (t) = 1,000e0.05t .
P (3) = 1,000e0.05(3) = 1,000e0.15 ⇡ $1,161.83 P (t) P 0 (t) = 1,000(0.05)e0.05t = 50e0.05t .
P 0 (3) = 50e0.05(3) = 50e0.15 ⇡ $58.09/
.
H
y
f
f
6
lim f (x)
4
y = H(x) b
3 2 1 4
3
2
10 1
1
2
b
2 3
lim H(x)
lim H(x)
4 3
4
3
2
lim H(x)
lim H(x)
lim H(x)
lim H(x)
x!1
1
10 b 1
1)2
lim s(x)
4
x
y = f (x)
4
8 < x 2 f (x) = : x2
lim g(x)
x! 2
1 x+1 lim f (x)
1
lim f (x)
lim s(x)
x!1
x!1
f
lim f (x)
x!1+
x=1
f (x) =
x! 1
4 2
4 h!0 x(x + h) lim
3
x1 x>1
f (1)
1
x!1
x2 x!1 x
2
x!3
x2 + x 6 g(x) = x 2
lim
1
3
lim H(x)
x!1+
r
2
2
x!2
x!1
lim f (x)
x! 2
x= 2 y 6
x!2
x!2+
(x
f
lim H(x)
x! 2
lim f (x)
f ( 2)
lim H(x)
lim H(x)
1
x! 2+
x! 2
x! 2+
x! 2
s(x) =
4
x= lim f (x)
x
r
4
lim f (x)
x! 1
f ( 1) f
3
lim f (x)
x! 1+
x! 1
lim f (x)
x!1
x2 + x 2 x!1 x 1 x+1 lim x! 1 x2 1 lim
(2 + h)3 h!0 h lim
5(x + h)2 h!0 h lim
8 5x2
lim f (x)
x!1
lim
2 x+h
2 x
f (x) =
h
h!0
x2 + x + 4 x 2
f (x) = 5ex
xe
f (x) = 5 ln x2 + 1
t s(t) = 256 s(3)
16t2 0 t 4.
f (x) = 2x3 + 5x2 f (x) = 5e
s(1)
7
7
2)2
· (3x
x3 + 3x ex r 5x + 1 f (x) = 7x 3
s(1) 1
f (x) =
v(t)
2
f (x) = 10x2 + 6x 3 p f (x) = 3 2x + 1 · ln x
v(2)
d4 7 x dx4 p f (x) = x3
$2,000
t A(t) = 2,000(1.04)4t A(3)
A(0)
A(3) 3
A(0) 0
0 t 3.
dy du du dx
f (x) = 2x3
f 0 (4)
f 00 (4)
dy dx p u= x
t
f (x) = 2x3 3x2 +x 1 1;
· e2x+5
y = u3 + e u p y= u u = x2 + ln x
s(t) =
x=
3
x2 2
f (x) = x2 + 1 s(3) 3
0.04x
7e
x = 0;
v(t)
16t2 + 80t 0 t 5. a(t)
x = 1.
7x2 + 4x + 3
$2,000 f (x) = 5x4 3x 7ex 2 ln x + 8 ✓ ◆ p 8 f (x) = 3 x + ex + x2 + 4 x p 5x f (x) = +3 x 7
5 p 3 x2
t A(t) = 2,000(1.00375)12t . A0 (1)
A0 (5)
g lim g(x)
x! 1
g( 1) lim g(x)
x!3
g(3)
lim g(x)
x!0
y
6
4 3
4
3
r
2 1
10 b 1
2
2
1
2
3
4
x
y = g(x)
3 4
lim
x! 5
x+5 x2 25
6 x!0 x lim
lim
x!0
x+6 x2 + 3x 18 f (x) = 2x2 + 4x
h!0
y = x + (4/x)
y = x2 + 6x
dy dx
y=
p 2 + 2 4 x + 6ex 2 x
1 6 x 6
f 0 (x)
f (x) = 3x3
f 0 (x)
f (x) = (4x
f 0 (x)
f (x) =
y = x6
2x4 + 3x2
t
t1 = 3
s(3)
t2 = 5
t=3
4
7)1/3 ex
x2 + 4 3 x
s(t) = 2t3 + 2t s(5)
1
d3 y dx3
s
2
3 ln x
s(t)
dy du y=
3 + ln u u2
du dx
u = 2x + 1
$4,000 t A(t) = 4,000e0.04t .
A0 (3) A0 (3)
dy dx
A
t $60,000
$2,400 A
t=8 A(t)
$100,000
v
t
t
v
6
v(t)
15 10 5
0 5 10 15
1
2
3
4
t
y
r
6 f (b)
A A
r
f (c2 )
f (a) ⇣ f (c3 ) P f (c1 )
0
r
r` ``` `
a c1
# #
c2
#
r
c3
r
c4
y = f (x)
b
- x
y y = f (x)
x f 0 (150) = 3 f (150) = 160
f (151)
f (155)
f 0 (150) = 3
f (150) = 160 163 f (151) ⇡ f (150) + f 0 (150)(1) = 160 + 3(1) = 163. 3(5) = 15
175
f (155) ⇡ f (150) + f 0 (150)(5) = 160 + 3(5) = 175. 150
155
f (151)
x x = x2
x1 ,
x1 = x
x2 = x + h,
x x = (x + h)
x = h.
y = f (x)
x
x
y y = y2
y1 = f (x2 )
f (x1 ),
x1 = x
x2 = x + h,
y = f (x + h)
y 6 f (x)
r
r
6 y = f (x + h)
f (x)
? x
h-
- x x1 = x
x2 = x + h =x+ x
x
y
y = f (x) f (x + h) h
f (x)
, y = f (x)
dy f (x + h) = f 0 (x) = lim h!0 dx h
f (x + h) h
f (x)
=
dy = f 0 (x) = lim x!0 dx
f (x)
.
y , x
y . x
x dy ⇡ dx
y , x
f 0 (x) ⇡
y . x x
y ⇡ f 0 (x) x.
f (x).
y = f (x + f (x +
x)
x)
f (x)
f (x) ⇡ f 0 (x) x,
f (x +
x) ⇡ f (x) + f 0 (x) x.
y 6 f (x)
r ⌘⌘ 6 r⌘ ⌘ ⌘ y 6 f 0 (x)
m = f 0 (x)
⌘
⌘ ⌘ r ? ? ⌘ ⌘ x -
x
-x x
x
y = f (x) y f 0 (x) ⇡ x
x
y ⇡ f 0 (x) x f (x+ x) ⇡ f (x)+f 0 (x) x
y = f (x) = 3x2 x
y
x1 = 2
x2 = 2.5
x
y
x1 = 2
x2 = 2 + h
y
x=2
x = 0.1
2x
f 0 (2) x f (2.1)
f (2.1)
x = x2
x1 = 2.5
y = f (x2 ) = [3(2.5)2
x = x2
2 = 0.5,
f (x1 ) = f (2.5) [3(2)2
2(2.5)]
x1 = (2 + h)
f (2) 2(2)] = 13.75
8 = 5.75.
2 = h,
⇥ ⇤ ⇥ y = f (x2 ) f (x1 ) = f (2 + h) f (2) = 3(2 + h)2 2(2 + h) 3(2)2 ⇥ ⇤ ⇥ ⇤ = 3(4 + 4h + h2 ) 4 2h 8 = 8 + 10h + 3h2 8 = 10h + 3h2 . y = f (x + ⇥ = 3(2.1)2
x)
f (x) = f (2 + 0.1)
2(2.1)
f 0 (x) = 6x
2
⇤
⇥
3(2)2
f (2) = f (2.1)
⇤ 2(2) = 9.03
f 0 (2) = 6(2) y = 1.03
2(2)
⇤
f (2)
8 = 1.03. f 0 (2) x = 10(0.1) = 1
2 = 10
x=2
x = 0.1
f 0 (2) x = 1 f (2.1) = f (2 + 0.1) ⇡ f (2) + f 0 (2)(0.1) = 8 + 1 = 9. p f (x) =
p
f (2.1)
9.03
18
x
18
16
x = 16
x = 18
16 = 2
f (18) = f (16 + 2) ⇡ f (16) + f 0 (16)(2). p f (16) = 16 = 4 1 f 0 (x) = p , 2 x 1 1 f 0 (16) = p = 8 2 16 1 f (18) ⇡ f (16) + f 0 (16)(2) = 4 + (2) = 4 + 0.25 = 4.25. 8 p 18 = 4.24264 . . . 6 V (r) =
✓
4⇡ 3
◆
0.005 r
3
6 36 + 0.005 = 36.005 V (36.005)
3
= 36 V =
V (36)
V ⇡ V 0 (36) r = V 0 (36)(0.005). ✓ ◆ 4⇡ 4⇡ V (r) = r3 V 0 (r) = · 3r2 = 4⇡r2 3 3 V ⇡ V 0 (36)(0.005) = 5, 184⇡(0.005) ⇡ 81.4
3
81.4 ⇥ 20 = 1, 628
7
dy ⇡ dx
V 0 (36) = 4⇡(36)2 = 5, 184⇡
3
1
y . x y
x
dy dy du = · . dx du dx dy
y = f (x) dx =
3
231
dx
x x. y
dy = f 0 (x)dx.
dy
dx
dx
dy
y 6 f (x)
r ⌘⌘ 6 r⌘ ⌘ ⌘ y 6
m = f 0 (x)
⌘
⌘
r⌘ ⌘ dx
⌘
dy
? ?
x-
- x x
dx
dy
dy
x
y
y
y ⇡ dy. dy
dx
dy
y = ln x2 + 4 dy dy
x=1
dx = 0.02 dy dx
dy 2x = 2 . dx x +4 dy =
dy 2x · dx = 2 dx. dx x +4
x=1 dx = 0.02 2(1) dy = (0.02) = (0.4)(0.02) = 0.008. (1)2 + 4
dx
f (8) = 300 C = f (T ) C
T
f 0 (8) = 3,800
f (36) = 4.50
f (8.4) 0
f (36) = 0.25 q = f (p)
p q
38
f (120) = 500 33
f 0 (120) = 10
C = f (w)
w
f (124)
C f (100) = 30
R = f (a)
a
f 0 (100) = 0.15
R f (10) =
50 f 0 (10) =
105
2
97
f (10.8)
$1,000 B = f (t)
t
B
H = f (t) f (12) =
1,095
t
H f (25) = 95
f 0 (12) = 8
f 0 (25) = 5
f (12.5)
C = f (r) C
28
r
32
0.5x
y = f (x) = 2e x = 0.01
x = 0
y = f (x) = e0.4x x = 0 H = f (t)
t
x = 0.01
y = f (x) = ln(x + 1) x = 0 x = 0.1
H f (3) =
y = f (x) = ln(3x + 1) x = 0 x = 0.1
120 f 0 (3) =
y = f (x) = x3 x = 1
10
x = 0.04
y = f (x) = x x3 x = 1
x = 0.02
3.5 ln 1.02 e0.05
4
y = f (x) = x2 + 2x x
y
x1 = 2
x2 = 2.5
x
y
x1 = 2
x2 = 2 + h
y
x=2 f 0 (2) x
p
y = f (x) =
4x2 + x
x
y
x1 = 2
x2 = 2.5
x
y
x1 = 2
x2 = 2 + h
x=2 f 0 (2) x
15.6
29
p 3
5
p
p 3
25.5
2 p 50 p 4
y y = f (x) = 3x + 1 x = 3 y = f (x) = x = 0.2
f 0 (x) x x = 0.5
4x + 3 x = 5
y = f (x) = 1/x x = 1 2/x x = 2
y = f (x) = 1/x2 x = 1 y = f (x) = 3/x
p
28
4.5
p 3
26.2
2 p 17
x = 0.01
f (2.01) f (2.01)
y = f (x) =
8.8
p 3
x = 0.1
f (2.1) f (2.1)
y
p
2
x=1
x = 0.3 x = 0.2 x = 0.04 x = 0.03
1 15.5
1 p 3 7.5
y = (2x + 1)3 dx = 0.01
dy
x=2
y = (3x + 1)2 dx = 0.02
dy
x=1
y = 3x3 + 2x + 1 y x=1 x = dx = 0.01 y = (x2 3)3 x = dx = 0.01 f (x) = x3 2x f (2.1) f (x) = xex 5 f (5.1)
y 2
dy
dy x=2
C(x) C(21)
R(x)
C(20)
R(21)
x R(20)
C(21)
C(20) =
C ⇡ C 0 (20)(21
20) = C 0 (20)
R(21)
R(20) =
R ⇡ R0 (20)(21
20) = R0 (20).
C 0 (20)
C(x) R(x) x
R0 (20)
C(21)
C(20)
R(21)
R(20)
P (x) M C(x) = C 0 (x) M R(x) = R0 (x)
x 0
M P (x) = P (x) (x + 1) M C(x) ⇡ C(x + 1)
M P (x) = M R(x)
C(x) = 0.01x3
C(x), M R(x) ⇡ R(x + 1)
M C(x),
P 0 (x) = R0 (x)
R(x),
M P (x) ⇡ P (x + 1)
P (x).
C 0 (x).
0.5x2 + 12x + 1,500
x
x
R(x) = 120x0.98 . x 100 100
x C(x) = 120x0.98
P (x) = R(x)
0.01x3 + 0.5x2
12x
1,500.
x = 100 C(100) = 0.01(100)3
0.5(100)2 + 12(100) + 1,500 = 7,700,
R(100) = 120(100)0.98 = 10,944.13, P (100) = R(100)
C(100) = 10,944.13
7,700 = 3,244.13.
100
C(x) 0
C (x) = 0.03x
2
R(x)
x + 12,
R0 (x) = 120(0.98)x
0.02
= 117.6x
0.02
.
x = 100 0
C (100) = 0.03(100)2 R0 (100) = 117.6(100) P 0 (100) = R0 (100)
(100) + 12 = 300 0.02
100 + 12 = 212,
= 107.25.
C 0 (100) = 107.25
212 =
104.75.
100
C(100) R(101) R(100) 7,700 R(100) = 10,944.13 C(101) = 0.01(101)3
0.5(101)2 + 12(101) + 1,500 = 7,914.51,
R(101) = 120(101)0.98 = 11,051.40, C(101)
C(101) C(100) =
P (101) P (100) P (100) = 3,244.13
C(100) = 7,914.51
P (101) = R(101)
C(101) = 3,136.87,
7,700 = 214.51,
R(101) R(100) = 11,051.40 10,944.13 = 107.24,
|212 214.51| ⇡ 0.012 = 1.2%. 214.51
P (101) P (100) = 107.24 214.51 =
107.27.
C(x) R(x)
P (x)
$8,000 C 0 (200) = 35
C(200) = 8,000
C(204) ⇡ C(200) + C 0 (200)(204
200) = 8,000 + 35(4) = 8,140.
C(x) = 1,000 + 15x + 0.1x2 + 200 ln(x + 1) x x R(x) = 80x. x 50 50 52
x P (x) = R(x)
C(x) = 80x
1,000
15x
0.1x2
200 ln(x + 1).
x = 50 C(50) = 1,000 + 15(50) + 0.1(50)2 + 200 ln(50 + 1) = 2,786.37, R(50) = 80(50) = 4,000,
P (50) = R(50) 50
C(50) = 4,000
2,786.37 = 1,213.63.
C(x) C 0 (x) = 15 + 0.2x +
200 , x+1
R(x)
R0 (x) = 80.
x = 50 C 0 (50) = 15 + 0.2(50) +
200 ⇡ 28.92, 51
R0 (50) = 80.
P 0 (50) = R0 (50)
C 0 (50) = 80
28.92 = 51.08.
50
C(52) ⇡ C(50) + C 0 (50)(52
50) = 2,786.37 + 28.92(2) = 2,844.21. R(x)
R(52) = R(50) + R0 (50)(52
50) = 4,000 + 80(2) = 4,160. $4,160
$5,600 $2,844.21 = $1,315.79
x
C(x) R(x) x
P (x)
x AP (x) =
AC (x) =
P (x) x
C(x) = 0.01x3
0.5x2 + 12x + 1,500
R(x) = 120x0.98 . x 100
C(x) R(x) AR (x) = x x
100
AC (x) =
C(x) 0.01x3 = x
AR (x) =
R(x) 120x0.98 = = 120x x x
AP (x) =
P (x) = AR (x) x
x = 100
0.5x2 + 12x + 1,500 = 0.01x2 x 0.02
0.5x + 12 +
1,500 , x
,
AC (x) = 120x
0.02
0.01x2 + 0.5x
12
AC (100) = 77 AR (100) = 109.44
1,500 . x
AP (100) = 32.44
AP (x) A0P (x)
d = dx
✓
0.02
120x
= 120 · ( 0.02)x =
2.4x
1.02
2
0.01x + 0.5x
1.02
12
0.01 · (2)x1 + 0.5
0.02x +
◆
1,500 x2
1,500 . x2 A0P (100) =
x = 100
1,500 x
1.87
100
100 100 C(100)/100
x
100
x y 6
y 6
R(x)
C(x) P (x)
0
-x
x⇤
0
-x
x⇤
MC
MR
MP
C(x)
x⇤
R(x)
M C(x⇤ ) = M R(x⇤ ) M C(x) < M R(x)
x < x⇤ ,
x > x⇤ ;
M C(x) > M R(x)
M P (x⇤ ) = 0 x < x⇤ ,
M P (x) > 0
C(x) = 0.01x3
M P (x) < 0
x > x⇤ .
0.5x2 + 12x + 1,500
x x R(x) = 100x.
M C(x) = C 0 (x) = 0.03x2
M C(x) = M R(x)
x + 12
M R(x) = R0 (x) = 100.
x 0.03x2
x=
0.03x2
x + 12 = 100
1±
p 1
x
88 = 0.
p 4(0.03)( 88) 1 ± 11.56 1 ± 3.4 = = . 2(0.03) 0.06 0.06 x⇤ ⇡ 73
M P (x) = P 0 (x) = R0 (x) M P (72) = 4.48
C 0 (x) =
M P (74) =
M P (x) > 0
0.03x2 + x + 88 . 2.28
x < 73
M P (x) < 0
x > 73. x = 73
x⇤ = 73 P (73) = R(73)
C(73) = 3,698.33
C(x) = 1,000 + 15x + 0.1x2 + 200 ln(x + 1) x x R(x) = 80x
M C(x) = C 0 (x) = 15 + 0.2x +
200 x+1
M R(x) = R0 (x) = 80.
M C(x) = M R(x) x 15 + 0.2x +
200 = 80 x+1
0.2x +
200 x+1
65 = 0.
x+1 0.2x(x + 1) + 200
x=
64.8 ±
65(x + 1) = 0
0.2x2
64.8x + 135 = 0.
p p 64.82 4(0.2)(135) 64.8 ± 4091 64.98 ± 63.96 = = . 2(0.2) 0.4 0.4
x1 ⇡ 2
x1 ⇡ 322
M C(1) = 115.2 > M R(1) = 80. M C(x) < M R(x)
x < x⇤
M C(x) < M R(x)
x < 322,
M C(x) > M R(x)
x > 322. x = 322
x⇤ = 322 P (322) = R(322)
C(322) = 8,406.07 p x
x = D(p) x
p x
C(x) = 4,000 + 33x + 0.25x2 x x = D(p) = 120
0.2p
p p
x
R(x)
x = 120
0.2p,
p p=
p = 600
120 x = 600 0.2
5x. R = p·x
5x
R(x) = p · x = (600
5x) · x = 600x
M C(x) = C 0 (x) = 33 + 0.5x
5x2 .
M R(x) = R0 (x) = 600
10x.
M C(x) = M R(x) 33 + 0.5x = 600
M P (x) = 567
10x
x
10.5x = 567
10.5x .
x = 54.
M P (53) = 10.5
M P (55) =
10.5.
x = 54
x C(x) = 1,500 + 12x + 0.1x2 . p
x
p = mx + b. m= 60 =
2/4 =
0.5
x = 120
p = 60
0.5(120) + b, b = 120
p=
0.5x + 120. R(x) p = 0.5x + 120
R=p·x R(x) = p · x = ( 0.5x + 120) · x =
0.5x2 + 120x.
M C(x) = C 0 (x) = 12 + 0.2x
M R(x) = R0 (x) =
M C(x) = M R(x) 12 + 0.2x =
M P (x) = 108
x + 120
1.2x .
x + 120.
x
1.2x = 108
x = 90.
M P (89) = 1.2
M P (91) =
1.2. x = 90
x = 90 p=
0.5(90) + 120 = 75.
x x C(x) = 1,200 + 53x4/5 M C(x) = C 0 (x)
100
M C(77) 100
C(77)
M C(77)
102 x
C(x) = 0.01x3 0 x 80
0.6x2 + 13x + 200 C(x) = 4,000 + 36x + 0.5x2
M C(x) = C 0 (x)
x
M C(50)
x
R(x) = 500x
4x2 x
C(50)
M C(50) 40 $1,400 40
41
$2,000
C(x) = 1,500 + 20x + 0.2x2 + 100 ln(x + 5) x x C(x) = 3,000 + 125x + 0.05x2
300e
R(x) = 90x
0.02x
x
x x 60 R(x) = 400x
60
x R(x) = 80x + 40xe
0.04x
.
63
C(x) = 1,200 + 18x + 0.1x2 x x
R(x) = 120x x
70 70
x C(x) = 0.02x2 + 15x + 4,500 x
74 R(x) = 200x. x C(x) = 1,200 + 53x4/5 . x p C(x) = 285 + 0.55 x, x
x C(x) = 800 + 22 ln(x + 5).
x
R(x) = 5.5x0.6 .
C(x) = 25,000 + 125x + 0.1x2 p 3 R(x) = 120 x2 .
x x R(x) = 350x
1,000
1,000
x C(x) = 2x2 +15x+1,500 x = D(p) = 420
0.5p
p p x R(x)
C(x) = 1,500 + 20x + 0.2x2 x x R(x) = 90x x
C(x) = 285 + 0.5x + 0.01x2
x = D(p) = 2,000 C(x) = 1,000 + 18x + 0.1x2
400p
p p
x
x x
R(x)
R(x) = 120x
x 1,500
C(x) = 3,000 + 5x + 0.01x2 p
x
1,500 R(x)
C(x) = 1,000 + 10x + 0.1x2 p
x R(x)
x
I I (a, b) (b, c) y
6
y = f (x)
R
✓
0
a
b
>
c
- x
d
f
(c, d)
f x 1 < x2 ,
(a, b)
x1
x2
(a, b)
(a, b)
x1
x2
(a, b)
f (x1 ) < f (x2 ).
f x 1 < x2 ,
f (x1 ) > f (x2 ). (a, b)
(a, b)
y y
y
6
f
r
0
6 r
f 0 (x) = 0
@r f (x) < 0 @ @ @ 0
f 0 (x) > 0
- x
x
0
f
f0
- x
x
f
f x
(a, b)
f
(a, b)
f 0 (x) < 0
x
(a, b)
f
(a, b)
f 0 (x) = 0
x
(a, b)
f
f0
0
f
(a, b)
0
f
(a, b)
3x + 1 f0 0
f
- x
x
f0
f 0 (x) > 0
f (x) = x3 f
6 @ @
f
f0
f !x""x3 !3x#1 y
20 10
!3
!2
1
!1
2
3
x
!10
!20
f (x) = x3
3x + 1
( 1, 1)
f
(1, 1) f0
( 1, 1)
0
(1, 1) f
( 1, 1) f0 0 f (x) = x3 f0 > 0
f0 < 0
f
( 1, 1)
f 0 (x) = 3x2
3x + 1
3
f
f '!x""3x2 !3 y
20 10
!3
!2
1
!1
2
3
x
!10
!20
f 0 (x) = 3x2
3 x
f0 > 0
( 1, 1)
(1, 1) f
( 1, 1) x
( 1, 1) f
( 1, 1)
(1, 1) f0 < 0
y
r
6 f (b)
A A
r
f (c2 )
f (a) f (c3 ) ⇣ P f (c1 )
0
r
r` ``` `
a c1
# #
#
c2
r
r
c3
y = f (x)
c4
f c1 c2
c3
f (c2 ) (c2 , f (c2 ))
f c2 = 5
- x
[a, b]
f
f (c2 )
f (5) = 20
f
b
20 f
x = 5 x=5
f
(5, 20) f (c1 )
f (c3 )
(d, e)
f
c
(d, e) f (c)
f (c)
f (x)
x
f (c)
f (c) f (x)
x
(b, f (b)) (c1 , f (c1 ))
f
(d, e) (a, c4 )
d c2
c4
x = c2
f (5) = 20
x = c2 a
c2
e
(d, e)
f (c) c
@
@ @
⇥ @
⇥
⇥
(d, e)
⇥⇥ B
B
B
BB
y 6
y 6
f SS
SS
a ! aa !! a!
◆◆
◆◆
◆◆
◆◆
SS
-x
0
f0
f0
(a, b) f0
x
f
(a, b)
f
f (a, b)
f0
(a, b)
(a, b) f0
(a, b)
p, f (p)
f 0 (c)
c, f (c) c, f (c)
SS f
-x
0
!a !! aa ! a
f 0 (c)
f 00 00
f (x) > 0
(a, b)
f
f 00 (x) < 0
(a, b)
f
f
(a, b)
(a, b) (a, b)
f 00
f 00
y 6
f
y 6
f
r
f 00 < 0
r
f 00 (p) = 0
f 00 < 0
f 00 (p)
f 00 > 0
f 00 > 0
- x
0
f (x) = x4
6x2
f 0 (x) = 4x3 p 12 = 0
12(x
1)(x + 1) = 0
x
1=0
x=1
x+1=0 x=
1. f 00
f 00 ( 2) = 12( 2)2 f 00 (0) =
f
12 = 48
12 = 36 > 0,
12 < 0,
f 00 (2) = 12(2)2
12 = 48
12 = 36 > 0,
00
f (1) = (1)4
6(1)2 = 1
f ( 1) = ( 1)4
p
g(x) = xe2x
f 00 12x2
- x
0
p
6=
6( 1)2 = 1
5, 6=
5.
( 1, 5)
(1, 5)
12x
f 00 (x) = 12x2 f 00 (p) = 0
12
g 0 (x) = e2x + 2xe2x = (1 + 2x)e2x .
g 00 (x) = 2e2x + 2(1 + 2x)e2x = 4(1 + x)e2x . g 00
g 00 (p) = 0 1+x = 0 x= 1
p g 00
x=
g 00 ( 3) = 4( 2)e
6
1
< 0,
g 00 (0) = 4(1)e0 > 0. g 00 ( 1, e
p= 2
1
g( 1) =
e
g
) f (x) = (x
f 0 (x) =
2 (x 3
f 00 (x) =
2 1 (x 3 3
f 00 x=5
f f 00 x=5
2
5)
1/3
5)
5)2/3
, 4/3
=
9(x
2 . 5)4/3
x=5
f 00 (5)
x=5 f 00 (x) = 0
f 00 (4) =
2 2 = < 0, 9 9( 1)4/3
f 00 (6) =
2 2 = < 0. 9 9(1)4/3
f 00
x=5
f (x) = x3
f 00
f
3x + 1
x f f f
f f
f 0 (x) = 3x2
3. f 0 (x) = 0
3x2
x2
3=0
x=1
x=
1=0
(x
1)(x + 1) = 0.
1
1
1 ( 1, 1) ( 1, 1) f0
(1, 1) f 0 ( 2) = 3( 2)2 f0 ( 1, 1)
3 = 9 > 0; f 0 (0) =
3 < 0; f 0 (2) = 3(2)2
( 1, 1)
(1, 1)
f0
(1, 1)
( 1, 1)
f 1+3+1=3 f c2 = 1 f (1) = (1)3
f
f c1 =
3(1) + 1 = 1
3 = 9 > 0.
( 1, 3) 3+1= 1 f
( 1, 1)
1
f ( 1) = ( 1)
3
3( 1) + 1 =
f (1, 1)
f 00 (x) = 6x. f 00
f 00 (p) = 0
p
p=0
f f 00 ( 1) =
6 < 0,
00
0
f 00 (1) = 6 > 0,
f 00 f (0) = (0)3
3(0) + 1 = 1.
f
(0, 1)
f 00
(0, 1)
f 00 f (x) = x3
( 1, 0)
(0, 1)
f
( 1, 0)
f
3x + 2
y 4 3
6 y = f (x)
2 1 4
3
2
10 1 2 3
1
2
3
4
x
y 4
y
6
4
3
3
2
2
y = f (x)
1 4
3
2
10 1
1
2
3
1
4 x
4
3
2
2
3
3
4
y
6
4
3
3
2
2
1 4
3
2
10 1
1
2
3
4 x
4
3
2
y = g(x)
4
2
3
4
x
4
x
4
x
y = f (x)
6 y = h(x)
10 1
1
2
3
3
y
6
4
3
3
2
2
1 3
2
2
3
y
1
1
2
4
10 1
2
y
6
6 y = h(x)
1
10 1
1
2
3
4 x
4
y = g(x)
2
3
2
10 1
1
2
3
2
3
3
x y 4
6
h
3
h
2 1 4
3
2
10 1 2
1
2
3
4
x
h h
y = f (x)
3
h h h h
y 4
6
3 2
y = g(x)
1
y 4
4
6
3
2
10 1
2
3
4
x
4
x
4
x
2
y = f (x)
3
1
3
2 1 4
3
2
10 1
1
2
3
4
x
2
y
3
4
y 4
2 1
2
y = s(x)
4
1 3
2
y = f (x)
3
6
3
4
6
10 1
1
2
3
4
3
2
x
10 1
1
2
3
2 3
2 3
y 4
6
3 2
y = g(x)
1 4
y 4 3
3
2
6
10 1
1
2
3
2
y = f (x)
3
2 1 4
3
2
10 1 2 3
1
2
3
4
x f (x) = x2
5x + 6
f (x) = 2x2 + 3x f (x) = x4
5
2x2
f (x) = 3x4
4x3
h(x) = 13 x3
3x2 + 5x + 6
h(x) =
x3
y = x2 e
x
3x2 + 15x + 3
y = xe3x
y
6
4 3
y = f (x)
2 1
f 4
3
2
1
f
0
1
2
3
4
x
1 2
f0
3
f
f
x f f
f f
f
f f f y
6
f
4 3
y = f (x)
2
f
1
4
3
2
1
0 1 2
1
2
3
4
x
f (x) = x3
12x
f (x) = x3 + 3x2 f (x) = x4
2x3
f (x) = x4
2x2
3
4 45x + 1
y
r
6
r
f (c2 )
f (c3 ) f (c1 )
0
r
r`
# # #
``` `
a c1
c2
r
r
c3
c4
y = f (x)
b
- x
(c, f (c)) f 0 (c) = 0 c
(a, b) (a, b)
f
c
c
f (0 c)
f 0 (c) = 0 (c, f (c))
f
c1 c2 c3
y
y
6
6 r
f
c
6 r
0
f (c) = 0
0
y
r
0
c4
f (c)
f
- x 0
c
f
- x 0
c f 0 (c)
- x
f
f (c, f (c))
c
f x=c
f 0 (c) = 0
(a, b) f f 0 (c)
f 0 (x) = 4x3
16x
c
f (x) = x4
8x2
g(x) = xe2x
0
f0
f (x) = 0 4x3 x3
16x = 0 4x = 0
x(x
2)(x + 2) = 0
x=0
x
2=0
x=0
x=2
x+2=0 x=
2. 2 0
2
g 0 (x) = e2x + 2xe2x = (1 + 2x)e2x . g0 1 + 2x = 0 c=
1 2
5)2/3
f (x) = (x
f 0 (x) =
2 (x 3
5)
1/3
g(x) =
=
f0 c=5 f 0 (x) = 0
f
g 0 (x) =
2 · (x2 + 1)
g0 1
x2 = 0
(x2 + 1)
2 3(x
5)1/3
2x +1
. f 0 (5) f
f
2
=
2 1
g
x2
x=5
2x · 2x
g 0 (x) = 0 x = 12
x2
(x2 + 1)
2
x=5 f0 c=5
. c
g 0 (c) = 0
(1 1
x)(1 + x) = 0 x=0
x=1
1+x=0 x=
1.
g
1
1 f (x) = x +
f 0 (x) = 1
(1, 6)
9 . x2
f0 1
9 x
x=0
0
f 0 (x) = 0
f
9 =0 x2
9 =1 x2 9 = x2 x=3
x=
3.
3
(1, 6)
f
c=3
(1, 6)
f (c)
c
c
f (c)
x = c1
x = c3 x = c2 x = c4
f f f f f0 x=c f
c x
f0
0
f f0
x c
x=c f
f0 x=c x
c
f
x = c
f0
A
f a f0
c
A AU
f
U A b
a
x -
c
b
f0
+
+
f0
f0
f
A AU
f a
f0
c
+
b
a
x -
A AU c
b
x -
f0
+ f0
f0
f
(a, b)
c f 0 (x) > 0
(a, c)
f 0 (x) < 0
(c, b)
f (c)
f 0 (x) < 0
(a, c)
f 0 (x) > 0
(c, b)
f (c)
f 0 (x)
x -
(a, c)
(c, b)
f
(a, b)
f f0
f (x) = x3
f 0 (x) = 3x2
3.
3x + 1
0
f0
f0
f 0 (x) = 0
3x2
3=0
x2
1=0
(x
1)(x + 1) = 0
x=1
x=
1. 1
1 ( 1, 1) ( 1, 1) f0
f 0 ( 2) = 3( 2)2
3 = 9 > 0; f 0 (0) =
f0
3 < 0; f 0 (2) = 3(2)2
( 1, 1)
A
f
f (1) = (1)3
+
( 1, 1)
-x +
3( 1) + 1 =
3(1) + 2 = 1
f (x)
(1, 1)
1
f0
f ( 1) = ( 1)3
3 = 9 > 0.
AU
1 f0
(1, 1)
f (x) = x3
3x + 1
1 + 3 + 1 = 3,
3+1=
1.
( 1, 3)
(1, 1)
x = c f 0 (c) = 0 x = c2
x=1
f 0 (c2 ) = 0 f x = c3 f 0 (c3 ) = 0 f (x) = x3 3x + 1 x= 1
x = c2 x = c3 x= x=c
1 x=1
f 0 (c) = 0
x=c x=c f
00
f 00
f 00
f f 00 (c)
f 0 (c) = 0 f
x=c f 00 (c)
f c
f 0 (c) = 0 00 f (c) > 0
f (c)
f 00 (c) < 0
f (c)
f
x=c
x=c
(a, b)
f 00 (c) = 0
f (c)
f 00 (c)
c f 00 (c) > 0
f (c)
c f 00 (c) < 0 c f 00 (c) = 0
f (c)
f (c)
f 00 (c)
f 0 (c)
f 00 (c) = f 0 (c) = 0 x=c f (x) = x4 + 1 f 0 (0) = f 00 (0) = 0 f x=0 f (x) = x4 + 1 f 0 (0) = f 00 (0) = 0 f x=0 f (x) = x3 + 1 f 0 (0) = f 00 (0) = 0 f x=0
f !x""x4 #1
f !x""!x4 #1
y
3
5
4
2
4 3
1
2 1
!2 1
!1
y
5 3
!2
f !x""x3 #1
y
2
1
!1 !2
!2
!3
1 !2
f 00 (0) = 0
3x + 1
f 00 (x) = 6x.
3
1
1
f 00 (1) = 6(1) = 6 > 0.
6 0. (0, 2)
A
U A 2
f0
( 2, 0) ( 1, 2)
f A
+
0
(0, 2)
AU
0
2
0
0
f0
f (x) = x4
-x +
8x3 + 6 2
f 00 ( 2) = 12( 2)2 f 00 (0) =
16 = 32 > 0
16 < 0
f 00 (2) = 12(2)2
f (x) =
10
f (0) = 6 16 = 32 > 0
4 + 8x3
f 0 (x) = 24x2
12x3 = 0
f (2) =
10
3x4 .
12x3 ;
f 0 (x) = 24x2 24x2
f ( 2) =
f 00 (x) = 48x
36x2 .
f0 12x3 = 0 12x2 (2
f 0 (x) = 0 x) = 0,
(2, 1)
16( 1) = 12 > 0;
(2, 1)
( 2, 0)
8( 2)2 + 2 =
x=0
x = 2.
f 00 (x)
0 4 + 64
2
f (0) = 4 (0, 4) (2, 12)
48 = 12
f (2) =
4 + 8(2)3
3(2)4 =
f 00 (x) f 00 (0) = 0,
f 00 (2) = 48(2)
36(2)2 = 96
144 =
58 < 0,
f (2) = 12
0
f 0 ( 1) = 24( 1)2 f 0 (1) = 24(1)2
12( 1)3 = 24 + 12 = 36 > 0
12(1)3 = 24
12 = 12 > 0, 0
f (0) ( 1, 2)
f (2, 1)
A
f
AU
0 f0
+
2
+
0
f0
2
f (x) = 3(x
1) 3
f 0 (x) = 2(x
1)
f
1 3
=
2 (x
1)
f 0 (1)
f
0
4 + 8x3
f (x) =
3x4
2.
1 3
;
f 00 (x) =
x 0
-x
x=1 f
2 (x 3
1)
4 3
=
2 4
3(x
1) 3
.
f 0 (x) = 0 f 0 (x) 0 f (x) = 0 f0 f c=1 2 c=1 f (1) = 3(1 1) 3 (1, 2)
x=1 2
2 = 3(0) 3
f 0 (1) x=1 f 0 (0) =
2 (0
1) f
1 3
=
2 ( 1)
1 3
=
2 < 0,
f 0 (2) = 1 ( 1, 1)
2 (2
2 1 3
1
1) (1) 3
= 2 > 0,
f (1) =
2 (1, 1)
2=
2
f A
U A 1
-x
f0
+ f0
1)2/3
f (x) = 3(x
2
1
f (x) = 3(2x + 3) 3 + 1 .
f 0 (x) = (2x + 3)
2 3
·2=
2 (2x + 3)
2 3
f 00 (x) =
;
4 (2x + 3) 3
5 3
·2=
8
f f ( 3/2) = 3 2( 3/2) + 3
1/3
5
3(2x + 3) 3
c=
.
3/2
f0
+ 1 = 3( 3 + 3)1/3 + 1 = 3(0)1/3 + 1 = 1 . ( 3/2, 1) f 0 ( 3/2) x=
2
f 0 ( 2) =
2( 2) + 3
2 3
=
2 2
( 1) 3
= 2 > 0,
2( 1) + 3 3/2
f ( 3/2) ( 1, 1)
f
f 1.5 f0
+ f0
f (x) = xe2x .
2
f 0 ( 1) =
-x + f (x) = 3(2x + 3)1/3 + 1
2 3
=
3/2 2 2
(1) 3
= 2 > 0,
f 0 (x) = e2x + 2xe2x = (1 + 2x)e2x .
f 00 (x) = 2e2x + 2(1 + 2x)e2x = 4(1 + x)e2x .
f0
f 0 (x) = 0
f 0 (x) = (1 + 2x)e2x = 0. 1 + 2x = 0 x = 1/2 f ( 1/2) = ( 1/2)e2( 1/2) = 1/(2e)
1/2 1/2, 1/(2e) x= f 00 (
f 00 (x)
1/2
1 2( 1 ) = 4(1 + ( ))e 2 2 f ( 1/2) =
f A
1 2)
= 2e
1
> 0,
1/(2e) ( 1/2, 1)
( 1, 1/2)
f
U A 1
f0
0
f0
-x + f (x) = xe2x
h(x) = 13 x3
f (x) = x2
5x + 6
f (x) = 2x2 + 3x g(x) =
2x + 1
g(x) = 5x
6
5
h(x) =
x3
y = x2 e
x
3x2 + 5x + 6 3x2 + 15x + 3
y = xe3x 2x +4 1 f (x) = 2 x +1
f (x) =
x2
f (x) = x2
g(x) = 3(x + 3)2/3 g(x) =
4x + 5
2)2/5
f (x) = 2x2 + 3x
5
4 x (0, 1)
g(x) =
x2 + 5x
6
2(x
f (x) = x +
1 x (0, 1)
f (x) = 9x+ f (x) = x4
18x2 + 2 ( 4, 1)
f (x) = 3x4
8x3 + 4 (1, 5)
g(x) = 2 h(x) = 13 x3 h(x) =
6x2
3x
3x2 + 5x + 6
x3
3x2 + 15x + 3
K(x) = x2 e
x
K(x) = xe3x s(x) = x4
x3
s(x) = 2x3
x4
f (x) = x4
8x2 + 3
f (x) = x4
18x2
K(x) = xe3x
f (x) = x2
5x + 6
s(x) = x4
f (x) = 2x2 + 3x
f (x) = x2
4x + 5
f (x) = 2x2 + 3x
5
x2 + 5x
6
g(x) = g(x) = 2
6x2
3x
h(x) = 13 x3 h(x) =
3x2 + 5x + 6
x3
3x2 + 15x + 3
K(x) = x2 e
s(x) = 2x3
x
x3 x4
f (x) = x4
8x2 + 3
f (x) = x4
18x2
g(x) = 2(x g(x) = 4
1
2)2/3 3(x + 1)2/3
h(x) = 1 + 3(x h(x) = 2
1
2)1/3
(x + 1)1/3
5
g(x) = x3
3x2 + 4
g(x) = x3
3x + 2
h(x) = 13 x3 h(x) =
x3
f (x) = x3
3x2 + 5x + 6 3x2 + 15x + 3 12x
f (x) = x3
27x 3
s(x) =
x + 3x2 + 1
s(x) =
x3 + 6x2 + 3
f (x) = x4
6x2
f (x) = x4
2x2 + 3
g(x) =
4
x4 + 8x2 + 2
x4 + 18x2 + 3
g(x) =
h(x) = x4
2x3
2
h(x) = x4
4x3
5
f (x) = x
4
f (x) = x2 e
x
f (x) = xe3x g(x) = 3(x + 3)2/3 + 2
4x + 2
f (x) = x4
32x
s(x) = 5x3
1
3x5
s(x) = 20x3
3x5
2)2/3 + 3
g(x) =
2(x
h(x) =
3(x + 3)1/3 + 2 2)1/3 + 3
h(x) = 2(x
(x1 , f (x1 )) (x2 , f (x2 )) . . .
f (x) = (x
20)4
2(x
20)2 + 10 f !x""!x!20"^4!2!x!20"^2#10
f !x""!x!20"^4!2!x!20"^2#10
f !x"!!x"20"^4"2!x"20"^2#10
y
y 50 40
y
400
20
300
15
200
10
100
5
30 20 10 !15
!10
5
!5 !10
10
15
x !20
!10
10
20
x
18
x y
20
22
24
x
f 0 (x)
f 00 (x)
f f 0 (x) = 0
f 0
f (x)
f f 00 (x) = 0
f 00 (x)
x
y
f (x) = x4
f 0 (x) = 4x3
16x;
f 00 (x) = 12x2
(0, 6) ( 2, 10) 00
16. (2, 10)
f (x) = 12x 12x2
f
✓
2 ±p 3
◆
16 = 0
x2 =
16 4 = 12 3
f 00 2
16 = 0
2 x = ± p ⇡ ±1.15 . 3
p x = ±2/ 3 f (x) ✓ ◆4 ✓ ◆2 2 2 16 32 26 = ±p 8 ±p +6= +6= ⇡ 2.9 9 3 9 3 3 ( 1.15, 2.9) (1.15, 2.9) [ 3, 3] [ 15, 15]
8x2 + 6
f !x""x4 !8x2 #6 y
15 10 5
!3
!2
1
!1
2
x
3
!5 !10 !15
f (x) = x4 f (x) =
4 + 8x3
3x4
f 0 (x) = 24x2
12x3 ;
f 00 (x) = 48x
(0, 4)
36x2 .
(2, 12) f 00
00
f (x) = 48x
2
36x = 0
36x2 = 0
48x 4x
8x2 + 6
3x2 = 0
x(4
3x) = 0
x=0
x = 4/3. x=0
f (0) =
x = 4/3
4 + 8(0)3
f (x)
3(0)4 = 0,
f (4/3) =
4 + 8(4/3)3
(0, 4)
3(4/3)4 ⇡ 5.48.
(1.33, 5.48) [ 1, 3]
[ 15, 15]
f !x""!4#8x3 !3x4 y
15 10 5
!1
x
1
2
3
f (x) =
4 + 8x3
3x4
!5 !10 !15
f (x) = 3(x
f 0 (x) = 2(x
1)
1 3
=
2 (x
1
1) 3
f 00 (x) =
;
2 (x 3
1)
4 3
=
2 3(x
4
1) 3
2
1) 3
2
.
(1, 2) f 00
f 00 (x) = 0
f 00
x=1
x=1 (1, 2) [ 2, 3] [ 4, 6]
f !x""3!x ! 1"2#3 !2 y
6 4 2
!2
1
!1
2
3
x
!2 !4
f (x) = 3(x
2
1) 3
2 1
f (x) = 3(2x+3) 3 +1
f 0 (x) = (2x + 3)
2 3
·2=
2 (2x + 3)
2 3
;
f 00 (x) =
4 (2x + 3) 3
5 3
·2=
8 5
3(2x + 3) 3
.
( 1.5, 1) f 00
f 00 (x) = 0
f 00 x=
x=
1.5
( 1.5, 1) [ 3, 1] [ 4, 6]
1.5
f !x""3!2 x # 3"1#3 #1
y
6 4 2
!3
!2
1
!1
x
!2 !4 1
f (x) = 3(2x + 3) 3 + 1 f (x) = xe2x f 0 (x) = e2x + 2xe2x = (1 + 2x)e2x , f 00 (x) = 4(1 + x)e2x
f 00
00
2x
f (x) = 4(1 + x)e = 0 1+x=0 x= 1 f (x) f ( 1) =
e
2
⇡
1/2, 1/(2e)
0.14
x= ( 1, 0.14)
1
[ 2, 0.5] [ 0.5, 1.5]
f !x""x!e"2 x y 1.5
1.0
0.5
!2.0
!1.5
!1.0
0.5
!0.5 !0.5
f (x) = xe2x
x
f 0 (x)
f 00 (x)
f 0
f
f (x) = 0
f 0 (x)
f
c f f
c c f 00 (x) = 0
f 00 (x)
00
f 00 (x)
00
f (x) > 0
12x3 ; (0, 4)
f
f (x) < 0
f (x) =
f 0 (x) = 24x2
f (c) f (c)
f 00 (x) = 48x
4 + 8x3
3x4
36x2 . ( 1, 2)
(2, 12) f
(2, 1) f
(2, 12)
(0, 4) f 00
(1.33, 5.48) 0 < x < 1.33 f (0, 1.33)
x 1.33 (1.33, 1)
4 r (0,
1
2
3
x
4)
8 12
f (x) =
4 + 8x3
3x4
f 00 f
f !x""!4#8x3 !3x4 y
x
15
f (x)
10 5 1
!1
2
3
x
!5 !10 !15
f (x) =
4 + 8x3
3x4
f (x) = xe2x
f 0 (x) = e2x + 2xe2x = (1 + 2x)e2x , f 00 (x) = 2e2x + 2(1 + 2x)e2x = 4(1 + x)e2x .
( 0.5, 0.2) f ( 0.5, 1) f
( 0.5, 0.2)
( 1, 0.5) ( 1, 0.1)
f 00
x< 1 ( 1, 1) f
( 1, 1)
y
x>
6
0.5
2
r
1
( 0.5,
-
0
r
1
x
0.2) 0.5
f (x) = xe2x
1 f ( 1, 0.1)
f !x""x!e"2 x
x
y
f (x)
1.5
1.0
0.5
!2.0
!1.5
!1.0
0.5
!0.5
x
!0.5
f (x) = xe2x f (x) = xe2x x
g(x) =
x4 + 8x2 + 2
g(x) =
x4 + 18x2 + 3
h(x) = x4
2x3
2
h(x) = x4
4x3
5
f (x) = x
4
4x + 2
f (x) = x4
32x
s(x) = 5x3 f (x) = x2
5x + 6
f (x) = 2x2 + 3x
3x5
s(x) = 20x3
3x5
5
f (x) = x2 e
x
g(x) = x3
3x2 + 4
f (x) = x2 e
2x
g(x) = x3
3x + 2
f (x) = xe2x
h(x) = 13 x3 h(x) =
x3
3x2 + 5x + 6 3x2 + 15x + 3
f (x) = x3
12x
f (x) = x3
27x
s(x) = s(x) =
x3 + 3x2 + 1 x3 + 6x2 + 3
f (x) = x4
6x2
f (x) = x4
2x2 + 3
4
1
f (x) = xe3x g(x) = 3(x + 3)2/3 + 2 2)2/3 + 3
g(x) =
2(x
h(x) =
3(x + 3)1/3 + 2
h(x) = 2(x
2)1/3 + 3
h(x) =
3(x + 3)1/3 + 2
h(x) = 2(x
2)1/3 + 3
f (0) = 1 f (x) ( 1, 3) (3, 1) f (x) = x2
f (1) = 2 f (x) ( 1, 1) ( 1, 1)
5x + 6
f (x) = 2x2 + 3x g(x) = x3
3x2 + 4
g(x) = x3
3x + 2
h(x) =
1 3 3x
h(x) =
x3
g(0) = 1 g(x) ( 1, 2) ( 2, 1)
5
3x2 + 5x + 6 3x2 + 15x + 3
f (x) = x3
12x
f (x) = x3
27x
s(x) =
x3 + 3x2 + 1
s(x) =
x3 + 6x2 + 3
f (x) = x4
6x2
f (x) = x4
2x2 + 3
4
g(x) =
x4 + 8x2 + 2
g(x) =
x4 + 18x2 + 3
h(x) = x4
2x3
2
h(x) = x4
4x3
5
f (x) = x4
4x + 2
f (x) = x4
32x
s(x) = 5x3
f (x) = x2 e
f (0) = 1 f (x) ( 1, 2) ( 2, 1)
( 1, 0) (0, 1)
f (1) = 2 f (x) ( 1, 1) (1, 1) ( 1, 1) ( 1, 1)
1
(1, 1)
K(1) = 2 K(x) ( 1, 1)
3x5
f (0) = 1 f 0 (x) > 0 f 0 (3) = 0 f 0 (x) < 0 (3, 1)
x
g(x) = 3(x + 3)2/3 + 2 2(x
s(0) = 1 s(x) ( 1, 1) (5, 1) (1, 5)
( 1, 1)
f (x) = xe3x
g(x) =
s(0) = 1 s(x) ( 1, 3) (4, 1) ( 3, 4)
K(0) = 1 K(x)
3x5
s(x) = 20x3
g(1) = 2 g(x) ( 1, 1) (1, 1)
2)
2/3
+3
f (1) = 2 f 0 (x) > 0 ( 1, 1) f 0 ( 1) = 0 ( 1, 1)
( 1, 1) ( 1, 3)
f 0 (x) < 0
g(0) = 1 g 0 (x) < 0 g 0 ( 2) ( 2, 1) g(1) = 2 g 0 (x) > 0 g 0 (1) (1, 1)
( 1, 2) g 0 (x) > 0
g 0 (x) < 0
( 1, 1)
s(0) = 1 s0 (x) > 0 (4, 1) s0 (x) < 0 ( 3, 4)
( 1, 3)
s(0) = 1 s0 (x) < 0 (5, 1) s0 (x) > 0 (1, 5)
( 1, 1)
f (0) = 0 f 0 (x) < 0 f 0 (x) > 0 f 00 (x) > 0 f 00 (x) < 0
C(x) = 1,500 + 20x + 0.2x2 0 x 400
x
x R(x) = 90x
. x
( 1, 2) ( 2, 1) ( 1, 0) (0, 1) x
f (1) = 3 f 0 (x) < 0 ( 1, 1) f 0 (x) > 0 00 (1, 1) f (x) < 0 f 00 (x) > 0
( 1, 1) ( 1, 1)
K(0) = 2 K 0 (x) > 0 K 00 (x) > 0 0 ( 1, 1) K (x) < 0 K 00 (x) > 0 (1, 1) K(1) = 0 K 0 (x) > 0 ( 1, 1) K 00 (x) > 0
K 00 (x) < 0 K 0 (x) > 0 ( 1, 1)
f (0) = 1 f 0 (x) > 0 ( 1, 1) f 0 (x) = 0 f 0 (x) < 0
( 1, 1) (1, 1)
f (1) = 0 f 0 (x) < 0 f 0 (x) = 0 (0, 2) 0 (2, 1)
( 1, 0) f 0 (x) >
g(0) = 0 g 0 (0) = 0 g 00 (0) > 0 g(2) = 4 g 0 (2) g 0 (x) < 0 (2, 1) g( 1) = 1 g 0 ( 1) = 0 g 00 ( 1) < 0 g(1) = 0 g 0 (1) g 0 (x) > 0 (1, 1) s(0) = 0 s0 (x) > 0 (4, 1) s0 (x) < 0 0 ( 3, 4) s ( 3) = s0 (4) = 0
( 1, 3)
C(x) = 1,200 + 30x + 0.5x2 0 x 250
x
x R(x) = 120x
.
s00 (0) = 0
s(0) = 2 s0 (x) < 0 ( 1, 1) (5, 1) s0 (x) > 0 (1, 5) s0 (1) = s0 (5) = 0 s00 (2) = 0
C(x) = 4,000 + 36x + 0.5x2
0 x 100
x x pE
R(x) = 500x
4x2
xE
.
x = D(p) = 300e x = S(p) = 5p x
x = S(p) = 2p
0 x 80
x = S(p) = p 20x2
.
R(x) =
x
p 2 p + 36
2
x = D(p) = 10 x = S(p) = 2p
0.05p
50
x = D(p) = 20
x
R(x) = 1,600x
20
x = D(p) = 200e
C(x) = 2x2 + 15x + 1,500,
0.1p
3
p p+1
P (x) an xn + an 1 xn 1 + . . . + a1 x + a0 = , bm 6= 0. Q(x) b m x m + bm 1 x m 1 + . . . + b1 x + b 0
2x + 1 f (x) = . x 1 x=1 1 x
x f
x=1 x = 1 1
x=1 lim f (x) =
x!1
x=1
1
1
lim f (x) = 1.
x!1+
f !x""!2x#1"#!x!1" y 15 10 5
!10
5
!5
10
x
!5 !10 !15
f (x) = (2x + 1)/(x
1)
x=a 1,
lim f (x) =
x!a
lim f (x) = 1,
x!a
1,
lim f (x) =
x!a+
lim f (x) = 1.
x!a+
x=a x=a x
x=a
a
x=a
x=a
f (x) = ln x
p f (x) = 1/ x
x = 0
x=1 2
f (x) =
x +2 x2 1
g(x) =
x x2
2
1 3x + 2
h(x) =
x2 x2
1 2x + 1
x=1 2
x +2 (1)2 1 = 0 x2 1 x=1
f (x) =
g(x) =
g(x) =
x x2 x2
x2
2
1 (1)2 3x + 2 x=1
(1)2 + 2 = 3 x=1
3(1) + 2 = 0
1 (x + 1)(x = 3x + 2 (x 2)(x
x=1 f
(1)2
1=0
1) x+1 = , x 6= 1. 1) x 2
x=1 h(x) =
x2 x2 x2
h(x) =
x2
g
1 (1)2 2x + 1 x=1
(1)2
2(1) + 1 = 0
1 (x + 1)(x = 2x + 1 (x 1)(x
1) x+1 = . 1) x 1
x=1
h
f (x) =
x2 x3
x 2x2
(x 2)(x + 1) x 2 = , x 6= x(x + 1)(x 3) x(x 3)
f (x) =
x=0 x=0
1=0
2 3x
1. x=3
x=3
y6
y6
f (x)
y6
f (x)
f (x)
-x
-x
x=a lim f (x) = x!a
y6
-x
x=a 1
lim f (x) = 1
x!a
f (x)
-x
x=a lim f (x) =
x!a+
x=a 1
lim f (x) = 1
x!a+
y=2 x lim f (x) = 2
x! 1
lim f (x) = 2.
x!1
y=2
y=b lim f (x) = b,
x! 1
lim f (x) = b.
x!1
y=b
x b
y=b x
y=0 y = a/d
a d
p f (x) = 1/ x
f (x) = ex
y6
y = 0
y6
y6
y6
f (x)
f (x) x!
y=b lim
x!
1
f (x) = b 1 y=b
lim f (x) = b
lim
x!1
y=b
f (x) = b
lim f (x) = b
x!1
f (x)
-x
f (x) =
y=b
2x2 + 5x 1 2 3x2
f (x)
-x
g(x) =
3x3 1 2x2 + 7
2/( 3) =
2 3
-x
h(x) =
-x
2 x2
4
y=
2 3
y=0 f (x)
lim
x!±1
x!1
lim
1
1 =0 xp p
x!1
x!
p = 1
1 = 0. x f (x) x
x!1
x!
1
x
x2 4 x!1 2x3 + 5x
2x + 1 x!1 x 3 lim
lim
x 1 x 3 x
2+ 2x + 1 (2x + 1)/x = = x 3 (x 3)/x 1
. |x|
1/x 2+ 2x + 1 = lim x!1 x x!1 1 3 lim
1 x 3 x
=
2+0 = 2. 1 0
x3 1 x2 4 (x2 4)/x3 = = x 3 3 3 2x + 5x (2x + 5x)/x 2+
.
1/xp
|x| 1 x2 4 x = lim x!1 2x3 + 5x x!1 2 +
lim
f (x) =
y=x
f (x) = x
4 x3 5 x2
x2
1 x
4 x3 5 x2
=
0 0 = 0. 2+0
.
x 1 x
y=x 1 = 0. x!1 x lim
y = x f !x""!x2 !1"#x y
10
5
!10
5
!5
10
x
!5
!10
y = mx + b m 6= 0
f (x)
f (x) = mx + b + q(x)
f (x) =
lim q(x) = 0,
x! 1
lim q(x) = 0.
x!1
3x2 2 . x+1 3
m=3
3x(x + 1) = 3x2 + 3x 3x 3x2
2 = 3x2 + 3x
f (x) = =
3x
2 = 3x(x + 1)
3x2 2 3x(x + 1) 3x = x+1 x+1
3x
2,
2
3x(x + 1) 3x 2 3x 2 + = 3x + . x+1 x+1 x+1 y = y = 3x
3
3
|x|
f 0 (x)
f 00 (x)
f f 0 (x) = 0
f 0
f (x)
f f 00 (x) = 0
f 00 (x)
x y
f (x) =
2x2 + 9 x2 9 x
x2
9 = (x
3)(x + 3) = 0,
x=3
x=
( 1, 3) [ ( 3, 3) [ (3, 1)
3
x=
3
x=3
x=
x=3 y=2
f 0 (x) =
4x x2
9 (x2
2x2 + 9 (2x) 9)
2
=
4x3
36x
4x3
(x2
2
9)
18x
=
54x (x2
9)
2
;
3
f 00 (x) =
54 x2
9
2
( 54x)2 x2 (x2
9)
9 (2x)
4
=
54 x2
(x2
0
x=0
9)
4
=
162 x2 + 3 (x2
9)
f 0 (x) = 0
x 54x
x2 + 9 + 4x2
9
f (x)
3
.
f 0 (x) 0
x=0
x x=
3
x=3 0
f (0) =
f 00 (x) = 0
x x
1
(0, 1)
f 00 (x) 162 x2 + 3 f 00 (x) = 0 x= 3
x f 00
x x=3 [ 5, 5]
[ 4, 6] f !x""!2x2 #9"#!x2 !9" y
5
x
5
!5
!5
f (x) = (2x2 + 9)/(x2
f (x) =
9)
2x2 4 x2 + 1
x x y=2
f 0 (x) =
f 00 (x) =
4x x2 + 1
2x2
(x2 + 1) 12 x2 + 1
2
4 (2x)
2
=
4x3 + 4x
(12x)2 x2 + 1 (2x) (x2 + 1)
4
f0
4x3 + 8x
(x2 + 1) =
2
12 x2 + 1
=
12x (x2 + 1)
x2 + 1
(x2 + 1)
4
2
4x2
;
=
12 1
3x2
(x2 + 1) f 0 (x) = 0
x x=0
3
.
f 0 (x)
0
f (0) =
p f 00
f 00 (x) = 12 1
x=p 12 1
3x2 = 0
1
p x = ±1/ 3 ✓ ◆ 2 1 4 1 f ±p = 13 = 3 3 +1
3x2 = 0
x2 = 1/3
4
(0, 4) f 00
00
f (p) = 0 3x2 = 0 p x = 1/ 3
p x = 1/ 3.
f (x) 10 = 4
5 . 2 ( 0.58, 2.5)
(0.58, 2.5) [ 6, 6]
[ 6, 6]
f !x""!2x2 !4"#!x2 #1" y
6 4 2
!6
!4
2
!2
4
6
x
!2 !4 !6
f (x) = (2x2
4)/(x2 + 1)
f (x) =
4x2 + 1 2x x=0 ( 1, 0) [ (0, 1)
x=0
x=0
f (x) = 2x + 1/2x f 0 (x) = 2
1 2x2
f 00 (x) =
1 . x3 x
f 0 (x) = 2
1 4x2 1 = . 2 2x 2x2
f 0 (x) = 0
f0
y
f 0 (x)
0
4x2
1
x = 1/2 1/2 1/2
f (x)
x = 1/2
x f ( 1/2) =
x=0 1/2 2
f (1/2) = 2
( 1/2, 2)
(1/2, 2) [ 3, 3]
[ 6, 6] f !x""!4x2 #1"#!2x" y
6 4 2
!3
!2
1
!1
2
3
x
!2 !4 !6
f (x) = (4x2 + 1)/(2x)
0 1/2
f 0 (x)
f 00 (x) f 0 (x) = 0
f 0
f (x)
f
f c f f
c f 00 (x) = 0
f 00 (x)
00
f (c) f (c)
c
f 00 (x)
f 00 (x) > 0
f
f (x) < 0
f (x) = ( 1, 3) [ ( 3, 3) [ (3, 1)
f (0) =
2(0)2 + 9 9 = = 2 0 9 9
2x2 + 9 x2 9 x= y=2
1.
(0, 1) 54x (x2
9)
2
;
f 00 (x) =
162 x2 + 3 (x2
9)
3
x=3 y
(0, 1)
f 0 (x) =
3
.
( 1, 3)
x=0 ( 1, 3) ( 3, 0) (0, 3) f0 54x (0, 3) (3, 1) ( 3, 0)
x=
0
A AU
AU
0
+
( 3, 0)
(3, 1)
(0, 3)
3
f0
f ( 1, 3)
f
A
+
x=3
f0
f
f0
3
(3, 1)
3
-x
0
f0 f (0) =
1 x
f ( 4) = f (4) =
2(16) + 9 ⇡ 6, (16) 9 ( 4, 6)
(4, 6)
f
( 4, 6)
(4, 6)
y
r
r
6
6 4
y=2
2
r
-
0
6
4
2 2
x=
3
2
4
6
x
x=3
4 6
f (x) = 2x2 + 9 / x2
f
00
f 00 x< 3
x< x>3
3
9
x>3 3 1
E(p) < 1
E(p) = 1 x
x = D(p) =
2.5p + 500,
p E(p) p = 96
p = 102 E(p) = 1
E(p) E(p) =
p · D0 (p) . D(p)
D(p) =
2.5p + 500
E(p) =
p · ( 2.5) 2.5p p = = . 2.5p + 500 500 2.5p 200 p
E(96) =
96 ⇡ 0.923. 200 96
E(102) =
D0 (p) =
2.5
102 ⇡ 1.04. 200 102
E(p) = 1 p =1 200 p p = (200
p),
2p = 200,
p = 100.
E(p) = 1
R(p)
R(p) = p · D(p). R0 (p) = 1 · D(p) + p · D0 (p). E(p)
p · D0 (p) = D(p)[1 D(p)
R0 (p) < 0
E(p) > 1 R0 (p) > 0
R0 (p) = D(p) 1 +
E(p) > 1 E(p) < 1
E(p)]. R0 (p) = 0
E(p) < 1
R0 (p) < 0 R0 (p) > 0 R0 (p) = 0
E(p) = 1
R 6
r
E(p) = 1
E(p) < 1
E(p) > 1
R
x = D(p) = 180,000 p
- p
p⇤
0
p
22
p, x E(p)
p = 16
E(p) = 1
E(p) p · D0 (p) . D(p) p D(p) = 180,000 22
E(p) =
180,000( 1) 90,000 p =p . 2 22 p 22 p
D0 (p) =
E(p) =
p
p·
p90,000 22 p
p
180,000 22
E(16) =
90,000p 1 p =p · p 22 p 180,000 22
p
=
p . 2(22 p)
16 16 4 = = . 2(22 16) 12 3
E(16) > 1 E(p) = 1 p =1 2(22 p) p = 2(22
p) = 44
2p,
3p = 44,
p ⇡ 14.67.
E=2 x = D(p) = 220 x = D(p) = 50 E = 0.5
x = D(p) = 250
5p 4p
p = 10 p=5
p2
p = 10
x = D(p) = 200 p2 p=8 p x = D(p) = 150 3p p = 30 p x = D(p) = 100 2p p = 25 x = D(p) = E(p)
100 p
x = D(p) =
500 p
x = D(p) = 4,500e
p = 30 p = 20 0.02p
p = 200
x = D(p) = 6,500e x = D(p) = 100e
x = D(p) = 50 x = D(p) =
p = 100
0.05p
x = D(p) = 3,000e x = D(p) = 30
0.04p
p = 40
0.03p
2p
p
p
2
p
200 (p + 2)2
p = 100
2
x = D(p) = 120
p=2
4p,
p x
p=4
E(p)
p=1
p = 18
300 x = D(p) = (3p + 10)2
p = 10
E(p) = 1
x p
x = D(p) = 270
x = D(p) = 600(5
p
2.5p,
p
p) 1,800
x E(p) p = 60
E(p) p=4
E(p) = 1
x p p
x
x = D(p) = 2,000 900
p 54,722
p R(p)
E(p) p = 150 E(p) p = 60
E(p) = 1
x = D(p) = 20
2p2
p x x E(p)
p R(p)
E(p) p = 60
E(p) = 1
x = D(p) = 30
p
p2
x = D(p) = 100 ln(150
p)
p
p
x
x
E(p) E(p)
x = D(p) = 150 ln(120
p)
p x E(p)
p = 90
E(p) = 1
x = D(p) = 2,000e
0.04p
x = D(p) = 5,500p
p x
0.8
p x E(p) p = 20
E(p) = 1
x = D(p) = 1,000e
0.01p
p x E(p)
E(p)
y 6 f (x)
r ⌘⌘ 6 r⌘ ⌘ ⌘ y 6 0
m = f 0 (x)
⌘
⌘
r⌘ ⌘
⌘
x
f (x) x
? ?
-
- x x
x) ⇡ f (x) + f 0 (x) x,
f (x + x
x
y = f (x +
dy ⇡ dx
y
x) ⇡ f (x),
y . x dy x
dx =
dx
x. y
dy
dy = f 0 (x)dx.
y ⇡ dy,
f (x + dx) ⇡ f (x) + dy. y = f (x) = ln(x2
3)
dy dy
x=2
dx = 0.01 f (2.01) dy dx
dy 2x = 2 . dx x 3
dx
y
dy =
dy 2x · dx = 2 · dx. dx x 3
x=2 dx = 0.01 ✓ ◆ 2(2) dy = (0.01) = (4)(0.01) = 0.04. (2)2 3
f (x + dx) ⇡ f (x) + dy. x = 2 dx = 0.01 f (2) = ln(22
3) = ln 1 = 0
dy = 0.04
f (2.01) = f (2 + 0.01) ⇡ f (2) + dy = 0 + 0.04 = 0.04. f (2.01)
0.039317 . . .
x
f
(a, b) f 0 (x) > 0
x
(a, b)
f
(a, b)
f 0 (x) < 0
x
(a, b)
f
(a, b)
y
y
6
f
r
0
f
f 0 (x) > 0
x
6 @@ @r f 0 (x) < 0 @ @ @
- x 0
- x
x
x f 0 (x) f (0 c)
x f 0 (c) = 0
x=c
y
r
6 f (b)
A A
r
f (c2 )
f (a) f (c3 ) ⇣ P f (c1 )
0
r
r` ``` `
a c1
# #
#
c2
r
r
c3
c4
y = f (x)
b
- x
[a, b] f (c2 ) f (b) c1 c2 c3
c4
f (c3 )
f (c4 )
f (x) = x3
f 0 (x) = 3x2
f (c1 ) f (c1 )
3x + 1
3.
f0
f 0 (x) = 0
3x2
3=0
x2
1=0
(x
1)(x + 1) = 0
x=1
x=
1. 1
1 ( 1, 1) ( 1, 1) f0
f 0 ( 2) = 3( 2)2 f0
3 = 9 > 0; f 0 (0) =
3 < 0; f 0 (2) = 3(2)2
( 1, 1)
(1, 1)
(1, 1)
3 = 9 > 0. ( 1, 1)
A
f
AU
1 f0
f ( 1) = ( 1)3 f (1) = (1)3
-x
+
+
3( 1) + 1 =
3(1) + 2 = 1
1 + 3 + 1 = 3,
3+1=
f (x)
1
1
1.
( 1, 3)
1 x=
(1, 1)
f 00 ( 1) = 6( 1) = 1 f
6 0 x=1
f
f !x""x3 !3x#1 y
20
10
!3
!2
1
!1
2
3
x
!10
!20
(1, 1) ( 1, 3) f 00 (p) = 0
f 0 (x)
f 00 (x)
f f 0 (x) = 0
f 00 (x) = 0
f 00 (x)
f 0 (x)
f 00 (p)
x y
f 0 (x)
f 00 (x)
f f 0 (x) = 0
f 00 (x) = 0
f 0 (x)
f 00 (x)
f (x) = P (x)/Q(x) y = 0 y = a/b
Q(x) = 0 P
Q
P
Q
a d
P
Q
f f !x""!2x2 #9"#!x2 !9" y
5
5
!5
x
!5
[a, b] [a, b] a
(a, b) b [a, b) (a, b) ( 1, b) (a, 1)
( 1, 1)
q p q = D(p) q
p
74
D(120) = 8,000 D0 (120) =
80
y = f (x) = 3x3 + 2x + 1 y
dy
x=1
x = dx =
0.01 f (1.01)
f
C(x) = 1,200 + 18x + 0.1x2 x
f
x
f
R(x) = 120x x
f f
70 70
y
6
4
y = f (x)
3 2 1 4
3
2
10 1
1
2
3
4
x
2
f (x) =
3
2 x2
4
f (x) = f (x) = 13 x3 3x2 +5x+6
4
4x2 +1
x2
g(x) =
x2
x+1 x 2
f x
f f
h(x) = 13 x3 y = x2 e f (x) =
f
y = x2 e f (x) =
f
x
2x3
2
f (x) = 3(x + 3)2/3 + 2
[0, 4]
2x +4
x2
x
[ 2, 2]
(0, 1)
2x x2 + 4
f (x) = x +
f (x) = x4
3x2 + 5x + 6
4 x
(0, 4) (0, 1)
[2, 6]
1,500
E(p) p = 18
E(p) = 1
x x 1,000 p x
R(x) x = D(p) = 120
4p
p x
y = f (x) = 2x3 + x2
y
dy
x
x=1
x = dx = 0.02 y= dy = f (1.02)
f (1.02) ⇡
C(x) = 900 + 80x x
5 2 1 3 x + x 9 135 x
R(x) = 160x
80
80
82
f (x) = 2x3
x4 + 1
f (x) = 2x3
x4 + 1
f (x) =
1 3x x 2
f (x) =
f (x) = 2x3
3x2
1 3x x 2
12x + 20; [ 2, 2.1]
x=
x=
f (x) = 2x3
3x2
12x + 20;
[0, 1]
x=
x=
f (x) = 4x2
x
3; ( 1, 1)
x=
x=
f (x) = 32x2 +
64 x ;
(0, 1)
x=
x=
Q = x2 + 2y 2
x
y=4
1,000
70,000 10,000
x = D(p) = 400
4p,
p
x E(p)
p = 20
E(p) = 1
M3 h
r
R
0 h 2R
R h
r 0hR
R h 2R
h
r
V = ⇡Rh2
⇡ 3 h 3
R
0 h 2R, 0 h 2R.
S = 2⇡Rh
R h C(h) h R r
h
r
y
y
6
6
y = f (x)
A= 0
Rb a
y = f (x)
A0 (x) = f (x)
f (x)dx - x
0
- x
y
y
6
6 y = f (x)
y = f (x)
A 0
a
A(x) b
- x
0
a
x
- x x
A x f (x)
x=a
x=b
[a, b]
f (x) x
x
A(x)
A A(x)
x
b
A(b) = A
f (x) A(x) f (x) A
f (x) [a, b]
0
v v(t) = 65, 0 t 2.5.
v
6 v(t) = 65
- t
0
t1
t2
⇥
65 ⇥ (t2 65(1 65(2
v
1) = 65 65(2.5
t1 ) 0.5) = 32.5 = 2.5
0) = 162.5
v
6
v
6
6
v(t) = 65
A 0
A = 32.5
-t
v(t) = 65
B 0
B = 65
-t
v(t) = 65
C 0
C = 162.5
-t
0
a t b
[a, b] v
6
v(t)
A t=a 0
t=b
a
b
- t
v
6 v(t)
- t
0
[0, 2]
1 (60 2
(60
)(0.25
)(1.5
) = 7.5
) = 90
.
. 7.5 + 7.5 + 90 = 105
v
6 v(t)
- t
0
0.8
M C(x) = 0.8, x
x
MC
6
x
0
0.8 0.8x [0, x]
M C(x) = 0.8
- x
x
MC
6 M C(x) = 0.8
A(x) = 0.8x - x
x
0
f
[a, b] f
[a, b] a
b
f (x) = x2 + 1 [0, 2] [0, 2] x = (2
0)/4 = 0.5
f !x"!x2 "1
y
f !x"!x2 "1
y
8
8
6
6
4
4
2
2 A 0.5
1.0
1.5
2.0
x
2.5
0.5
1.0
1.5
2.0
2.5
x
f !x"!x2 "1
y 6 5 4 3 2 1 I
II 0.5
III 1.0
IV 1.5
2.0
x
x = 0.5
f (0) = 1 f (0.5) = 1.25
f (1) = 2
f (1.5) = 3.25
A ⇡ L4 = = f (0) ·
x + f (0.5) ·
x + f (1) ·
x + f (1.5) ·
x
= (f (0) + f (0.5) + f (1) + f (1.5)) x = (1 + 1.25 + 2 + 3.25)0.5 = 3.75. 3.75 x = (2
A
0)/8 = 0.25
A ⇡ L8 = (f (0) + f (0.25) + f (0.5) + f (0.75) + f (1) + f (1.25) + f (1.5) + f (1.75)) x = (1 + 1.0625 + 1.25 + 1.5625 + 2 + 2.5625 + 3.25 + 4.0625)0.25 = 4.1875.
f !x"!x2 "1
y 6 5 4 3 2 1 0.5
v(t) = 240t
1.0
1.5
x
2.0
240t2 , 0 t 1,
t
v!t"
60 50 40 30 20 10 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
t
[0, 1]
v!t"
60 50 40 30 20 10 0 0.0
0.2
0.4
0.6
0.8
t = (1
1.0
1.2
t
0)/4 = 0.25
A ⇡ L4 = (v(0) + v(0.25) + v(0.5) + v(0.75)) t = (0 + 45 + 60 + 45)0.25 = 32.5
.
x M C(x) = 0.0000625x2
MC
0.05x + 25, 0 x 550.
MC!x"!0.0000625x2 "0.05x#25
25 20 15 10 5 0
[0, 500]
100
200
300
400
500
600
x
MC!x"!0.0000625x2 "0.05x#25
MC 25 20 15 10 5 0
100
200
300
x = (500
400
500
600
x
0)/5 = 100
A ⇡ L5 = (M C(0) + M C(100) + M C(200) + M C(300) + M C(400)) x = (25 + 20.625 + 17.5 + 15.625 + 15)100 = 9375. $9,375 $9,875
y 4
[0, 4]
[ 2, 2] y 4
[ 2, 0]
[ 4, 4]
6
1
y = f (x) 4
3
2
2 1 3
2
y 4 3
1
2
3
4
x
2
10 1
1
2
3
4
x
6 y = g(x)
2
6
1 4
y = f (x)
1 3
4 3
2
4
-
10 1
y
-
10 1
y = g(x)
3 2
3
4
6
1
2
3
4
x
3
2
10 1
1
2
3
4
x
[ 4, 4]
v
6
y 4
v(t)
6 y = f (x)
3 2
4
3
2
-
10 1
y 4
1
2
3
4
-t
0
1
x
v = v(t)
6 y = f (x)
3 2
[0, 8]
[0, 6]
[2, 4]
[4, 8]
1 4
3
2
-
10 1
1
2
3
4
x v
6
y 4
v(t)
6 y = h(x)
3 2
4
3
2
-
10 1
y 4
1
2
3
4
-t
0
1
x
6
M C(x)
y = h(x)
3
x
2
x
1 4
3
2
10 1
1
2
3
4
x MC
6 M C(x)
v = v(t)
- x
0
M C(x) x [0, 4]
[0, 2]
[1, 3]
[2, 4]
x
MC
y
6
6
M C(x)
y = f (x)
A = 200 - x
0
- x
0
x y x y
A P (0) = 0 y
x
6
y
y = f (t)
P (0) = 0
A = 200
x - t
0
y
t y
t y v(t) = 0.4 + 0.05t, 0 t 10, t t
y
t y
v(t) = 1.2 A
t
0.02t, 0 t 15,
x M C(x) = 54 0.05x 0 x 200 $1,200
v(t) = 44
4.4t, 0 t 10,
t x M C(x) = 15
v(t) = 44 + 2.2t, 0 t 20,
0.06x 0 x 100
M R(x) = 90
x 0.5x 0 x 120
t
M C(x) = 20
M C(x) = 18
0.02x 0 x 300 x
0.05x 0 x 100 x
M R(x) = 30
0.2x 0 x 100 x
v(t) = 0.4 + 0.5t t
0.025t2 , 0 t 10,
x M C(x) = 84
v(t) = 1.2
0.9x + 0.008x2 0 x 100
0.05t + 0.001t2 , 0 t 15,
t x 20
p 0.5 x 0 x 100
M C(x) =
r(t) = 200e0.04t 0 t 10 t v(t) = 44
0.44t2 , 0 t 10,
t
r(t) = 200e
0.04t
0t6 t
v(t) = 44 + 0.11t2 , 0 t 20, t
M P (x) = 90 0 x 120
x 10 ln(x + 1)
M P (x) = 30
2x1/3 0 x 100 x
P (0) = 0
A(x) A(x)
f (x)
y
6
y = f (x)
A(x) 0
a
- x
x A(x)
f (x)
a s(b)
b
s(a)
v(t) = s0 (t)
x C(x)
C(0)
C(x) M C(x) = C 0 (x)
[1, x]
A1 (x) A2 (x) [2, x]
f (x) = 3 A1 (x)
A01 (x)
A02 (x)
A2 (x)
f (x)
y
y
6
6 f (x) = 3
A1 (x) = 3(x
f (x) = 3
1)
A2 (x) = 3(x
x
0
A1 (x)
- x
2)
- x
x
0
f (x)
A2 (x)
f (x) (x
1)
3 A1 (x) = 3(x
1) = 3x
3.
(x A2 (x) = 3(x
2) = 3x
A1 (x)
2)
3
6.
A2 (x)
A01 (x) = A02 (x) = 3 = f (x).
[1, x]
A3 (x) [2, x]
A4 (x)
f (x) = 2x
y
y
6
6
2x
2
2x f (x) = 2x
f (x) = 2x
A3 (x) = 12 x · 2x
A4 (x) = 12 x · 2x
1 1 2
4
·2
x
0
A3 (x)
f (x)
- x
1 2 2
·4
x
0
A4 (x)
f (x)
- x
x 2x
1 1 x · 2x 2
A3 (x) =
1 1 · 2 = x2 2
2
1. 2
1 x · 2x 2
A4 (x) =
1 2 · 4 = x2 2
4.
A3
A4
A03 (x) = A04 (x) = 2x = f (x). f (x)
4
A(x) A0 (x)
[a, x] A0 (x) = lim
h!0
A(x + h) h
A(x)
h>0
. A(x + h)
A(x + h)
y
A(x)
y
6
6
y = f (x)
y = f (x)
A(x + h) A(x) ⇡ h · f (x)
A(x + h) 0
a
- x
x x+h
0
a
A(x + h) h
A(x + h)
x x+h
A(x + h)
- x
A(x)
A(x)
h
f (x) A(x + h) h
A(x) ⇡ h · f (x),
A(x + h)
A0 (x) = lim
h!0
A(x + h) h
A(x)
A(x)
⇡ f (x).
= f (x).
A(x) f
[a, b]
f
[a, x]
F (x)
a0
a kx e +C k
k 6= 0
Z
g(x) dx
Z Z =3
4
3x + 6x ✓
1 5 x 5
◆
2
2x + 8 dx = 3
+6
✓
1 3 x 3
◆
2
✓
Z
1 2 x 2
4
x dx + 6
◆
3x4 + 6x2
Z
x dx
+ 8x + C = Z ✓
Z ✓ =3
3e
Z
5 p 3 x2
2x
e
2x
=
3 e 2
2x
=
3 e 2
2x
dx 5· 5·
◆
5
Z
dx = 3 x
2 3
1
2 3
+1
Z
e
2x
dx
5
Z
2
2
2x + 8 dx Z
x dx +
3 5 x + 2x3 5 3e
2x
Z
8 dx
x2 + 8x + C.
5 p 3 x2
◆
dx
3 x5
◆
dx
1 p dx 3 x2
dx x
2 3 +1
1 1 x3 + C = 1/3
+C 3 e 2
2x
p 15 3 x + C. Z ✓
2 4 + x x3
Z ✓
◆ Z Z Z 2 4 3 1 1 1 + 3 dx = 2 dx + 4 dx 3 dx x x x5 x x3 x5 Z Z Z 1 3 =2 dx + 4 x dx 3 x 5 dx x 1 1 = 2 ln x + 4 · x 3+1 3 · x 5+1 + C ( 3) + 1 ( 5) + 1 = 2 ln x
2x
2
3 + x 4
4
+ C = 2 ln x
2 3 + 4 + C. x2 4x
F (x) = x2 + C
f (x) = 2x x2 + 3
x2
3 x2
1 x2 x2 + 2
2x 2x F (x)
F 0 (x) = 2x
F (1) = 3
F (x) = x2 + C
(1)2 + C = F (1) = 3, C=2 F (x) = x2 + 2. y
6
r 0
- x
f (x) = 2x F F 0 (x) = x3
F (2) = 1. F (x)
F (x) =
Z
x3 dx =
1 4 x + C. 4 F (2) = 1
C
1 4 (2) + C = F (2) = 1 4 4 + C = 1,
F (x) =
1 4 x 4
v(0) = 5 s(t) v(t)
C=
3.
3.
s(0) = 12 a(t)
s0 (t) = v(t);
a(t) = 6t s(t)
s00 (t) = v 0 (t) = a(t), v(t) v(t) =
a(t) Z
a(t) dt =
Z
6t dt = 3t2 + C1 . v(0) = 5
3(0)2 + C1 = v(0) = 5,
C1 C1 = 5.
v(t) = 3t2 + 5. s(t) v(t) Z Z s(t) = v(t) dt = (3t2 + 5) dt = t3 + 5t + C2 . s(0) = 12 (0)3 + 5(0) + C2 = s(0) = 12,
C2 C2 = 12.
s(t) = t3 + 5t + 12.
a(t) = v(0) = 0
s(0) = 500
v(t) v(t) =
a(t) Z
a(t) dt =
Z
32 dt =
32t + C1 .
v(0) = 0 v(t) =
32t.
s(t) v(t) Z Z s(t) = v(t) dt = ( 32t) dt = s(0) = 500 16(0)2 + C2 = s(0) = 500,
s(t) =
C1 = 0
16t2 + C2 . C2 C2 = 500.
16t2 + 500. s(t) = 0
16t2 + 500 = 0
t
32
2
t=
r
500 = 5.59 16
v(5.59) =
.
32(5.59) =
178.9
.
x M C(x) = 0.00006x2
0.04x + 25, 0 x 550. M C(x) =
0
C (x)
C(x) M C(x) Z Z C(x) = M C(x) dx = (0.00006x2 = 0.00002x3
0.04x + 25) dx
0.02x2 + 25x + C. C(0) = 500
C(x) = 0.00002x3
C = 500
0.02x2 + 25x + 500.
x = 300 C(300) = 0.00002(300)3
0.02(300)2 + 25(300) + 500 = $6,740. $6,740
A1 (x)
A2 (x) f (x) = 2 + 3x
[1, x]
[2, x] A1 (x)
A2 (x)
A01 (x)
A02 (x)
f (x) A3 (x)
A4 (x) f (x) = 3 + 0.5x
[1, x]
Z Z
Z
x3 dx 3 dx
Z
x1.4 dx
Z
x
⇡
Z
dx
(2x3
Z
p 3 x dx
Z
4) dx
p 5
12 x7 dx
Z ⇣
p 5
3 x3
2 dx x2
x4 dx Z 6 dx Z
Z
Z
[2, x]
⌘
dx
e
dx
3x4
Z
p 3
4 x dx
Z
p 7 x5 dx
Z
Z
3 p dx 3 x2
Z
1 dx x
Z
3 dx x
Z Z
p 3
5x2 + 2 dx
2 x2 dx 3 dx x4
4 x0.6 p 5
dx
Z
dx
Z
3 x3
Z ✓
1 4 + x3 x
Z ✓
1 p x
Z
Z
Z
2 p dx x
Z
x1.2 dx
x
Z
x2.5
dx
1 x2
3 p 3 x
◆
Z ✓
2 1 p + p 4 3 x x
dx
Z
Z
2e3x dx
Z
Z
5x
Z
+ 5 dx
Z
6e
Z
2 x/2 e dx 3
dx
Z ✓
3 + 5e x
10x
Z ✓
2 + 4e x
8x
Z ✓
3 4 p + x x
2
p 3
x2
3 x1.2
◆
dx
dx ◆
dx
dx
3e2x dx
3x4
5e2x + 2 dx
Z
3e
Z
1 x/3 e dx 4
◆
◆
3ex dx
dx
p ◆ + 3 x3 dx
2 e3x
2 x
3 dx xe
Z
dx
0.02x
x2
Z ✓
ex dx
2e
4
dx
Z Z
p 3
dx
◆
1 dx x⇡ 4
3 x0.4
0.04x
dx
Z ✓
5 p 4 x
2 3 + x e2x
◆
s(t)
dx
s(0) = 0 f (x)
f 0 (x) = 2x + 4 f (1) = 3 f 0 (x) = 3
4x f (2) = 1
f 0 (x) = x2 + 4 f (0) = 3 f 0 (x) = x2
3 f (1) =
f 0 (x) = 2x2 0
f (x) = 3x + 2x
f 0 (x) = 3x2
1
0.05t + 0.001t2 0 t 15
x + 5 f (0) = 2
2
f 0 (x) = x2
v(t) = 1.2 t=0
4 f (0) = 1
s(t) s(0) = 0
4x + 3 f (1) = 2 2x
5 f (1) = 3
f 0 (x) = 2e3x f (0) = 4 f 0 (x) = 3e2x f (0) = 2 p f 0 (x) = 3/ x f (1) = 3 p f 0 (x) = 2/ 4 x f (1) = 4
v(t) = 44
4.4t. s(t)
s(0) = 0
f 0 (x) = 3/x f (1) = 2 f 0 (x) =
2/x f (1) = 1
s(t) v(t) = 4t2 s(0) = 4 v(t) = 44 + 2.2t
v(t) = 3t s(0) = 5 a(t) = 3 v(0) = 2 s(0) = 5
s(t) s(0) = 0
a(t) = 5 v(0) = 1 s(0) = 3 a(t) = 4t v(0) = 3 s(0) = 10 a(t) = 2t v(0) = 2 s(0) = 6
M C(x) = 20 v(t) = 0.4 + 0.2t t
0.02t
2
0 t 10
0.02x 0 x 300 x
t=0 C(x)
R(0) = 0
M C(x) = 18
0.05x 0 x 100 x
V 0 (t) = r(t) = 200e0.04t 0 t 10 t C(x) $1,000,000 V (t) t
x M C(x) = 84
0.9x + 0.008x2
0 x 200
$1,200 C(x)
0.04t
0t7
t
x M C(x) = 20
V (t) = r(t) = 200e
$1,500,000
p 0.5 x 0 x 100
V (t) t
C(x)
x M R(x) = 90
0.5x 0 x 120 R(x)
M P (x) = 30
0.02x1/3 0 x 100 x
R(0) = 0 R(0) = 0
P (0) =
300
P (x)
M R(x) = 30
M P (x) = 30
0.2x 0 x 100
p 0.02 x 0 x 100 x
x
P (0) = R(x) R(0) = 0
200
S 0 ? = 0.00018p2 + 0.04p + 0.6 p x = S(p)
y
y
6
y = f (x)
y
6
y = f (x)
6
y = f (x)
A
A a
0
A(x) -x
b
0
d
a
A(b) A(a) b
x
-x
0
d
a
b
x
-x
[a, b]
A f (x) f (x) A(b)
[a, b]
A(x)
[d, x]
A(a)
A
[d, b]
[d, a] F (x)
F (x) = f (x) A = A(b) = F (a) + C
A(x) = F (x) + C
C
A(a) = (F (b) + C)
(F (a) + C)
F (b)
C = F (b)
f (x) A = A(b)
F (a).
C A f (x)
[a, b]
f (x) x=b
x=a
A(a)
f (x) = x2 + 1 [0, 2]
[ 1, 3] 2
f (x) = x + 1 Z
x2 + 1 dx =
1 3 x +x+C 3
F (x) = x3 /3 + x
x2 + 1
[0, 2]
1 3 F (0) = (2) + (2) 3
F (2)
1 3 2 (0) + (0) = 4 . 3 3
[ 1, 3]
1 3 F ( 1) = (3) + (3) 3
F (3)
1 1 ( 1)3 + ( 1) = 13 . 3 3
f !x"!x2 "1
y
y
f !x""x2 #1
8 15 6 10 4 5
2
0.5
1.0
1.5
2.0
2.5
[0, 2]
R
x !2
!1
1
2
3
[ 1, 3]
dx
f (x) = x2 + 1 [0, 2]
x
f !x"!x2 "1, n!4
y 6
6
5
5
4
4
3
3
2
2
1
1 0.5
1.0
f !x"!x2 "1, n!8
y
1.5
2.0
x
0.5
f !x"!x2 "1, n!16
y 6 5 4 3 2 1
0.5
A = lim
n!1
n X
1.0
x
2.0
f (xk ) x.
k=1
n f (xk )
1.5
x k
1.0
1.5
2.0
x
y = f (x) Z Z
[a, b]
b
f (x) dx, a
n X
b
f (x) dx = lim
n!1
a
Rb
•
a
Z
a
f (x) dx
a
f (x) dx
Rb
• f (x) dx =
f (x) dx
f R
R
x
b
f (xk ) x,
k=1
a
•
a
b
Rb
Z
f
b
f (t) dt = a
f (x)
f (x) b
S
dx
x
R
x Z
dx
a
x
f (r) dr. a
Rb a
[a, b]
f (x) dx
•
Rb a
Rb a
f (x) dx f (x)
f (x) dx
f (x) Rb a
f (x) dx Rb a
f (x) dx
n
f (x) dx
b
[a, b] f (x)
[a, b]
f Z
[a, b]
b
f (x) dx = F (b)
F (a)
a
F
F 0 (x) = f (x)
f
f (x) = x2 + 1
[0, 2]
A=
=
Z
Z
2 2
(x + 1) dx = 0
1 3 (2) + (2) 3
b a
0
1 3 2 (0) + (0) = 4 . 3 3
b
f (x) dx = [F (x)]a = F (b)
F (x) Z
2
1 3 x +x 3
F (a),
f (x) Z
1
x3 dx 2
Z
1
x3 dx = 2
4
p
x dx =
0 3 2 = (4) 2 3
p
x dx
0
1 4 x 4
1 = (1)4 4 Z
4
2 3 x2 3
2
2ex dx 0
1 2
1 1 ( 2)4 = 4 4
Z
1 (16) = 4
4 0
3 2 2 1 (0) 2 = (8) = 5 . 3 3 3
3 3 . 4
Z
2 0
2ex dx = [2ex ]20 = 2e2
2e0 = 2(e2
1).
f (x) Z
4
p
Rb a
f (x) dx
f (x) [a, b]
16 . 3
x dx =
0
f "x#!x1!2 y 2.5 2.0 1.5 1.0 0.5
A!16!3 1
2
3
4
5
x
"0.5
f (x) Z
1
15 . 4
x3 dx = 2
f (x) x=
2
x=1
x
R0
x Z
0 3
x dx = 2
1 4 x 4
1 4 (0) 4 Z 1 1 4 3 x dx = x 4 0
0
R1
1 ( 2)4 = 0 4
1 (16) = 4
2
x dx
R1 0
x3 dx
4,
1 0
1 4 1 (0) = 4 4
0=
1 . 4 B=4
3
x3 dx
2
=
1 = (1)4 4
2
x
B
A = 1/4 A
f !x""x3 y 2
!2.0
!1.5
!1.0
0.5
!0.5
x
1.0
!2 !4 !6 !8
•
f (x)
f (x) [a, b]
•
f (x)
•
f (x)
[a, b] [a, b] x
Rb a
Rb
f (x) dx
f (x) dx a f (x) [a, b] [a, b]
x [a, b] f !x"!3#x y 4
3
2
1
1
2
f (x) = 3/x Z 4 3 dx 1 x
Z
4 1
3 4 dx = [3 ln x]1 x
3
4
5
x
Rb a
f (x) dx x
= 3 ln 4
3 ln 1 = 3 ln 4 = 6 ln 2.
6 ln 2 [1, 4]
x f (x) = 1 Z
x
2
3
1
x2
dx
1
f !x""1 ! x2
y 2
0.5
1.0
1.5
2.0
2.5
3.0
x
!2 !4 !6 !8
Z
3
1 1
x2
dx = x
= (3) 6 23
1 3 x 3
3 1
1 3 (3) 3
1 3 (1) = 3
1
[1, 3]
6
2 = 3
2 6 . 3
x
f (x) = x2 Z
4
3
x2 0
4 dx
f !x""x2 ! 4
y 6 4 2
0.5
1.0
1 3 x 3
4x
1.5
2.0
2.5
x
3.0
!2 !4
Z
3
x2
4 dx =
0
=
1 3 (3) 3
3 0
4(3)
1 3 (0) 3
4(0) =
3
0=
3.
3 f (x) = x2
4
[0, 2]
[2, 3]
f (x) Z
2
x2
4 dx =
0
3
x2 2
1 3 x 3
2
4x 0
1 3 1 3 (2) 4(2) (0) 3 3 3 1 3 4 dx = x 4x 3 2 1 3 1 3 = (3) 4(3) (2) 3 3 =
Z
4(0) =
4(2) =
5
3
1 3
0=
✓
1 5 3
1 5 , 3
◆
1 =2 . 3
1 1 2 5 +2 =7 . 3 3 3
f f (x) dx a a b Rb
[a, b]
x=2
f (x) R 30 0
f (t)
f (t) dt
30t2 ,
v(t) = 40t t
v
s
s(3)
s(0) =
⇥ = 20t2
Z
10t3
3
v(t) dt = 0
⇤3 0
= 20(3)2
Z
3
40t
30t2
dt
0=
90.
0
10(3)3 90
x •
[a, b]
A(x) [a, x]
[a, x]
x
f (x)
A0 (x) = f (x). f (x) •
f (x) Z
•
f (x) = F (x) + C. Rb a
Z
b a
f (x) dx b
f (x) dx = [F (x)]a = F (b)
F (x) • [a, b] A=
F (a) f (x)
A Z
x
x
b
f (x) dx. a
f (x)
•
a
x
• a
Rb
b
Rb a
f (x) dx f (x) x
f (x) f (x) dx
[1, 3]
[ 2, 0]
[ 3, 1]
6 4 2
1
!1
2
3
x
4 3 2 1
f (x) = 2e
1
!1
2
3
f (x) = x + y
y 4 3 2 1
f (x) = 2e
1
!1
2
3
f (x) = 2 +
0.3x
y
y
!2
!1
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
x
x
2 x
x
x
3 x2
8
6 5 4 3 2 1 !3
3
6 5 4 3 2 1
5
!2
2
3 f (x) = 0.5x + p x
x
0.2x
!3
1
6 5 4 3 2 1
5
!2
p f (x) = 1 + 3 x y
f (x) = 0.2x2 + 2
!3
[1, 4]
8
y 35 30 25 20 15 10 5
!2
f (x) dx
[a, b]
f (x) = 3x2 + 1
!3
a
x
[1, 2] [0, 2]
Rb
[a, b]
6 4 2 1
2
3
x
x
Z
Z
3 2
(3x + 2) dx 0
Z Z
1
4x2 + 5x dx 1 9 1
Z Z
Z
t
2
⌘
dt
2e 2 5 1
3r
dr
Z
1 dx x2
4 1 e 1
Z
2 p dx x ✓
2 3x + x
◆
dx
Z
4 dx
3x4
5x2 + 2 dx
1 1 8
⇣p
t+1
3e
0.05r
3
1
2
x M C(x) = 84
0.9x + 0.008x2
0 x 200
3 dx x4 27 1 e 1
x
3 p dx 3 x2
✓
3 x
4x
R2 1
x2 ) dx
dr
2 dx x3
1
8/x3 ) dx
dt
3
R3
5/x2 ) dx
⌘
1
1
Z
2 dx x2
2x3
(1
0
Z Z
0
4x2 ) dx
2
Z
1
Z Z
⇣p
R2
M C(x) = 20
◆
p 0.5 x 0 x 100
dx
(2/x) dx v(t) = 0.4 + 0.2t t (4/x) dx
R2 1
R2 1
R3 0
0.02t2 0 t 10
t=0
(1
(1
v(t) = 1.2 t t=0
(4
0.06t + 0.005t2 0 t 12
v(t) = 44
4.4t,
t M R(x) = 30
0.2x 0 x 100 x
V 0 (t) = r(t) = 200e0.04t 0 t 10
v(t) = 44 + 2.2t, t
t $1,000,000
V 0 (t) = r(t) = 200e M C(x) = 20
0.02x 0 x 300 x
0.04t
0t7
t $1,500,000
M P (x) = 30
0.02x1/3 0 x 100 x
M C(x) = 18
0.05x 0 x 100
P (0) =
300
x
M P (x) = 40 x M R(x) = 90
0.5x 0 x 120
p 0.02 x 0 x 100 x
P (0) =
400
v(t) = 54t t
15t2 ,
v(t) = 60t
24t2 , v
t v
x
[a, b] [a, b]
Rb a
a0
2x g(x) = x2 p f (x) = 3 g(x) = x + 1 p f (x) = 4 g(x) = x 5 f (x) = 4
x0 x>0
8 < 2 0.5x f (x) = : 0.2x2 + 2 8 < 1 + x2 f (x) = : 1+x
f (x) = 4 g(x) = x2
3 3x
2 g(x) = x2 + 1
x0
f (x) =
x>0
f (x) = x
1 g(x) = x2
x0 x>0 f (x) = x + 2x2
x3 g(x) = 2 y
[0, 6]
4 3
8 < 2 x f (x) = : 0.2x2 + 0.8 8 < 1 + x2 f (x) = : 10 8 < 4 f (x) = : x2
8 < x2 + 3 f (x) = : 1 + 3x
2
g
2
x1 x>1
x3 x>3
1 !2
!1
1
2
3 f
!1
x
!2
f (x) = 2 +
p
x g(x) = 4
x
y
x2 x>2 x2 x>2
5 4 3 2 1 0 !1 !2
f
1
2
3
4 g 5
x
g(x) = x2
x2 + 6x + 2
6x + 12 f (x) =
y 15
5 0
2
3
4
g(x) = 0 f (x) = x2
5
y = x3
y = 4x
y=x y=x p y= x y=3 p y= x y=2
f 1
y = x2
3
g
10
y = x3
6
x
1
y=4
x2
y=4
x2
y=x 4
x=0
y = x2
4
y=4
4x
2
y =x +1 x=
y = x2 y = x2
f
3
2
x=0
1
x=2
2 x=0
x=2
2 1 !3
!2
!1
f 1
2
3
x
!1
f (x) = ex
!2
f (x) = e f
f (x) = 4
g
f (x) = x2
x
x2 2x + 1
f (x) = 2x + 1 f (x) = x + 1 g(x) = x2 + 3 f (x) = x2
2x
f (x) = 2x
1 g(x) = 1
[ 2, 2]
2 g(x) = 5
[ 1, 2]
x
[ 2, 3]
f (x) = x 1 g(x) = 2 [ 3, 1] p f (x) = x + 2 g(x) = x + 3 [1, 4] f (x) = ex + 1 g(x) = 1 x p f (x) = x + 2 g(x) = x 3 f (x) = 4 g(x) = 1
e
x
0.5x
3 x2 2 f (x) = 2 x p f (x) = x + 1 p f (x) = 4 x f (x) =
[0, 2] [1, 4] [ 1, 2]
2 f (x) = g(x) = 14 x2 x
[1, 4]
4 g(x) = 12 x x2
[1, 5]
f (x) =
f (x) = 2
t vA (t) = 0.4 + 0.2t vB (t) = 0.2 + 0.25t
0.02t2 0 t 10
0.02t2 0 t 10
t vA (t) = 0.4 + 0.2t
y=x y = x2 p y=8 x y = x2 y=
2x
y=3
vB (t) = 0.1 + 0.3t x2
0.02t2 0 t 5
0.02t2 0 t 5
s1 (t) = 0.6t + 4.5 t v(t) = t
1 3 180 t
+ 18 t2 + 50, 0 t 10, v
s2 (t) = 7.2e0.06t
$5,000
s1 (t) = 0.6t + 6.2 t
s2 (t) = 8.8e0.05t
$6,000 t 0.9t
f (t) = 80e
t f (t) = 150e
0.6t
f (t) =
v(t) = t
1 3 180 t
+ 18 t2 + 50, 0 t 15, v
1 3 3 2 t + t 200 20
3 t+210, 0 t 30 8
S(t) = 500e0.15t 0 t 10
t=2 f (t) =
1 3 t 160
3 2 1 t + t + 285, 0 t 30 20 4
t=7
t=1
r(t) = 200e0.04t 0 t 10
S(t)
t=6
t
t
S(t) = 800e0.1t 0 t 12 r(t) = 200e
R(t)
t
p f (x) = x x2 + 1
Z
0.04t
x 3x2 + 2 dx
Z
f (x) = ln x
p
x(2x + 1) dx
Z
x3 + 1
2
dx
0t7
t
x 3x2 + 2 3x3 + 2x Z Z 3 2 x 3x + 2 dx = 3x3 + 2x dx = x4 + x2 + C. 4 p
Z
x(2x + 1) 2x3/2 + x1/2 Z ⇣ ⌘ p 4 5 2 3 x(2x + 1) dx = 2x3/2 + x1/2 dx = x 2 + x 2 + C. 5 3 x3 + 1
Z
3
x +1
Z
Z
x6 + 2x3 + 1 dx =
Z
Z ✓
2 3x + x
x+1 p dx = x
6x
p
Z
x+1 p dx x
4 dx e2x
◆
dx =
3 2 x + 2 ln |x| + C. 2
p x 1 1 p + p = x+ p , x x x Z ✓
p
1 x+ p x
4 4e 2x e2x Z Z 4 dx = 4e 2x dx = e2x
Z
1 7 1 4 x + x + x + C. 7 2
3x2 2 2 + = 3x + , x x x
3x2 + 2 dx = x
x+1 p x Z
dx =
x6 + 2x3 + 1
3x2 + 2 dx x
3x2 + 2 x Z
2
2
x2 + 1 dx ?
◆
dx =
2e
2x
p 2p 3 x + 2 x + C. 3
+ C.
p 6x x2 + 1
F (x) = 2
p (x2 + 1)3 = 2 x2 + 1
3/2
,
F 0 (x) = 2
3 2 x +1 2
Z
q 3 + 1 dx = 2 (x2 + 1) + C.
6x
p
x2
1 2
· 2x = 6x x2 + 1
1 2
= 6x
p
x2 + 1.
F (x)
Z
1 ur+1 + C r 6= 1 r+1 Z Z 1 1 1 du = ln |u| + C du = ln u + C u>0 u u u u>0 R u e du = eu + C ur du =
u u
du =
x
u
du · dx = g 0 (x)dx dx
u = g(x)
y = f (u)
u = g(x)
dy dy du du = · = f 0 (u) . dx du dx dx
y = F (x) = f (g(x)),
F 0 (x) = f 0 (u)g 0 (x),
F 0 (x)dx = f 0 (u)g 0 (x)dx = f 0 (u)du. f (u)
F (x)
u = g(x)
du = 2x dx
u = x2 + 1 Z
6x
Z
p 3 u du = 3
p
x2 + 1 dx =
Z
Z
Z
6x
p x2 + 1 dx x2 + 1
du = 2xdx
u
2xdx
Z p p 3 x2 + 1 2x dx = 3 u du.
p 1 2 3 u 2 du = 3 · u 2 + C = 2 u3 + C. 3
u = x2 + 1 q p 3 2 u3 + C = 2 (x2 + 1) + C. Z q p 3 2 6x x + 1 dx = 2 (x2 + 1) + C.
u
du u
u
e u
u u
dx
x
du x
u = x3 + 5 Z 3 x3 + 5 dx =
du = 3x2 dx
Z
3x2
=
1 1 4 u + C = (x3 + 5)4 + C. 4 4
x3 + 5
3
Z
3x2 dx =
Z
Z
3x2 x3 + 5 Z
3
dx
u3 du
3ex dx ex + 2
u = ex + 2 du = ex dx Z Z 3ex 1 1 x dx = 3 e dx = 3 du ex + 2 ex + 2 u
= 3 ln u + C = 3 ln (ex + 2) + C.
Z u = x4 + 1
du = 4x3 dx
4 du = 4x3 dx
1 du = x3 dx. 4
x3 (x4 + 1)
2
dx x3 dx
du
Z
x3
Z
1 3 2 2 x dx (x4 + 1) (x4 + 1) Z Z 1 1 1 1 = du = du 2 u 4 4 u2 ✓ ◆ 1 1 1 = +C = + C. 4 u 4 (x4 + 1) dx =
u=
0.03x2
du =
Z
xe
0.03x2
0.06x dx
dx x dx
0.06 du =
Z
xe
0.03x2
dx =
Z
e
0.03x2
1 du = x dx. 0.06
x dx
Z 1 1 du = eu du 0.06 0.06 1 u 1 0.03x2 = e +C = e + C. 0.06 0.06 =
Z
1 du = x dx, 0.06
0.06x dx
eu
u = ln x Z
ln x dx = x
Z
=
1 2 1 u + C = (ln x)2 + C. 2 2
ln x
du =
1 dx = x
Z
1 dx x
ln x dx x
p
x2
u du
Z
4 0
1 du = x dx 2 Z Z x 1 p p dx = x dx 2 2 x +9 x +9 Z Z 1 1 1 1 p p du = du = 2 u 2 u p p 1 p = 2 u + C = u + C = x2 + 9 + C. 2
du = 2x dx
Z
x +9
dx Z
p
x dx x2 + 9
u = x2 + 9
Z
4
p
0
x
dx =
x2 + 9
p = 42 + 9
p
hp i4 x2 + 9 0
p
02 + 9 =
p
25
u = x2 + 9 u = 25
x=4
9=5
3 = 2. 1 du = x dx 2
du = 2x dx
x=0 x
u=9
u Z
4
p
0
= =
Z
x x2
25 9
dx =
+9
Z
4 0
1 1 1 p du = 2 u 2 25
1 p 2 u 2
=
p
p
Z
1 x2
25 9
25
+9
x dx
1 p du u
p
9=5
3 = 2.
9
Z Z
3
3x2 x3 + 2
Z
0
3x2 x3 + 2
3x2 x3 + 2
3
4
( 1)3 + 2
x=0 u=2 u Z 0 Z 3 3x2 x3 + 2 dx = 1
=
2
u3 du = 1
=
1 [16 4
1] =
1 4 u 4
2
= 1
0
x3 + 2
3
15 . 4
Z
x3 (x4 + 1)
1 0
dx =
2
x3
1
1⇥ 4 2 4
⇤ 1 14 = [16 4
(x4 + 1)
2
1 + C. 4 (x4 + 1)
dx =
du = 3x2 dx
3x2 dx
Z Z
dx
1 3 4 x + 2 + C. 4 0 1 3 4 dx = x +2 4 1
u = x3 + 2
Z
3
1
dx =
1
1 = (0 + 2)4 4
0
1 4 4 (x + 1)
1 0
1 0
1] =
15 . 4
x3 (x4 + 1)
2
dx
x= x
1
u=1
1 4(1 + 1)
=
1 = 4(0 + 1)
1 1 1 + = . 8 4 8 u = x4 + 1
u=1 Z
1
Z
(x4
0
=
Z
x
Z
x3
2 1
2
p
+ 1)
2
x=1 u Z 1 dx = 0
u=2 1 (x4
✓ ◆ 1 1 1 1 du = u2 4 4 u
3
4x + 3x
+ 1) 2 1
5 dx
Z
1) dx
Z
x + 2 (3x
2
1 = 4
1 2
( 1) =
x(2x + 1) dx
2x4
1
2
2x3 4 dx x2
Z
1 + 2x3 dx x
Z
2 dx e3x
Z
e2x + 2 dx ex
Z
Z
ln u
2x x2 + 1 2
6
4x 2x + 3
dx 4
dx
x
1 1 1 · = . 4 2 8
Z p
x=0
x3 dx
Z
u>0
1 du = x3 dx 4
du = 4x3 dx
dx
6x2 2x3
Z
3x2 x3
Z
⇣p p 3 3 x x4
5 2
8
5
dx dx
Z
⌘3 p ⇣p 3 x x +2 dx
Z
xe3x dx
⌘4 1 dx
2
Z Z
3
2x2 e4x dx
Z
4x3 dx x4 + 1
Z
1 dx 3x + 4
Z
Z
p 3 (2 x + 3) p dx x
Z
p 5 (3 x 2) p dx x
Z
ln 3x dx x
Z
3(ln x)2 dx x
4x3 (x4 + 1) 1 2x
3
2
dx
dx
Z
p
4 dx 4 x
Z
Z
3ex dx x (2e + 1)3
Z
Z
x
Z
p
Z
p
3 dx 2x + 1 4e
x
x
3e
dx
2
+ 4 dx
Z
x2
e2x dx e2x + 1
Z
2 dx x ln x + 4
p
x2
1 (3x + 5)
2
Z
dx
p
x3 + 2 dx
p
3 (2x
3)
p
2
dx
Z
ln x4 dx x
Z
Z
e
2x+1
Z
3e
Z
2ex/3 dx
Z
3ex/2 dx
Z
e x p dx x
Z
p
Z
1 dx x ln x
Z
1 dx x ln x3
Z
2 2/x e dx x2
Z
1 3/x2 e dx x3
Z Z
dx
p
ln
x
x
dx
3x 2
dx
1 p dx xe x
6
3x2 + 2x + 1
x3 + x2 + x + 1
6x2 + 8x + 5
2x3 + 4x2 + 5x + 1
Z ✓
◆ p ⌘3 p ⇣ 2 x 1 + x3 + 2xe3x dx
Z ⇣
x2
p
1 + x3 + 4xe
3x2
⌘
dx
dx 5
dx
Z
1
Z Z Z Z Z Z Z Z
(x2 + 1)
0
Z
x
dx
3
1
x x
2
1
5
dx
1
9 1
p
t 2 p t
1
2
dt
0
1 e 1 4 1
3 0 2
2 1
Z
Z
1 dx 3x + 4 ln x2 dx x
Z
2 p p 2 dx x ( x + 2)
Z
x p dx 2 x + 16
Z
1
(2x2 + 1)
0 2
x x2
1 dx (3x + 4)2
Z
2
dx
5
1
0 x 200
x
dx
C(x)
p 3
8 1
t+1 p 3 2 t
1
3
dt V 0 (t) = r(t) = 200te
3
3r2 er dr 0 2 1 e
2 dx 2x + 1 ln
p
x
x
1 27
p 3
1
p
0 t 10
x3 0 2
x2
$1,000,000 V (t) t
dx
3 p 2 dx 3 ( x + 1)
1 5x
1
1
0.04t2
t
V 0 (t) = r(t) = 100te
0.01t2
0t7
t 1
dx $1,500,000 V (t)
p x4 + 1 dx
t
2 dx (2x + 1)2 M P (x) = p
R 60 20
M C(x) = p
p 0.02x 0.0001x2 + 1
1
2
Z
M C(x) = 40
8x
10
p 3x x3 + 1 dx 2
0
Z
Z
2
2rer dr 2
Z
40x 0 x 100 + 100
x2
x
P (x)
C(x)
40x 15 x 100 2x2 400 x
M P (x) dx
0.2p D0 (p) = p 260,000 p2
p x = D(p)
Z p
4x2 + 5 dx
Z p p xp 2 a2 x2 ± a2 dx = x ± a2 ± ln x + x2 ± a2 + C. 2 2 x2
4
◆ Z p Z s ✓ 5 4x2 + 5 dx = 4 x2 + dx 4 r
x2 +
x2
5 5 + + ln x + 4 4
Z
2
=x
r
=
5 dx 4
a2 =
5 4
" r # r x 5 5/4 5 2 2 =2 x + + ln x + x + +C 2 4 2 4 r
x2 +
5 + C. 4
a2 = 5/4
Z
3 2x2
7x
dx = 3
Z
1 dx. x(2x 7)
Z
a 3 2x2 x
7x
dx
Z
xn dx =
Z
eax dx =
Z
1 ax e +C a
xeax dx =
Z
1 · eax (ax a2 xn eax a
xn eax dx =
Z
ln x dx = x ln x
n a
xn ln x dx = xn+1 ax dx = Z Z Z Z Z Z Z Z Z Z Z
1
p
1 ± a2 1
xn
1 ax
e
dx + C
n
ln x n+1
Z
(ln x)n
1
dx + C n > 1
1 + C n 6= (n + 1)2
1
1 x a ln +C 2a x+a p dx = ln |x + x2 ± a2 | + C
dx =
a2
p
Z
ax + C a > 0 a 6= 1 ln a
x2 x2
1) + C
x+C
(ln x)n dx = x(ln x)n
Z Z
1
1 dx = ln x + C x > 0 x
Z
Z
1 xn+1 + C n 6= n+1
x a2 ± x2
dx =
x a dx = 2 a + bx b
1 a+ ln a
x b
p
a2 ± x2 +C x
a ln |a + bx| + C b2
x a 1 dx = 2 + 2 ln |a + bx| + C 2 (a + bx) b (a + bx) b 1 1 x dx = ln +C x(a + bx) a a + bx 1 1 1 x dx = ln + +C x(a + bx)2 a(a + bx) a2 a + bx p p xp 2 a2 x2 ± a2 dx = x ± a2 ± ln x + x2 ± a2 + C 2 2 p 2 x a + bx dx = (3bx 2a)(a + bx)3/2 + C 15b2 Z p p 2 n n 3/2 x a + bx dx = x (a + bx) na xn 1 a + bx dx + C b(2n + 3) p x 2 p dx = 2 (bx 2a) a + bx + C 3b a + bx
Z
1 1 x dx = ln + C. x(a + bx) a a + bx a=
Z
3 2x2
=3
Z
7
✓
7x ◆ 1 ln 7
1 x2
b=2 Z dx = 3
a2
x +C = 7 + 2x
1 x2
Z
1 dx x( 7 + 2x)
3 x ln + C. 7 2x 7 Z
5
dx
p 1 x 5 p + C. dx = p ln 5 2 5 x+ 5 Z
Z
1 x2
1 x a ln + C. 2a x+a p a= 5
dx =
a2 = 5 Z
1 dx = 3 x(2x 7)
ln x dx = x ln x
ln 4x dx
x + C. x
Z
ln 4x dx =
1 4
Z u
Z
ln u du =
ln u
u = 4x
1 1 du = 4 4
1 [u ln u 4
Z
ln u du.
u] + C =
u ln u 4
u + C. 4
4x
Z
1 ln 4x dx = 4
=
u ln u 4
Z
ln u du
u + C = x ln 4x 4
x + C. Z
Z
n
(ln x) dx = x(ln x)
n
n
Z
(ln x)n
1
(ln x)2 dx
dx + C
du = 4 dx
1 4
du = dx
n=2 Z (ln x)2 dx = x(ln x)2 Z
2
Z
(ln x)1 dx + C.
(ln x)2 dx
= x(ln x)
2
2
= x(ln x)2 = x[(ln x)
Z
(ln x)1 dx + C
2[x ln x
2
x] + C
2 ln x + 2] + C. Z
Z
n ax
x e
n=2 Z x2 e =
Z
xn eax dx = a a=
4x
1 2 x e 4
dx = 4x
xeax dx =
a= 4 Z x2 e
+
n a
Z
dx
=
1 2 x e 4
4x
+
=
1 2 x e 4
4x
+
=
1 2 x e 4
4x
1 ax
dx + C
x1 e
4x
e
4 x2 e
4x
2 4
4 1 2
Z
x1 e
4x
1 · eax (ax a2
4x
xn
x2 e
1 2
Z
x1 e
1 1 ·e 2 16
Z
dx + C
dx + C
1) + C
4x
4x
1 (4x + 1)e 32
dx + C ( 4x 4x
+ C.
1) + C
4x
dx
Z Z Z Z Z Z Z Z Z Z
Z Z
2xe
3x
2 0.5x
x e
Z
dx
Z
dx
Z
ln(5x) dx
Z
ln(7x) dx x ln(2x) dx x ln(3x) dx 2x ln x2 + 1 dx 3x2 ln x3 + 2 dx
5x dx 2x 3 1 dx x x2 + 4 p
3 p dx x 25 + x2 Z p 4x2 9 dx Z p 2x2 + 5 dx p
1
x2 + 4
Z Z Z Z Z
x dx 3x + 4
Z
Z
Z
dx
Z
p
2 dx 4x2 + 1
2x dx x4 9 3 x2
16
dx
x dx (3x + 4)2 x dx (5 3x)2 ln x dx x3 ln x dx x2 2 3x2
5x 3
5x2
2x
dx dx
4 p dx x 4 x2 p x 1
3 4x2
dx
p 3x 2x + 1 dx
Z
p 2x 5x
Z
p x2 x + 1 dx
4 dx
Z
p x2 x + 4 dx
Z
p
x 2x
p
x dx x+4
Z
1
dx
R
Rb a
f (x) dx F 0 (x) = f (x) f Z
b
[a, b]
b
f (x) dx = [F (x)]a = F (b)
a
f (x)
x
[a, b]
F (x) + C F
f
F (a).
[a, b]
x •
f (a) dx
[a, x] Z
[a, x]
A(x) [a, x]
f (x) b
f (a)dx a
x
f (x)
A0 (x) = f (x). f (x) •
f (x) Z
•
f (x) = F (x) + C. Rb a
Z
b a
f (x) dx b
f (x) dx = [F (x)]a = F (b)
F (x) • [a, b]
•
a
Z
x
f (x) dx. a
Rb a
b
Rb a
f (x)
b
x
•
F (a) f (x)
A
A=
x
f (x) f (x) dx
f (x) dx f (x) x
[a, b] x [a, b]
Rb a
f (x) dx
y
y
6
6
y = f (x)
A= a
0
RAb a
f (x)dx
y = f (x)
A(x) A0 (x) = f (x)
- x
b
a
0
- x
x
f !x"!3#x y 4
3
2 fav 1
1
f
2
=
3
1 b
a
Z
4
5
x
b
f (x) dx a
y y
6
6
f
y = f (x)
A Rc a
0
Z
a b
f (x) dx = a
Rb
f (x) dx
c
c Z
c
f (x) dx + a
b Z
g
f (x) dx
- x
0
- x a
b
f (x) dx c
A=
b Z
b
[f (x) a
g(x)] dx
Z Z
k dx = kx + C
1 xr+1 + C r+1
xr dx =
Z
k r 6= Z
1 dx = ln |x| + C x
Z
aekx dx =
Z
[f (x) ± g(x)] dx =
a kx e +C k Z Z kf (x) dx = k f (x) dx
Z ✓
2x
2e
Z ✓
Z
1 + p 3 x2
f (x) dx ±
4 x
1 4 2e + p 3 2 x x Z Z = 2 e 2x dx + x 2 e 2
= =
e
2x
2x
+
1
2 3)
(
+ 3x
1 3
dx,
2
◆
dx
+1
dx
4
2 3 +1
x
Z
x>0
g(x) dx
◆
6x + 3 2 3
1 dx = ln x + C x
2
6x + 3
2x
Z
1
x>0
1 dx x
4 ln x
6
Z
x2 dx +
1 6 x3 + 3x + C 3
Z
3 dx
2x3 + 3x + C.
4 ln x
C F (x) F 0 (x) = f (x) = 2e
e
2(1)
1
+ 3(1) 3
C F (x) =
2x
1 + p 3 x2
e
2x
1
+ 3x 3
Z
0
p
F (1) = 4,
2
4 ln x
4
6x2 + 3
2(1)3 + 3(1) + C = 4.
4 ln(1) C=e
4 x
f 2x3 + 3x + e
x + 1 dx
2
.
Z
4
p
x + 1 dx =
0
=
3 2 (4) 2 + 4 3
4
2 3 x2 + x 3
0
3 2 2 1 (0) 2 + 0 = (8) + 4 = 9 . 3 3 3
f (x)
x
[a, b] f !x""x2 ! 4
y 6 4 2
0.5
1.0
1.5
2.0
2.5
3.0
x
!2 !4
f (x) = x2 Z 3
4 x2
4 dx
0
Z
3
(x
2
4) dx =
0
=
1 3 (3) 3
4(3)
3
1 3 x 4x 3 0 1 3 (0) 4(0) = 3
3
0=
3.
3 f (x) = x2
4
[0, 2]
f (x) Z
3
x2
4 dx =
0
3
x2 1
1 3 x 3
2
4x 0
1 3 1 3 (2) 4(2) (0) 3 3 3 1 3 4 dx = x 4x 3 2
= Z
4(0) =
5
1 3
0=
1 5 , 3
[2, 3]
x=2
=
1 3 (3) 3
4(3)]
1 [ (2)3 3
4(2) =
3
✓
5
1 3
◆
1 =2 . 3
1 1 2 5 +2 =7 . 3 3 3
Z
5ex dx 3ex + 2
u = 3ex + 2 du = 3ex dx Z Z 5ex 1 dx = 5 ex dx 3ex + 2 ex + 2 Z Z 1 1 5 1 =5 · du = du u 3 3 u =
1 3 du
= ex dx
5 5 ln u + C = ln (ex + 2) + C. 3 3
R
Rb a
v(t) = 80t t
f (x) dx
3t2 , v
e
x2
dx
s
s(1)
s(0) =
⇥ = 40t2
t3
Z
⇤1 0
1
v(t) dt = 0
= 40(1)2
Z
1
80t
3t2
dt
0
(1)3
0 = 39. 39
$2,000
A(t) = 2,000e0.055t . A(t)
A
=
1
=
1
0
Z
1
2,000e0.055t dx 0
2,000 0.055t e 0.055
1
= 0
2,000 ⇣ 0.055(1) e 0.055
⌘ 2,000 e0.055(0) = e0.055 0.055
1 ⇡ $2,056.02. 0.015(2056.02) ⇡ $30.84
M C(x)
A x
x x
MC
y
6 M C(x)
0
- x
x y
y
6 y = f (x)
v(t) = 0.4 + 0.2t t t=0
-x
0
0.02t2 0 t 10
p y = 8 x
x M C(x) = 20
p 0.5 x
y = x2
0 x 100
s1 (t) = 0.6t+4.5 t
s2 (t) = 7.2e0.06t
Z
6x5 + 8e3x
Z ✓ Z Z
9 1 e 1
4 dx
3
1 4 p + 3+ 5 3 x x x
⇣p
✓
◆
Z
dx x > 0
⌘ 2 dt
t
3 x
4x
◆
dx
f 0 (x) = 3e2x
f (x) f (0) = 2
f (x) = 4 x [0, 2]
Z
[1, 2]
[ 2, 2]
3
4 0
x2
dx.
2
p
x(2x + 1) dx
Z
2x3 4 dx x2
Z
6x2 2x3
Z
2
4x3 dx x4 + 1
Z
Z
5
xe3x dx
Z
Z
5
3ex (2ex + 1) 1
x3 0 1 0
p
dx
3
x4 + 1 dx
8x (2x2
+ 1)
2
dx
dx
Z Z
2xe
3x
dx
ln(5x) dx
Z
2x 5x
Z
x2 4x3 + 3x
Z
x2 2x3 + 7
p
4 dx
Z
x dx 3x + 4
Z
ln x4 dx x
Z
ln x dx x3
Z
xe
Z
2xe
Z
1 dx 3x + 4
4x2
dx
$5,000
6
5 dx
dx
4x
dx
A x
y
x
y P (0) = 0
y
6 y = f (x)
A = 180 - x
0
Z
3x4
Z ✓
Z
Z
9 1
e 1
2e3x + 2 dx
3
1 4 p + 3+ 5 3 x x x
⇣p ✓
t
4x
⌘ 2 dt
3 x
◆
dx
◆
dx x > 0
0.3x2
f (x) = 2 [ 1, 2]
[0, 1]
Z
y = x2
3
4
x2
dx
0
4
y =x+2
Z Z
x x2
6x x2
Z p Z
Z
4
2
dx
15 dx 5x
x2 + 4 dx
x (2
x)2
dx
(ln x)2 dx
Z
t2 e
Z
(ln x)2 dx x
t3
dt
$3,000
x M C(x) = 30
p 0.2 x 0 x 100
v
t
t = 4.5
Z v
4.5
v(t) dt 0
6
v(t)
15 10 5
0 5 10 15
1
2
3
4
t
p 6
pE
p = S(x)
t
xE , p E
p = D(x)
- x 0
xE
x = D(p)
x = S(p) E(p)
x = D(p)
p = D(x) p = D(x) p = S(x)
pE
qE qE
p 6
p = S(x)
r
pE
xE , p E p = D(x)
- x
0
xE
p = D(x) p 66
x 0
p 10
10
30
30
60
60
90
90
100
p = D(x)
5 4 3 2 1
0
20
40
60
80
- x
100
5(10) + 4.5(20) + 4(30) = $260. 4(60) = $240. x0 [0, x0 ] p0 p0 ⇥ x 0
x0
p0 p = p0
[0, x0 ]
p 6
p0 p = D(x) 0
- x
x0
p0 Rb
[a, b]
a
p = D(x) Z
(x0 , p0 )
x0
D(x) dx. 0
=
Z
p0 ⇥ x 0 x0
D(x) dx 0
p = D(x) = (x x=3
f (x) dx
(p0 ⇥ x0 ) .
5)2
x=3 p0 = 4
x0 = 3 =
=
Z
D(x)
3
5)2 dx
(x 0
4⇥3=
Z
p=4
3
x2
10x + 25 dx
12
0
3
x3 3
5x2 + 25x
12 = (9
45 + 75)
0
12 = 27.
0
p = D(x) =
14.4 ln(0.07x + 0.018)
x
p
x0 = 2 p = D(2) =
14.4 ln(0.07(2) + 0.018) ⇡ 26.57. p0
$26,570
p0 ⇥ x0 ⇡ 26.57( Z
Z Z
x0
D(x) dx = 0
= $53.14 (
2
14.4 ln(0.07x + 0.018) dx. 0
ln x dx = x ln x
x + C.
1 u = 0.07x + 0.018 du = 0.07 dx 0.07 du = dx Z Z 1 14.4 14.4 ln(0.07x + 0.018) dx = 14.4 ln u du = ln u du. 0.07 0.07
14.4 0.07 u Z
Z
⇥ 2(
Z
ln u du =
14.4 [u ln u 0.07
u] + C =
14.4 u(ln u 0.07
0.07x + 0.018
14.4 14.4 ln(0.07x + 0.018) dx = 0.07
Z
ln u du
1) + C.
.
14.4 u(ln u 1) + C 0.07 14.4 = (0.07x + 0.018)[ln(0.07x + 0.018) 0.07 =
Z
2
14.4 ln(0.07x + 0.018) dx = 0
=
1440 [(0.158)(ln 0.158 7 73.9
1)
1] + C.
14.4 (0.07x + 0.018)(ln(0.07x + 0.018) 0.07 1)] ⇡ $73.90 (
(0.018)(ln 0.018
53.14 = $20.76
2
1) 0
.
.
p !thousand $" 70 60 p!D!x"
50 40 30 20 10 0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
x !million cars"
p = S(x) p x
6
p
p = S(x)
4
0
3
200 300 400
2 1
0
100
x = 200p
3.50(300) = $1,150.
200
400
300
400
- x
500
2 + 3.5 · 300 = $825. 2 $1,150 p
p = S(x)
6
p0
- x
x0
0
p0 x0 [0, x0 ] p0 p0 ⇥ x 0
x0
p0 p = p0
[0, x0 ]
p = S(x) Z
(x0 , p0 )
x0
S(x) dx. 0
(p0 ⇥ x0 )
Z
p0 ⇥ x 0 x0
S(x) dx. 0
p = S(x) = x2 + 2x + 1 x=3 x0 = 3 = 16 ⇥ 3
x=3 p0 = 16 Z
S(x)
3
x2 + 2x + 1 dx = 48 0
p = 16
x3 + x2 + x 3
3 0
= 48
(9 + 9 + 3
0) = 27.
p = S(x) = 1,500e0.04x x
p 2,500 2,500 2,500 2,500
x0 = 25 p = S(2) = 1,500e0.04(25) ⇡ $4,077
.
p0
$4,077 2,500 2,500
p0 ⇥ x0 ⇡ $40.77
Z
Z
x0
S(x) dx = 0
=
1,500 0.04x e 0.04
⇥ 2,500
= $101,925.
25
1,500e0.04x dx 0
25
= 0
1,500 h 0.04(25) e 0.04
i e0 ⇡ $64,435.6. 2,500
$101,925
$64,435.6 = $37,489.4
.
p !$" 6000 p!S!x"
5000 4000 3000 2000 1000 0
0
5
10
15
20
25
30
x !hundred cell phones"
2,500
(xE , pE ) pE ⇥ x E
[0, xE ]
p 6
p = S(x)
t
pE
xE , p E
p = D(x)
- x 0
p = D(x) = (x
xE
5)2 ;
p = S(x) = x2 + 6x + 4,
(xE , pE )
D(x) = S(x) 5)2 = x2 + 2x + 1,
(x
24 = 12x,
x2
10x + 25 = x2 + 2x + 1,
xE = 2.
x=2
S(x)
pE = S(xE ) = 22 + 2(2) + 1 = 9. (2, 9) x=2 =
=
x3 3
Z
2
(x
5)2 dx
0
(9 ⇥ 2) =
2
5x2 + 25x
18 = 0
✓
8 3
Z
p=9
2
x2
10x + 25 dx
18
0
20 + 50
◆
0
2 18 = 14 . 3
x=2 Z
=9⇥2 = 18
✓
2
x2 + 2x + 1 dx = 18 0
◆ 1 0 =9 . 3
8 +4+2 3
p = D(x) = 6.5
x3 + x2 + x 3
p=9
2 0
0.004x
x
p
p = S(x) = 0.005x + 2. (xE , pE )
S(x) = D(x) 0.005x + 2 = 6.5
0.004x,
x = 500
S(x)
0.009x = 4.5,
xE = 500.
pE = S(xE ) = 0.005(500) + 2 = $4.5. (500, $4.5) 1 (500)(4.5 2
2) = $625.
1 (500)(6.5 2
4.5) = $500. 625 + 500 = $1,125 p !$" 7 p!S!x"
6 5 4
p!D!x"
3 2 1 0
0
100
200
300
400
500
600
700
x !chocolate bars"
$0.25 (xE , pE )
p p p
0.25
0.25 = 0.005x + 2. p
p = S(x) = 0.005x + 2.25. S(x) = D(x) 0.005x + 2.25 = 6.5 x = 472
0.004x,
0.009x = 4.25,
S(x)
pE = S(xE ) = 0.005(472) + 2.25 = $4.61. (472, $4.61)
1 (472)(4.61 2
1 (472)(6.5 2
2.25) = $556.96.
4.61) = $446.04.
556.96 + 446.04 = $1,003 p !$" 7 p!S!x"
6 5 4
p!D!x"
3 2 1 0
0
100
200
300
400
500
600
700
x !chocolate bars"
xE ⇡ 472.
(xE , pE )
p (1.09)p
9%p (1.09)p = 6.5
0.004x. p
p = D(x) ⇡ 5.9633
0.00367x. S(x) = D(x)
0.005x + 2 = 5.9633 x = 457
0.00367x,
0.00867x = 3.9633,
S(x)
pE = S(xE ) = 0.005(457) + 2 = $4.29. (457, $4.29)
1 (457)(4.29 2
2) = $523.27.
1 (457)(5.9633 2
4.29) = $382.35.
523.27 + 382.35 = $905.61 p !$" 7 p!S!x"
6 5 4
p!D!x"
3 2 1 0
0
100
200
300
400
500
600
700
x !chocolate bars"
xE ⇡ 457.
1 (400)(4 2
1 (400)(6.5 2
2) = $400.
4.9) + (400)(4.9
4) = 320 + 360 = $680.
400 + 680 = $1,080 p !$" 7 p!S!x"
6 5 4
p!D!x"
3 2 1 0
0
100
200
300
400
500
600
700
x !chocolate bars"
p0 p 6
S(x)
p1 p1
p0 p0 D(x) 0
- x
p1 p1
p 6
12
S(x) = 2 + 0.2x
10 8 6 D(x) = 12
4
0.24x
2 0
- x 10
20
30
p
40
50
x
6 S(x)
p1 p0
D(x)
- x
0
D(x) = 360
3x x = 80
D(x) = 800
0.8x x = 200
0.8x x = 400 p D(x) = 50 x + 4 x = 21 p D(x) = 40 x + 5 x = 11 D(x) = (x
S(x) = 3 + 0.16x
10 8 6 D(x) = 11
4
0.22x
2 0
3x x = 40
D(x) = 800
p 6
12
D(x) = 360
-x 10
20
30
40
50
6)2 x = 2
D(x) = 1600
0.08x2 x = 120
D(x) = 1200
x2 x = 30
D(x) = (x
90)2 x = 40
D(x) = 300e
0.02x
x = 100
D(x) = 200e
0.04x
x = 80
D(x) = 360e
0.01x
x = 200
D(x) = 540e
0.02x
x = 150
p=8 x p = 8 S(x) = 700 + 1.5x x = 200 S(x) = 160 + 0.4x x = 60 S(x) = 120 + 0.1x x = 100 S(x) = 40 + 0.08x x = 200 p S(x) = 4 x + 5 x = 4
p S(x) = 6 x + 16 x = 9 S(x) = 20 + 0.01x2 x = 30 S(x) = 40 + 0.03x2 x = 20 S(x) = x2 + 2x + 8 x = 3 S(x) = x2 + 4x + 1 x = 2
x = D(p) = 1, 500
S(x) = 20e0.02x x = 40 S(x) = 30e
0.01x
20p,
p x
x = 120
S(x) = 10e0.02x x = 100 S(x) = 40e0.01x x = 80
(xE , pE ) p x
D(x) = 320
3x S(x) = 120 + 5x
D(x) = 368
0.4x S(x) = 17 + 0.5x
D(x) = 94
0.25x S(x) = 24 + 0.75x
D(x) = 300
1.7x S(x) = 20 + 0.3x
D(x) = 220
0.04x2 S(x) = 26.4 + 0.06x2
p = D(x)
D(x) = 90
0.01x2 S(x) = 15 + 0.02x2
D(x) = (x
4)2 S(x) = x2 + 2x + 6
D(x) = (x
6)2 S(x) = x2 + 4x + 4
D(x) = 6
x
D(x) = 8 x
p x p
0 x 6 S(x) = x + 6 p 0 x 8 S(x) = 2x + 8
p = D(x)
p 81 D(x) = p S(x) = x + 1 x+1
p 200 D(x) = p S(x) = 2 x + 3 x+3
45p + 50x = 3, 000,
p = D(x) = 200
0.1x,
p = S(x) = 20 + 0.05x
p x
p
x
p = D(x) = 112
0.04x,
p = S(x) = 0.06x + 42 p
x
t B(t) = 1, 000e0.06t . $5,000 P (5) = 5,000e
0.06(5)
⇡ $3,704.09.
$1,000 3
⇣ r ⌘nt B(t) = P0 1 + . n
P0 = 1,000 r = 0.06 n = 365 t=3 ✓ ◆365⇥3 0.06 B(3) = 1,000 1 + ⇡ $1,197.20. 365
B(t) = P0 ert . P0 = 1,000 r = 0.06
t=3
B(3) = 1,000e0.06(3) ⇡ $1,197.22.
$7,300 7,300/365 = $20
20e0.05(10) . 20e0.05(10
1/365)
.
20e0.05(10
2/365)
.
20e0.05(10
9 364 365 )
= 20e0.05(1/365) .
20e0.05(10) + 20e0.05(10
1/365)
+ 20e0.05(10
2/365)
. . . 20e0.05(10
t = 1/365 n X
7,300e0.05(10
(k 1)/365)
t=
k=1
n = 10 · 365
20 = 7,300 t
f (tk ) t.
7,300e0.05(10
tk )
tk = (k
1)/365 n
10
7,300e0.05(10
t)
dt.
0
du = Z
.
k=1
f (tk ) Z
n X
1 = 365 t
9 364 365 )
0.05 dt
du/( 0.05) = dt
10
7,300e
0
146,000 [eu ]0.5
0
u
u = 0.05(10 t) t = 10 u=0 u
u = 0.5
1 7,300 dt = 7,300e du = 0.05 0.05 0.5 ⇥ ⇤ = 146,000 1 e0.5 ⇡ $94,713.
0.05(10 t)
0
=
Z
t=0 t
Z
0
eu du 0.5
$94,713
R(t) r%
RT
T
0
R(t)
t r% T
Z
T
R(t)er(T 0
t)
dt.
T R(t)er(T
t)
dt
R(t)
=
S rT e r
S
1 .
R(t) = S Z
T
Ser(T
du =
dt
T
Ser(T
t)
dt =
0
=
u = r(T
1 du = dt r
r dt Z
t)
0
S ⇥ 0 e r
Z
t = 0
u = rT t
0
Seu rT
⇤ S ⇥ erT = 1 r
1 S du = r r
t = T
u = 0 u
Z
0
eu du = rT
⇤ S rT erT r = e r
S u0 [e ]rT r
1 .
$2,000 T = 18 1 = 17 S = 2,000 2,000 ⇣ 0.05(17) e 0.05
r = 0.05
⌘ 1 ⇡ $53,586.
$1,000,000
S r = 0.06
1,000,000 = S
S ⇣ 0.06(47) e 0.06
67 T = 47
⌘ 1 = S(262.95).
S = 1,000,000/262.95 ⇡ 3803. $3,803
3,803/52 = $73.14 R(t)
20 = 47 $1,000,000
t)
R(t) = 3,600 + 180t
Z
T
R(t)er(T
t)
dt.
0
r = 0.06 T = 47 Z
t
R(t) = 3,600 + 180t
47
(3,600 + 180t)e0.06(47
t)
dt .
0
u = 0.06(47 t = 47
t)
du =
u=0 t = 47
t
1 du = dt 0.06 3,600 + 180t
0.06 dt
t=0
u = 2.82 u
3,600 + 180t = 3,600 + 180(47
u/0.06) = 12,060
3,000u.
t Z
47
(3,600 + 180t)e0.06(47 Z
dt =
Z
2.82
201,000
2.82
eu du =
1 · eax (ax a2
1) + C.
0
ueu du = 50,000 [eu (u 2.82
⇥ = 50,000 e0 ( 1) $3,171,150 1.6
1) + C
⇤ e2.82 (1.82) ⇡
0
1)]2.82 $1,576,690.
$1,576,690 = $1,594,450,
1 du 0.06
0
eu du + 50,000 2.82
e2.82 ⇡ $3,171,150.
201,000 1
2.82
a=1 x=u Z ueu du = eu (u Z
Z
0
xeax dx =
50,000
3,000u)eu
(12,060
( 201,000 + 50,000u)eu du =
Z
u
0
0
201,000
Z
t)
0
=
u = 0.06(47
u/0.06
Z
0
ueu du. 2.82
t)
$3,650 3,650/365 = $10
10e
0.06(1/365)
.
10e
0.06(2/365)
.
10e
0.06(3/365)
.
10e
0.06(10)
10e
0.06(1/365)
.
+ 10e
0.06(2/365)
+ 10e
0.06(3/365)
3,650e
0.06(k/365)
t=
k=1
.
f (tk ) t.
k=1
f (tk ) Z
n X
0.06(10)
n = 10 · 365
t = 1/365 n X
. . . 10e
3,650e
0.06tk
10
3,650e
0.06t
dt = 3,650
0
3,650 h = e 0.06
0.06(10)
e0
i
n 1 e 0.06
3,650 ⇥ = 1 0.06 r% RT R(t)e 0
R(t)
10 0.06t 0
e
0.6
⇤
⇡ $27,447. R(t) T
rt
dt
t r%
T Z
T
R(t)e 0
rt
dt.
R(t)
=
D 1 r
rT
e
D
.
R(t) = D Z
T rt
De
dt =
0
D ⇥ = e r
2,000 ⇣ 0.05
rT
1
e
D e r
⇤ 0
D⇥ = 1 r
T rt 0
e
rT
⇤
.
T = 17 D = 2,000 ⌘ e 0.05(17) ⇡ $22,903.
r = 0.05
P0 = 22,903
$22,903 $53,586
22,903e0.05(17) = $53,586.
D = 300(12) = 3,600 r = 0.12 3,600 ⇣ 0.12
T =5 1
e
0.12(5)
⌘
= $13,535.7.
$250,000
D $250,000
250,000 =
D ⇣ 1 0.05
e
0.05(30)
⌘
= D(15.5374).
r = 0.05
T = 30
D D = 250,000/15.5374 ⇡ 16,090. $16,090
16,090/12 = $1,340.85
1,340.85 + 150 = 1,490.85
250,000 =
17,890.2 1 0.05
0.05T
e
1,490.85(12) = 17,890.2
. T
250,000 = 357,804 1 0.69871 = e
0.05T
T =
e
250,000 =1 357,804
=1
0.05T
e
0.05T
0.69871 = 0.301297
ln 0.301297 ⇡ 24 . 0.05 $16,090
16,090(30) = $482,700 17,890.20(24) = $429,364.80
$17,890.20
$53,335.20
$150 $840,000 $100,000 $100,000
T = 90
P = 100,000e
0.04(25)
$36,788
D
D ⇣ 1 0.04
e
$840,000 = 803,212 r = 0.04
0.04(25)
⌘
= D(15.8).
D = 803,212/15.8 = 50,836. $50,836 $5,000
r = 0.04
⇡ $36,788.
$803,212
803,212 =
65 = 25
$4,236
$36,788 = T = 25
D = 5,000(12) = 60,000 r = 0.04 60,000 ⇣ 1 0.04
e
0.04(50)
⌘
T = 50
= 1,296,997.
$1,296,997
S
= 1,296,997 r = 0.06
1,296,997 = S
S ⇣ 0.06(40) e 0.06
⌘ 1 = S(167.053).
S = 1,296,997/167.053 ⇡ 7,764. $7,764
7,764/12 = $645
T = 65
25 = 40
$2,000 5%
$680,000 $8,000 6%
$600,000
4%
4.5% $760,000
$2,500 5%
$80,000 $4,000 5.5%
R(t) = 3,600 + 160t t
$200,000 4.5% R(t) = 3,200 + 220t $100,000
t
4.2%
$160,000 $580,000 $2,500
$8,000
$26,000 $3,800 $5,000
$220,000
$360,000
$300
$540,000
$50,000 $38,000 $680,000
$80,000
$3,000
$80,000
$4,500
$100,000
R1 a
f (x) dx
Rb
R1
f (x) dx 1
f (x) dx 1
$1,000,000
D T 1,000,000 =
Z
T 0.05
De
dt =
0
D 1 0.05
e
0.05T
. T
D 1 T !1 0.05 lim
0.05T
lim e
T !1
lim
T !1
D 1 0.05
1,000,000 =
e
0.05T
=
D lim 1 0.05 T !1
0.05T
=
D . 0.05
= 0,
e
D 0.05
D D = 1,000,000(0.05) = 50,000. $50,000
f (x) =
1 x1.1
g(x) =
1 x0.9
[1, 1) Z
b
f (x) dx 1
Z
b
g(x) dx. 1
,
e
0.05T
.
[a, b] ( 1, b] [a, 1)
( 1, 1)
Z
Z
b
f (x) dx = 1
b
g(x) dx = 1
Z
Z
b
x1.1
1
b 1
1
1 x0.9
dx =
dx =
1 x 0.1
1 0.1 x 0.1
b 0.1
= 1
b 1
⇥ 10 b
⇥ = 10 b0.1
0.1
⇤ 1 = 10
⇤ 1 .
b!1
b!1
⇥ lim 10 b0.1
10 = 10 b0.1
lim 10
b!1
f (x) [1, 1)
g(x)
2.0
1.5
f !x"! 1#x1.1
1.0
0.5
0.0
0
2
4
6
8
10
12
14
10
12
14
x
y 2.0
1.5
g!x"! 1#x0.9
1.0
0.5
0.0
0
2
4
6
8
⇤ 1 = 1.
x
[1, 1)
10 , b0.1
Z
1
f (x) dx = lim
b!1
a
Z
1
Z
Z
b
f (x) dx a
b
f (x) dx = lim 1
f (x) dx =
1
a! 1
Z
c
f (x) dx + 1
Z
Z
b
f (x) dx a
1
f (x) dx
c
Z
c Z
1
c
f (x) dx 1
f (x) dx
c
Z
1
1 x1.1
1
Z
dx = 10
1 1
f (x) f (x)
Z
1 dx x0.9
lim
b!1
1
1 dx x0.9
1
Z
b
f (x) dx a
f (x) Z
1
f (x) dx
a
Z
1
x1.2 dx
5
x1.2 ! 1
x ! Z 11 x1.2 dx
x
f (x) = x1.2 6! 0
x ! 1
5
Z
1
e
0.2x
Z
dx
0
1
Z
ln x dx
1
1 1
Z 1 4 1 dx dx x3 x 2 Z b e 0.2x dx 0
Z
b
e 0
0.2x
dx =
1 e 0.2
b 0.2x
= 0
⇥ 5 e b
0.2b
⇤ e0 = 5
5e
0.2b
. e0.2b
e
0.2b
= 1/e0.2b
0 lim
b!1
Z
b
e
0.2x
b 1
0
Z
4 dx = x3
lim
b!1
Z
b 2
Z
5e
b!1
Z
Z
dx = lim 5
b 1
1
e
0.2x
b!1
b 2
5 f (x) = ln x 6! 0
x!1
ln x dx
1
Z 4 1 2 x2
b
= 1
✓ 4 dx = lim 2 b!1 x3
1 b dx = [ln x]2 = ln b x
Z
dx
ln x ! 1
2
b 1
4 dx x3
1 b2
1 =2 12 b!1
◆ 2 = 2. b2 Z 1 4 dx x3 1 Z b 1 dx x 2 ln 2 = ln b
1 dx = lim (ln b b!1 x
ln 2) = 1. Z
1 2
1 dx x
2 . b2 2/b2 ! 0
2
ln 2. b
lim
= 5.
0
x!1
1
0.2b
ln b
R(t)
t 100r%
Z R(t) Z
1
rt
R(t)e
dt.
0
R(t) = D 1
De
rt
dt =
0
D . r
D 100r% Z
1
De
rt
dt = lim
T !1
0
Z
T rt
De
D⇥ 1 T !1 r
dt = lim
0
e
rT
⇤
=
$5,000
D = 5,000(12) = 60,000 r = 0.04 60,000 = 1,500,000. 0.04 $1,500,000 = 1,500,000 r = 0.06
1,500,000 = S
S T = 40
S ⇣ 0.06(40) e 0.06
⌘ 1 = S(167.053).
S = 1,500,000/167.053 ⇡ 8,979.19. $8,979
8,979/12 = $740.27
D . r
y = 3/x4 Z Z Z Z Z
1 1 1 1
2 dx x2
Z
2 dx x
Z
1
1 x1.4
1 1 4 1
dx
1 p dx x 2e
2x
dx
0
Z Z Z Z Z
1 0 1 0 1
Z Z Z
1 1 0
xex dx
2e
2x
dx
1 9
Z
Z Z Z
4 dx x2 2x dx 2 (x + 1)2
Z
x5 dx
Z
1
1
1
Z
3
1
1
Z
x1.4 dx
1
2
1 dx x+1
0
Z
1
2x dx x2 + 1
0 1
Z
2
Z
1 1
Z
1
x dx
Z
[2, 1)
5 dx x2
y = 1/x1.5 [1, 1)
5 dx x
y = 2xe
y = 2x/ y=0
3 p dx x3 3e
0.5x
V 0 (t) = r(t) = 200e 3x2 dx x3 + 2
0 1
5 dx x+2
0
p
(x2 + 16)3 x = 3
dx
0 1
0.04t
t $1,000,000 V (t) t
1
x0.8 dx
1 1
x2 ex dx
0 1
3e0.05x dx
V 0 (t) = r(t) = 100e
0.01t
t
0
$1,500,000 2 1
1 dx x4
1
(x2
0
4x dx + 4)3
1
x4 dx 1 1 1
x3 dx
x = 0
y=0
1 dx x0.8
1
0.2x2
V (t) t
$800,000
$500,000
$4,500
$3,000
p 0p1 P(
P(
) = 0.5
P(
) = 0.5
) = P(
) = 0.5
) = 487/1,000 = 0.487 P(
t P (0 t 12) = 1 1/3
1 P (0 t 4) = 3
f (t) =
1 12 ,
0 t 12 1/3 = P (0 t 4)
- t 0
4
12
f (t) = 1/12, 0 t 12 t x
x f (x)
0
f (x) x f x
P (c x d) =
Z
[c, d] d
f (x) dx. c
t 1 f (t) = , 0 t 12 12
t f
f (t) Z
12
f (t) dt = 0
Z
12 0
0
1 1 dt = t 12 12
t
12
1 (12) 12
= 0
[0, 12] 1 (0) = 1. 12
f
P (9 t 12) =
Z
12
f (t) dt = 9
Z
12 9
1 1 dt = t 12 12
12
= 9
1 (12) 12
1 3 (9) = = 0.25. 12 12 25%
g(x)
A
f (x)
g(x)
f (x) = kg(x)
k = 1/A
t 0 t 12
f (t) = kt,
k
R 12 0
f (t)
[0, 12]
f (t) dt = 1 1=
Z
12
f (t) dt = 0
Z
12
kt dt = 0
k 2 t 2
12
= 0
k [144 2
0] = k · 72.
k = 1/72 f (t) =
1 t, 72
0 t 12 .
P (9 t 12) = =
1 (144) 144
Z
12
f (t) dt = 9
Z
12 9
1 1 2 t dt = t 72 144
12 9
1 63 (81) = ⇡ 0.44. 144 144 44% p f (t) = k t, 1 t 4 k
t t
R4 1
f (t)
[1, 4]
f (t) dt = 1 1=
Z
4
f (t) dt = 1
Z
4 1
p 2 3 k t dt = k t 3 p
4
=k 1
2 hp 3 4 3
p i 14 13 = k · . 3
k = 3/14 f (t) =
3p t, 1 t 4. 14
P (3 t 4) = =
1 hp 3 4 7
=
7
23
4 3
Z
2 1
4 3
3p 3 2p 3 t dt = t 14 14 3
1t2 2 1
p i 13 ⇡ 0.2612.
P (1.6 t 2.4) = =
3t4
p i 33 ⇡ 0.40.
P (1 t 2) = 1 hp
Z
3p 3 2p 3 t dt = t 14 14 3
1 hp (2.4)3 7
p
Z
2.4 1.6
3p 3 2p 3 t dt = t 14 14 3
2.4 1.6
i (1.6)3 ⇡ 0.242.
[a, b] f (x) =
1 b
a
, a x b
f (x) =
1 b
a,
a x b.
- x a
b
x
x
[0, 6]
f (x) =
f (t) =
1 8
f (t)
1 = ,0 t 8 0 8
0t2
P (0 t 2) =
0.25 Z
2 0
1 1 dt = t 8 8
2
= 0.25. 0
2t3 P (2 t 3) =
0.125 Z
3 2
1 1 dt = t 8 8
3
= 0.125. 2
1 b
a
axb
b
a 2
kx
f (x) = ke Z
1
kx
ke
dx = lim
b!1
0
= lim
b!1
⇥
k e k
b
0 Z
f
b
ke
kx
dx
0
kx 0 kb
= lim ( e b!1
, x
)
y
⇤ ( 1) = 1.
2.0 1.5 f !x"! 2e"2 x
1.0 0.5 0.0 0.0
0.5
1.0
1.5
2.0
k k
x t
k f (x) =
1 e 150
x/150
k f (t) = 2.5e
2.5t
, x
k = 1/150 0.
k = 5/2 = 2.5 , t
0.
2.5
3.0
x
k = 3/2 = 1.5 f (t) = 1.5e
1.5t
P (0 t 1) = =
1.5 e 1.5
1
Rb a
1.5 e 1.5
Z
0.
1
f (t) dt = 0
1 1.5t
= 0
P (0 t 2) = =
, t
Z
e
f (t) dt = 0
= 0
h
e
1
1.5e
Z
1.5t
dt
0
i e1.5(0) = 1
1.5(1)
2
2 1.5t
h
Z
e
1.5
e
3
⇡ 0.777.
2
1.5e
1.5t
dt
0
1.5(2)
i e1.5(0) = 1
⇡ 0.950.
0.95 = 0.05 = 5%
[a, b] f (x) dx
1 [1, 4] 3 1 f (x) = [0, 5] 5 1 f (x) = x [0, 2] 2 1 f (x) = x [1, 3 4 f (x) =
f (x) =
1 2 x [0, 3] 9
f (x) =
1 2 x [1, 4] 21
f (x) =
3 [1, 3] 2x2
f (x) =
4 [1, 4] 3x2
f (x) =
1 [1, e] x
f (x) =
1 [e, e2 ] x
f (x) =
3 2 x [ 2, 2] 16
f (x) =
3 2 x [ 1, 1] 2
f (x) = 0.2e f (x) = 3e
0.2x
3x
t
[0, 1) t
[0, 1)
f (x) =
3 [1, 1) x4
f (x) =
24 [2, 1) x4 k
f (t) =
k f (x) = 2 [1, 2] x f (x) =
t
k [2, 4] x2
1 t, 0 t 10 . 50
t
f (x) = kx [1, 5] f (x) = kx [0, 3]
f (t) = k(30
f (x) = kx2 [ 1, 2]
t), 0 t 30 . k
f (x) = kx2 [ 2, 1] f (x) = kex [0, 2] f (x) = ke2x [0, 1] t
f (t) = kt2 , 0.5 t 2, t t t f (t) =
1 , 0 t 10 . 10
k
x
t 0
12
6 µ µ
x
f (x)
- x µ
x x
f (x) [a, b] •
x µ
•
x
µ=
Z
b
xf (x) dx a
x
=
s Z
a
s
µ
x
b
(x
µ)2 f (x) dx =
Z
f
b
x2 f (x) dx a
µ2
f1 (x)
f2 (x)
-x
µ
- x
µ
µ f (x) 2
= Z
=
Z
=
Z
b
µ)2 f (x) dx =
(x a
b
x2 f (x) dx
2µ
a b
x2 f (x) dx
Z
Z
b
x2
b
xf (x) dx + µ2 a
2µ · µ + µ2 · 1 =
a
1 f (t) = , 12
t
f (t) > 0 Z
µ=
=
Z
12
tf (t) dt = 0 2 12
1 t · 12 2
= 0
12
t2 f (t) dt = 0
=
1 t3 · 12 3
=
s Z
12
= 0
Z
t 0
12
t2 0
f (x) dx a
x2 f (x) dx
µ2 .
a
t 0 t 12
f
1 dx 12 ⇤ 0 = 6.
⇤ 0 = 48.
12 0
b
b
1 dx 12
1 ⇥ 3 12 36
t2 f (t) dt
Z
Z
[0, 12]
12
1 ⇥ 2 12 24
Z
2µx + µ2 f (x) dx
a
µ2 =
p 48
62 =
p
12 ⇡ 3.46.
[0, 12]
3p f (t) = t, 1 t 4 14
µ=
Z
3 14
=
4
tf (t) dt = 1
Z
Z
4
t3/2 dt = 1
4
t 1
3 h 5/2 = (4) 35
(1)5/2 =
Z
Z
4 2
t f (t) dt = 1
3 = 14 =
Z
4
t
5/2
1
3 h 7/2 (4) 49
=
s
Z
4
t2 1
4 1
3 (31) ⇡ 2.657. 35
3p t dx 14
3 2 7/2 dt = t 14 7
4 1
i 3 (1)7/2 = (127) ⇡ 7.7755. 49
4
t2 f (t) dt 1
t
3p t dx 14
3 2 5/2 t 14 5 i
t
µ2 =
p
7.7755
µ = 1/k
(2.657)2 ⇡ 0.846.
f (t) = ke = 1/k
kt
, 0 k2
f (t) = kt2 , 0.5 t 2, t k
t t 1 0 t 10. f (t) = 10
t
t
w
w
t µ = 11,000 = 2,500
µ = 9,600 = 1,500
x2
x=
2y = 0
y = Ce2x
2, 2
( 2)2 (2)2
y0
4=0
(Ce2x )0
4=0
= C(2e2x )
4=0
v(t) = 2t + 1, s(t) 0
s (t) = v(t) = 2t + 1.
s(t) =
Z
v(t) dt =
Z
2(Ce2x )
v(t) (2t + 1) dt = t2 + t + C.
2Ce2x = 0
x
r
dP = rP. dt P 0 (t)
P (t) P (t) = P0 ert , P 0 (t) = P0 rert = r P0 ert = rP (t).
dy 2x = 2 , y 6= 0. dx 3y dx
dy
dy/dx y
x dy 2x = 2 dx 3y dy = 2x dx 3y 2 dy = 2x dx. 3y 2
Z
3y 2 dy =
Z
2x dx
y 3 = x2 + C,
y=
p 3 x2 + C.
y=
p 3
x2 + C
dy 2x = 2 , y 6= 0. dx 3y y y
y0
dy d p d 1/3 3 = x2 + C = x2 + C dx dx dx 1 2 2x 2/3 = x +C (2x) = . 2/3 2 3 (x + C) 2x 2x 2x = p = . 2/3 3y 2 3( 3 x2 + C)2 (x2 + C)
•
dy/dx = f (x)g(y) y0 = x + y
•
y=
p 3
x2 + C
x
y
C
dy = 4xy dx
y(0) = 3 dy = 4xy dx 1 dy = 4x y dx 1 dy = 4x dx. y Z
1 dy = y
Z
4x dx
ln |y| = 2x2 + C1 ,
|y| = e2x
2
+C1
2
= e2x · eC1 .
C 2 = e C1 2
|y| = C2 e2x ,
2
y = ±C2 e2x .
±C2
C = ±C2 2
y = Ce2x . x=0
y=3
C
2
3 = Ce2(0) = Ce0 = C. C=3 2
y = 3e2x .
y>0 C
y
y0 =
3x2 , y 6= 0, y y(1) = 2 y
0
dy/dx
dy 3x2 = dx y y
dy = 3x2 dx
y dy = 3x2 dx. Z
y dy =
Z
3x2 dx
1 2 y = x3 + C, 2 C = 2C1 p y = 2x3 + C
y 2 = 2x3 + 2C1 .
y=
p 2x3 + C.
x=1
y=2
C 2=
p 2(1)3 + C
4 = (2)2 = 2 + C, C=2 p y = 2x3 + 2. y0
y0
x + 2xy = 0
y0 = x
2xy
dy = x(1 dx
2y). x
dy = x(1 dx 1 Z
2y)
dy = x dx. 2y dy = 1 2y
Z
x dx.
y
x + 2xy = 0
y(1) > 0
u=1
2y
du =
2dy
1 du = dy 2
Z
Z dy 1 = dy 1 2y 1 2y ✓ ◆ Z Z 1 1 1 1 = du = du u 2 2 u =
1 1 ln u + C = ln(1 2 2
1 ln(1 2
2y) =
2y) + C.
1 2 x + C. 2
y 1 ln(1 2 ln(1
2y) =
1 2 x +C 2
x2
2y) = 2
2C
2y = e x 2C = e ✓ ◆ 1 1 2C y= + e e 2 2 1
x2
·e
x2
.
2C
C
y=
1 + Ce 2
x2
.
P0 t dP = 0.04P. dt
$2,000
dP = 0.04P dt dP = 0.04 dt P Z Z dP = 0.04 dt P
ln P = 0.04t + C1 P = e0.04t+C1 = eC1 e0.04t = Ce0.04t .
P = Ce0.04t . t = 0 P = P0 P0 = Ce0.04(0) = Ce0 = C.
P (t) = P0 e0.04t .
P0 = 2,000
t=2
P (2) = 2,000e0.04(2) ⇡ $2,166.57. T T dT = dt
k(T
C
C). T (0) = T0
dT = dt dT T Z
C
=
dT T
ln(T T
k(T
C
C)
k dt.
=
Z
C) =
k dt kt + C1
kt+C1
C=e
= e C1 e
kt
= ae
kt
,
a = e C1 T = C + ae
kt
. T (0) = T0
T0 = C + ae a = T0 T (t) = (T0
k(0)
= C + ae0 = C + a.
C C)e
kt
+ C.
C
E(p) = 2
D(2) = 3
x = D(p) E(p) pD0 (p) . D(p)
E(p) =
D0 (p) = dx/dp
x = D(p)
E(p) = 2
p dx = 2. x dp
p dx =2 x dp dx dp = 2 x p Z Z dx dp = 2 x p ln x =
2 ln p + ln C = ln p ln C
x=
2
+ ln C = ln
✓
C p2
◆
C
C . p2
D(2) = 3 3=
C , 22
C = 12 x = D(p) =
12 . p2
y = xe
y=x y=e
1/x x
y = e2x y=e y=
3x
1 x+1
xy 0 + y = 2x
y0 + y = 0 y 00 + y 0 y 00 + y 0
6y = 0 6y = 0
y0 + y2 = 0
x
y 00 + 2y 0 + y = 0
dy x = dx y dy = 3x2 y dx 4y 3
dy = 4x dx
dy y = dx x
dy 2x = dx y dy = 4x3 y dx 2x2 y
dy =1 dx
dy 2y = 2 dx x
dy ex = 3 dx 4y
dy 2e2x = 2 dx y
E(p) =
p 100 p
D(50) = 500
dy 3 = dx y
dy 4 = dx y
E(p) =
2p 400 p
D(300) = 50
dP = 0.02P dt
dP = dt
0.04P $10,000
y 0 = 4x + xy y 0 = 3x
xy
y0
2y = 0
xy
P
y 0 + 3x + xy = 0
dy x = dx y
dP = 0.04P dt
y=2
dy 2x = dx y
x=1
y=5
dy = 3x2 y dx
x=2 $30,000
y(0) = 3
dy = 4x3 y y(0) = 2 dx dy 3 = 2 dx y
dP = 0.05P dt
P 2,000
y(0) = 2
dP = 0.02P dt dP = dt
$2,000
y(0) = 1
dy 4 = 2 dx y
800
0.04P
P (0) = 100 P (0) = 200
y 0 = 4x + xy
y(0) = 5
y 0 = 3x
y(0) = 2
xy
$24,000
x = D(p) $50,000 E(p) = 1 E(p) = 0.5
p>0 p>0
E(p) =
2 p
D(2) = 5e
E(p) =
5 p
D(5) = 3e
D(2) = 3 D(4) = 6
x0
p0 [0, x0 ]
p = p0
p = D(x) (x0 , p0 ) Z
x0
(p0 ⇥ x0 ) .
D(x) dx 0
x0 p0 [0, x0 ]
p = p0
p = S(x) (x0 , p0 ) Z x0 (p0 ⇥ x0 ) S(x) dx. 0
(xE , pE ) pE ⇥ x E [0, xE ]
p 6
pE
0
p = D(x) = 6.5
0.004x
p = S(x)
r
xE , p E p = D(x)
- x
xE
x
p
p = S(x) = 0.005x + 2. (xE , pE )
S(x) = D(x) 0.005x + 2 = 6.5
0.004x,
x = 500
0.009x = 4.5,
xE = 500.
S(x)
pE = S(xE ) = 0.005(500) + 2 = $4.5. (500, $4.5)
4.5(500)
5
00(0.005x + 2) dx = 2,250 0
[0.0025(500)2 + 2(500)] = 2,250
= 2,250
Z
Z
500
(6.5
0.004x) dx
0
⇥ = 6.5(500)
⇥ 4.5(500) = 6.5x
0.002(500)2
⇤
2,250 = 2,750
⇥
0.0025x2 + 2x 1,625 = $625.
0.002x2
⇤500 0
⇤500 0
2, 250
2,250 = $500.
R(t) r% Z
T
T
R(t)er(T
t)
dt.
0
R(t) Z
T
Ser(T 0
t
t)
dt =
S rT e r
S
1 .
$2,000 T = 18 1 = 17 S = 2,000 2,000 ⇣ 0.05(17) e 0.05
r = 0.05
⌘ 1 ⇡ $53,586.
$1,000,000
S r = 0.06
1,000,000 = S
S ⇣ 0.06(47) e 0.06
67 T = 47
20 = 47 $1,000,000
⌘ 1 = S(262.95).
S = 1,000,000/262.95 ⇡ 3,803. 3,803/52 = $73.14
R(t) r% Z
T
T
R(t)e
rt
dt.
0
R(t) Z
T
De
rt
0
dt =
D⇥ 1 r
e
D rT
⇤
.
D = 300(12) = 3,600 r = 0.12 3,600 ⇣ 1 0.12
e
0.12(5)
⌘
= $13,535.7.
T = 5
t
R(t)
t r%
Z R(t) Z
Z
Z
1
1
rt
R(t)e
dt.
0
R(t) = D 1
De
rt
D . r
dt =
0
f (x) dx = lim
b!1
a b
f (x) dx = lim
Z
a! 1
1
Z
1
b
f (x) dx a
Z
b
f (x) dx a
f (x) dx =
1
Z
c
f (x) dx + 1
Z
1
f (x) dx
c
Z
c
Z
1
e
0.2x
Z
dx
0
1
Z
ln x dx
0
1 2
1 dx x Z b e
0.2x
Z
c
f (x) dx 1
1
f (x) dx
c
dx
0
Z
b
e
0.2x
dx =
0
1 e 0.2
b 0.2x
= 0
⇥ 5 e0.2b
⇤ e0 = 5
5e
0.2b
. e0.2b
b 0 lim
b!1
Z
b
e
0.2x
dx = lim 5
5e
b!1
0
Z Z
x!1
1 0
ln x dx
1
e
0.2b
0.2x
= 5.
dx
5
0
ln x ! 1
f (x) = ln x 6! 0
x!1
e
0.2b
= 1/e0.2b
Z
Z b 2
1 b dx = [ln x]2 = ln b x
b
1 dx x
2
ln 2 = ln b
ln 2. b
lim
b!1
Z
b 2
1 dx = lim (ln b b!1 x
ln b
ln 2) = 1. Z
1 2
1 dx x
x f (x)
f (x)
0
x f
P (c x d) =
x Z
[c, d] d
f (x) dx. c
g(x) x
A
x f (x) = k · g(x)
k = 1/A µ
µ
x x
f (x) [a, b] • •
x
µ
x
µ=
Z
b
xf (x) dx a
x
=
s
Z
b
(x a
µ)2 f (x) dx =
s
Z
b
x2 f (x) dx a
µ2
f t
p f (t) = k t, 1 t 4
t
k
R4 1
f (t)
[1, 4]
f (t) dt = 1 Z
1=
4
f (t) dt = 1
Z
4 1
p 2 3 k t dt = k t 3 p
4
=k 1
2 hp 3 4 3
k = 3/14 3p t, 1 t 4. 14
f (t) =
µ=
Z
4
tf (t) dt = 1
Z
4
t 1
3p t dx 14
3 h 5/2 = (4) 35
4 3 2 5/2 t 14 5 1 i 3 (1)5/2 = (31) ⇡ 2.657. 35
Z
Z
3 14
=
Z
4
t3/2 dt = 1
4
t2 f (t) dt = 1
3 = 14 =
Z
4
t
5/2
1
3 h 7/2 (4) 49
=
s
Z
4
t2 1
3p t dx 14
4 3 2 7/2 dt = t 14 7 1 i 3 (1)7/2 = (127) ⇡ 7.7755. 49
4
t2 f (t) dt 1
µ2 =
p
7.7755
(2.657)2 ⇡ 0.846.
p i 14 13 = k · . 3
P (3 t 4) =
Z
P (1 t 2) =
Z
3p 3 2p 3 t dt = t 14 14 3
4 3
3p 3 2p 3 t dt = t 14 14 3
2 1
P (1.6 t 2.4) =
Z
2.4 1.6
4
= 3
2
= 1
3p 3 2p 3 t dt = t 14 14 3
1 hp 3 4 7
1 hp 3 2 7
2.4
= 1.6
3t4
p i 33 ⇡ 0.40. 1t2
p i 13 ⇡ 0.2612.
1p [ (2.4)3 7
p (1.6)3 ] ⇡ 0.242.
[a, b]
f (x) =
1 b
a
f (x) = ke
, a x b.
kx
, 0 x. µ = 1/k
k
1/k
µ
f (x) =
1 p e 2⇡
1 x 2(
µ 2
)
,
1 < x < 1. µ=0
=1 1 f (x) = p e 2⇡
P (0 x z) =
x2 2
Z
,
z 0
1 < x < 1.
1 p e 2⇡
x2 2
dx
z
0.5 0.4 0.3 0.2 0.1 0.0
x
z
!0.1
x x µ X x=
X
µ
µ
,
x P (A X B) = P (a x b),
a=
A
µ
b=
B
µ
.
t
µ = 10,000
= 2,000
P (12,000 t) a=
x = (t
10,000)/2,000
x
12,000 10,000 = 1, 2,000
P (12,000 t) = P (1 x) = P (0 x) = 0.5
0.3413 = 0.1587.
P (0 x 1)
t 9,000
P (t 9,000) b=
12,000 t
9,000 10,000 = 2,000
P (t 9,000) = P (x = 0.5
0.5, 0.5) = P (0.5 x) = P (0 x)
0.1915 = 0.3085.
P (0 x 0.5)
v(t) = 2t + 1, s(t) s0 (t) = v(t) = 2t + 1. v(t) s(t) =
Z
v(t) dt =
Z
(2t + 1) dt = t2 + t + C. C
C s(0) = 3 s(t) = t2 + t + 3
y0 =
3x2 , y 6= 0, y y(1) = 2 y
0
dy/dx
dy 3x2 = dx y dy = 3x2 dx y dy = 3x2 dx. y
Z
y dy =
Z
3x2 dx
1 2 y = x3 + C, 2 C = 2C1 p y = 2x3 + C
y 2 = 2x3 + 2C1 .
y=
p 2x3 + C.
x=1
C 2=
p 2(1)3 + C
y=2
y(1) > 0
4 = (2)2 = 2 + C, C=2 p y = 2x3 + 2.
(xE , pE ) p 6
S(x)
p1
$2,000 5%
p0 D(x) 0
- x
$2,500 5%
p0
$80,000 p1
$580,000
p0
p1 p1
$220,000
D(x) = (x 4)2 S(x) = x2 + 2x + 6
$3,000
Z
1 1
Z
Z
2 dx x2
1
3e
0.5x
Z
dx
0
1 2 1
5 dx x x1.4 dx
1
x
y = 3/x4 [2, 1) $3,000
P (0 x 2.13) P ( 1.46 x) P (1.25 x)
P ( 1.58 x 0) P ( 2.3 x 0.4) w
k
f (x) =
k x2
[2, 4]
t t
f (t) =
⇢
1 10
0
0 t 10 t 10
t µ = 11,000 = 2,500
E(p) = 1
p>0
x = D(p) D(2) = 3
$10,000
dy = 3x2 y dx dy 2x2 y =1 dx y 0 = 4x + xy
dy x = dx y dy 3 = 2 dx y
y(1) = 2 y(0) = 1
dP = 0.02P P (0) = 100 dt
P
dP = 0.04P dt
800
D(x) = (x
$300,000
$4500 $P0 $P0
5)2
S(x) = x2 + x + 3
$250,000
Z Z
1
e
3x
dx
1 1 2
3 dx x k
f (x) = 2x/9
f (x) = k
[0, 3] µ
1 x2
[1, 2]
x P (0.44 x 1.47) P ( 1.69 x)
y0 =
x 6y
y 0 = 4x
xy y = 2
$6,000
P dP = 0.03P dt
500.
x=0
$2,600 $80,000
$3,500
$80,000 $2,600
$3,500
$4,600
D
x
y
C = f (x, y) = 30x + 0.2y.
C = f (6, 400) = 30(6) + 0.2(400) = 180 + 80 = $260. z = f (x, y)
xy
z = f (x, y) = 3x2 y 5 + 2x3 y 2 + exy , f (0, 1) f (2, 1)
f (1, 1)
f (0, 1) = 3(0)2 (1)5 + 2(0)3 (1)2 + e(0)(1) = 0 + 0 + e0 = 1; f (2, 1) = 3(2)2 (1)5 + 2(2)3 (1)2 + e(2)(1) = 12 + 16 + e2 = 28 + e2 ; f (1, 1) = 3(1)2 (1)5 + 2(1)3 (1)2 + e(1)(1) = 3 + 2 + e1 = 5 + e. z = f (x, y)
(a, b, c)
c = f (a, b)
z = 2x + 3y 1
y 0 !1 2.0 1.0 z
0.5
1
1.5 z 1.0 0.5
0.0
0 y !1
0.0 !1
x
0 !1 1
z=e
(x2 +y 2 )
x
0 1
z=
p 4
x2
y2
y
1
x
!1
0
0
y 0
1
!1
1
!1
x 0
1
!1 2
4
z 0
z 2
!2 0
z = y2
z = x2 + y 2
x2
z = f (x, y) x y
z
z x
z
y
C = f (x, y) = 30x + 0.2y.
C = f (5, y) = 30(5) + 0.2y = 150 + 0.2y = h(y). y
h0 (y) = $0.2/
C = f (x, 1,000) = 30x + 0.2(1,000) = 30x + 200 = g(x). x
g 0 (x) = $30/
z = f (x, y)
f
fx (x, y) = lim
h!0
f (x + h, y) h
f (x, y)
,
.
f fy (x, y) = lim
k!0
f (x, y + k) k
y f (x, y)
,
.
z = f (x, y)
x
@z , @x
@f . @x y z = f (x, y)
y @z , @y
@f . @y x (a, b) (a,b)
f (x, y) = 3x2 y fx (x, y) =
x @f @z @ = = (3x2 y) = 6xy. @x @x @x f
fy (x, y) =
y
@f @ = (3x2 y) = 3x2 . @y @y x @f @x
fx (3, 2)
fx (x, y) = fx (3, 2) =
@f @x
x=3
y=2
. (3,2)
@f = 6xy @x = 6(3)(2) = 36. (3,2)
y
x
f0
f (x) f f 0 (x, y)
z = x2 y + 3x
@z @x
4y 3
@z @y
@z @ = x2 y + 3x @x @x
4y 3 =
@ @ x2 y + (3x) @x @x
@ 4y 3 = 2xy + 3 @x
@z @ = x2 y + 3x @y @y
4y 3 =
@ @ x2 y + (3x) @y @y
@ 4y 3 = x2 + 0 @y
z = f (x, y) = 3x2 y 5 + 2x3 y 2 + exy
fx (0, 1)
0 = 2xy + 3. 12y 2 = x2
12y 2 .
fy (1, 1)
fx (x, y) = 6xy 5 + 6x2 y 2 + yexy , fy (x, y) = 15x2 y 4 + 4x3 y + xexy .
fx (0, 1) = 6(0)(1)5 + 6(0)2 (1)2 + (1k)e(0)(1) = 0 + 0 + e0 = 1, fy (1, 1) = 15(1)2 (1)4 + 4(1)3 (1) + (1)e(1)(1) = 15 + 4 + e1 = 19 + e.
x = a f 0 (a) z = f (x, y)
y = f (x) (a, b) fx (a, b)
(a, f (a)) fy (a, b) z = f (x, y) = 4
fx (x, y) =
2x
x2
fy (x, y) =
y2
2y.
(1, 0.5) fx (1, 0.5) =
2(1) =
2
fy (1, 0.5) = y
z = f (x, 0.5) = 4
x2
y=
( 0.5)2 = 4
x2
2( 0.5) = 1. 0.5
0.25 = 3.75
x2 .
x y=
0.5 z = f (1, y) = 4
x=1 x (1)2
y2 = 4
1
y2 = 3
fx (1, 0.5) = x=1 y2 .
2
y
x=1 y=
0.5
fy (1, 0.5) = 1 z
P
z = f (x, y) fx (a, b)
y=b
(a, b, f (a, b)) y=b y
(a, b) x z = f (x, b) z = f (x, b) z
x=a
fy
P
(a, b)
z = f (a, y)
(a, b, f (a, b)) x
=
a y (a, b) x z = f (a, y)
V (r, h) = ⇡r2 h h
r
Vr (2, 5) Vh (2, 5)
Vr (r, h) = 2⇡rh
Vr (2, 5) = 2⇡(2)(5) = 20⇡ 1 20⇡ 3
h=5 Vh (r, h) = ⇡r2
Vh (2, 5) = ⇡(2)2 = 4⇡ 1 4⇡ 3
h=5
p b = 160 pb
0.15x
r=2 5
0.05y
p l = 220
0.15x
r = 2 2
0.15y,
pl
x y R(x, y) Rx (200, 400)
R(x, y) = p b · x + p l · y = (160
0.15x
= 160x
0.15x
= 160x + 220y
0.05y)x + (220 2
0.05yx + 220y 0.2xy
Rx (x, y) = 160 Rx (200, 400) = 160
Q = f (x, y) = Cxk y 1
0.15x
k
0.2y
0.15x2
0.15xy
0.15y 2
0.15y 2 .
0.3x
0.2(400)
0.3(200) = 20.
, 0 < k < 1,
Q
x
fx (x, y)
0.15y)y
y
fy (x, y)
Q = f (x, y) = 15x0.6 y 0.4 , Q
x y
f (200, 100) fx (200, 100) fy (200, 100)
f (200, 100) = 15(200)0.6 (100)0.4 = 2,273.57. $100,000
0.4 0.4
fx (x, y) = 15(0.6)x fx (200, 100) = 9
✓
y
100 200
=9
◆0.4
⇣ y ⌘0.4 x
.
⇡ 6.82 $100,000 $100,000
fy (x, y) = 15(0.4)x
fy (200, 100) = 6
0.6
✓
y
0.6
200 100
◆0.6
z = x2 y + 3x @z = 2xy + 3 @x
@z = x2 @y
✓ ◆0.6 x =6 . y ⇡ 9.09 $100,000 $101,000
4y 3 12y 2 . x
@ @x @ @x
✓ ✓
@z @x @z @y
◆
◆
=
@ @ (2xy + 3) = 2y, @x @y
✓
=
@ x2 @x
@ @y
12y 2 = 2x,
z = f (x, y)
@z @x ✓
◆ @z @y
y
=
@ (2xy + 3) = 2x; @y
◆
=
@ x2 @y
12y 2 =
24y.
f (x, y) fxx fxy fyx
@ 2f @ 2z @ 2f @ 2z @ = = = = = 2 2 @x@x @x@x @x @x @x ✓ ◆ @ 2f @ 2z @ @z = = = @y@x @y@x @y @x ✓ ◆ @ 2f @ 2z @ @z = = = @x@y @x@y @x @y
fyy =
@ 2f @ 2z @ 2f @ 2z @ = = = = 2 @y@y @y@y @y @y 2 @y
✓
✓
@z @x
@z @y
◆
◆
f (x, y) = 3x2 y 5 + 2x3 y 2 + exy
(a) fxx =
@ 2f @ = 6xy 5 + 6x2 y 2 + yexy 2 @x @x = 6y 5 + 12xy 2 + y 2 exy ,
(b) fxy =
@ 2f @ = 6xy 5 + 6x2 y 2 + yexy @y@x @y = 30xy 4 + 12x2 y + exy + xyexy ,
(c) fyx =
@ 2f @ = 15x2 y 4 + 4x3 y + xexy @x@y @x = 30xy 4 + 12x2 y + exy + xyexy ,
(d) fyy =
@ 2f @ = 15x2 y 4 + 4x3 y + xexy @y 2 @y = 60x2 y 3 + 4x3 + x2 exy .
@ 2f @ 2f = , @y@x @x@y
z = f (x, y)
f (x, y)
fxy = fyx .
fxy
fyx
fxy = fyx
f (x, y) = ex
fx (x, y)
2
+y
fy (x, y)
fxx (x, y) 7 3
f (x, y) = 2x y
f (x, y) =
p f (x, y) = 3 xy f (x, y) = 2xey
2
f (x, y) = 2x2 e3y
f (x, y) = 2x2 y 3 + ex+y
1 x + 2y
f (x, y) = @z @x
z=
z = e2x
z = 5x4
@z @y
2
z= 10x2 y + 6ex
f (x, y) = x2 ey + ln y f (x, y) = ln(2x + 3y) y
f (x, y) = ln(3x + 5y) f (x, y) = x2
3yex +
y x2
p z = 2 3 xy
5xey +
x2 y3
3
f (x, y) =
x5 y3
4
f (x, y) =
f (x, y) = y 3 exy f (x, y) = ex
2
f (x, y) = x2 exy
y
f (x, y) = exy
V (r, h) =
⇡ 2 r h 3 h
r
Vr (3, 5) Vh (3, 5)
4y 2 + y
2y 3 + 5x2
2xy
V (x, h) = x2 h x
h Vx (2, 3) Vh (2, 3) P
f (x, y) = ln y 2 + x2
f (x, y) = ex
2
x2
t t
y
2
3y 2
fy (1, 2)
f (x, y) = ln y 3
y2 x7
y x
fx (1, 2) f (x, y) = 2x4 + x3 y 2
y2
f (x, y) = 2x2 + y 3
2
p z = 3 xy
f (x, y) = 3x4
f (x, y) = 5x4 y 2 + exy
7yex + 8y 4
3x ln y
2xy
y
z = x2 ln(xy)
x y
z = 2x3 ey
2x
2y 3 + 5x2
f (x, y) = y 3 ex + ln x
1
z = x3 y 4 + 2xy 3
z = ex ln(xy)
4y 2 + y
f (x, y) = 3x4
3 f (x, y) = 4 2 x y
2
fyy (x, y)
f (x, y) = 2x4 + x3 y 2
f (x, y) = ln x2 y 2
z = x2 y + 3x4 y 2
fxy (x, y)
f (x, y) = 5 xy
2 f (x, y) = 2 yx
z = e2xy
4x y p 3
f (x, y) = ln x2 + y 2
f (x, y) =
5 2
f (P, t) = P (1 +
0.04 12t 12 )
fP (2,000 ft (2,000 P t
p b = 120
0.5x
0.3y
p l = 150
0.5x
0.5y,
pb
t
pl x
f (P, t) = P e0.04t
y
fP (2,000 ft (2,000
R(x, y) Rx (65, 35)
$10,000 r%
Ry (65, 35)
t t f (t, r) = 10,000(1 +
0.01r 12t 12 )
ft (5, 3) fr (5, 3) $10,000 r% t t
p s = 150
0.5x
0.2y
p l = 250
0.1x
0.5y,
ps
pl x y
f (t, r) = 10,000e0.01rt ft (5, 3)
R(x, y)
fr (5, 3) x
Rx (50, 30) Ry (50, 30)
y R(x, y) = 1.5x2 + 6y C(x, y) = 1,000 + 10x + 40y.
Rx (5, 2)
P (x, y)
Ry (5, 2) x
Px (65, 35) Py (65, 35)
y R(x, y) = 2x2 + 4y Rx (4, 1)
C(x, y) = 800 + 30x + 40y. P (x, y)
Ry (4, 1) Px (50, 30) Py (50, 30)
Q x y Q = f (x, y) = 20x0.6 y 0.4 , $100,000
Q x y f (250, 120) fx (250, 120) fy (250, 120)
Q = f (x, y) = 140x0.8 y 0.2 ,
Q = f (x, y) = 180x0.7 y 0.3 ,
Q
Q
x
x
y
y
f (500, 250)
$50,000
fx (500, 250) fy (500, 250)
Q = f (x, y) = 160x0.65 y 0.35 ,
y = f (x)
f f (x) x = x0
f (x, y)
f (x0 ) x f (x)
x0
x = x0 f f (x0 ) x
f (x, y) • f
R
(x0 , y0 ) f (x, y) f (x0 , y0 ) (x, y)
(x0 , y0 )
• f
(x0 , y0 ) f (x, y)
f (x0 , y0 )
(x, y)
(x0 , y0 )
• f
(x0 , y0 ) f (x, y) f (x0 , y0 ) R
(x, y) • f
(x0 , y0 ) f (x, y)
f (x0 , y0 ) R
(x, y)
y 0 x 0
y 1 !1 0
1
!1
x 0
1
!1 1 2
!1 4 z
z 0
2 0
!2
z = x2 + y 2 z = y2 (x0 , y0 ) = (0, 0)
x2
z = f (x, y) = x2 + y 2
f (0, 0) = 0 (0, 0) y = f (x) f 0 (c)
x=c f 0 (c) = 0 (x, y)
R
S
R
R
c
S R
R
(x, y)
f (x, y) fx (x, y) fy (x, y) R
2x 4y
R
R
(x0 , y0 )
fy (x0 , y0 ) = 0.
z = f (x, y) = x2 + 2y 2
⇢
(x, y)
R
(x, y) (x0 , y0 )
fx (x0 , y0 ) = 0
fx (x, y) = 2x
R
4xy + 2x + 4y + 5,
4y + 2
fy (x, y) = 4y
4x + 4.
4y + 2 = 0 4x + 4 = 0.
x x = 2y
4y
1.
4(2y
1) + 4 = 0,
4y + 8 = 0.
y=2 x = 2(2)
x = 2y
1
1 = 3. (3, 2)
z = f (x, y) (x0 , y0 )
R (x0 , y0 )
R (x0 , y0 ) (x0 , y0 )
R
fx R
fy f
(x0 , y0 ) f
f x = c z = f (x, y)
f
(x0 , y0 )
(x0 , y0 )
x
z = f (x, y) = y 2 x2 h(y) = f (0, y) = y 2 y 0
0
(0, 0) y=0
h g(x) = f (x, 0) = x2
g
x = 0
z = f (x, y) (x0 , y0 )
z = f (x, y) (x0 , y0 ) y = f (x)
x=c
x=c
z = f (x, y) fx (x0 , y0 ) = fy (x0 , y0 ) = 0
(x0 , y0 ) D = fxx (x0 , y0 )fyy (x0 , y0 )
[fxy (x0 , y0 )]2 .
•
D>0
fxx (x0 , y0 ) > 0
f
(x0 , y0 )
•
D>0
fxx (x0 , y0 ) < 0
f
(x0 , y0 )
•
D 0,
f
( 1, 0)
( 1, 2) D = 6( 1) · (6(2)
(1, 2)
6)
02 =
36 < 0.
fxx ( 1, 0) =
6 < 0.
f
( 1, 2)
(1, 0) D = 6(1) · (6(0)
02 =
6)
f
36 < 0. (1, 0)
(1, 2) D = 6(1) · (6(1)
02 = 36 > 0,
6)
f
fxx (1, 0) = 6 > 0.
(1, 2) f
( 1, 2)
pb = 160 pb
( 1, 0)
(1, 2)
(1, 0)
0.15x
0.05y
pl = 220
0.15x
pl
0.15y, x
y R(x, y) R(x, y) = 160x + 220y
0.2xy
0.15x2
0.15y 2 .
R(x, y) Rx (x, y) = 160
0.2y
0.3x,
Ry (x, y) = 220
0.2x
0.3y;
Rxx (x, y) = ⇢
160 220
0.3, Rxy (x, y) =
0.2y 0.2x
0.2, Ryy (x, y) =
0.3.
0.3x = 0 0.3y = 0.
y y = 800
1.5x. x
220 0.25x
0.2x
0.3(800
20 = 0,
1.5x) = 0, x = 80.
y = 800 y = 800
1.5x
1.5(80) = 680.
R
(80, 680) ( 0.2)2 = 0.09
D = ( 0.3) · ( 0.3) R
D
(80, 680)
0.04 = 0.05 > 0,
Rxx (80, 680) =
(80, 680)
R(80, 680) = 160(80) + 220(680)
0.2(80)(680)
x
R(80, 680)
0.15(80)2
0.15(680)2 = $81,200.
y
C(x, y) = 5,000 + 60x + 100y. P (x, y)
P (x, y) = R(x, y)
C(x, y)
= 160x + 220y
0.2xy
0.15x2
0.15y 2
(5,000 + 60x + 100y)
= 160x + 220y
0.2xy
0.15x2
0.15y 2
5,000
0.2xy
2
2
= 100x + 120y
0.15x
0.15y
60x
100y
5,000.
P (x, y) Px (x, y) = 100
0.2y
0.3x,
Py (x, y) = 120
0.2x
0.3y;
Pxx (x, y) = ⇢
100 120
0.3, Pxy (x, y) =
0.2y 0.2x
0.2, Pyy (x, y) =
0.3.
0.3x = 0 0.3y = 0.
y y = 500
1.5x. x
120
0.2x
0.3(500
1.5x) = 0,
0.3 < 0.
0.25x
30 = 0,
x = 120. y = 500
y = 500
1.5x
1.5(120) = 320.
P
(120, 320)
D = ( 0.3) · ( 0.3)
( 0.2)2 = 0.09
P
D
(120, 320)
0.04 = 0.05 > 0,
Pxx (120, 320) =
(120, 320)
P (120, 320) = 100(120) + 120(320)
2
2
10y + 8
2
2
6y + 4
f (x, y) = x + y + 6x f (x, y) = x + y + 10x 2
f (x, y) = x + xy + 10y
0.2(120)(320)
0.3 < 0.
P (120, 320)
0.15(120)2
0.15(320)2
5,000 = $20,200.
f (x, y) =
x2
y 2 + 6x + 4y + 2
f (x, y) =
x2
y 2 + 4x + 6y + 1
f (x, y) = x3 + y 3
3x2
12y + 2
f (x, y) = x3 + y 3
6y 2
3x + 5
f (x, y) = x3 + 3xy
y3
2
f (x, y) = x + 10x + xy f (x, y) = y
3
f (x, y) = x
f (x, y) = x3
4xy + 8x
3
4xy
6y
f (x, y) = x2 + y 2 + xy
8y + 1
2
2
6x + 2
3
3
f (x, y) = x + y + xy f (x, y) = x + y 3
f (x, y) = x + y
3x 3
2
6y
12y + 2 2
3x + 5
3xy + y 3
f (x, y) = 4y 3
3x2
12y 2 + 6x
f (x, y) = 4x3
3y 2
24x2 + 6y + 1
f (x, y) = 2x3 y
24x + 16y + 1
f (x, y) = 2xy 3 + 4y f (x, y) = ey
2
f (x, y) = ex
2
5
+x2 +1 y2
2x + 1 f (x, y) = e3
x2 y 2
y = exy
D
f (x, y) = x2 + y 2 + xy
y+1
f (x, y) = x2 + y 2 + xy
5x + 4
f (x, y) = x2 + xy + 6y f (x, y) = x2 + 4x + xy f (x, y) = 2xy
y3
x2
f (x, y) = 4xy
x3
y2
x2
P (x, y) = 2,000
xy
1.5y 2 + 80x + 90y
x y x
y
P (x, y) = 560x + 20xy x
20x2
6y 2
y
pb = 200
1.5x
0.3y
pl = 300
0.1x
0.8y
pb
pl x y
R(x, y)
px = 600
0.4x
x
y
py = 400
0.2y
C(x, y) = 60,000 + 30x + 40y R(x, y)
ps = 150
0.5x
0.2y
pl = 250
0.1x
0.5y
ps
pl x y
P (x, y)
C(x, y) = 1000 + 40x + 80y. px = 500
0.2x
x
py = 600
0.3y
y P (x, y)
C(x, y) = 100,000 + 100x + 180y + 0.2xy R(x, y) P (x, y) C(x, y) = 8,000 + 30x + 40y.
CA (x) = 9 + 0.04x2 x
C(x, y) = 15 + 2x2 + 3y 2
CB (y) = 6 + 0.04y 2
x
y
y
q =x+y p = D(q) = 65
0.04q P (x, y)
P (x, y)
(x, y) z = f (x, y) = 3x2 y 5 + 2x3 y 2 + exy f (2, 1) = 3(2)2 (1)5 + 2(2)3 (1)2 + e(2)(1) = 12 + 16 + e2 = 28 + e2 .
z = f (x, y) z
x
z z
x y
fx (x, y) fy (x, y)
@z @y
@z @x
@f @y
@f @x
x x
@f @z @ = = 3yx2 = 4yx. @x @x @x z
fy (x, y) =
y
@f @z @ = = 3yx2 = 3x2 . @y @y @y x z = f (x, y)
y f f
z = 3yx2 fx (x, y) =
y x
y
z = f (x, y) fxx fxy fyx
@ 2f @ 2z @ 2f @ 2z @ = = = = = 2 2 @x@x @x@x @x @x @x ✓ ◆ @ 2f @ 2z @ @z = = = @y@x @y@x @y @x ✓ ◆ @ 2f @ 2z @ @z = = = @x@y @x@y @x @y
fyy =
@ 2f @ 2z @ 2f @ 2z @ = = = = 2 @y@y @y@y @y @y 2 @y fxy
✓
✓
@z @x
@z @y
fyx
z = f (x, y) = 3x2 y 5 + 2x3 y 2 + exy
(a) fx =
@f @ = 3x2 y 5 + 2x3 y 2 + exy @x @x = 6xy 5 + 6x2 y 2 + yexy ,
fy =
@f @ = 3x2 y 5 + 2x3 y 2 + exy @y @y = 15x2 y 4 + 4x3 y + xexy .
(b) fxx =
@ 2f @ = 6xy 5 + 6x2 y 2 + yexy @x2 @x = 6y 5 + 12xy 2 + y 2 exy ,
fxy =
@ 2f @ = 6xy 5 + 6x2 y 2 + yexy @y@x @y = 30xy 4 + 12x2 y + exy + xyexy ,
fyx =
@ 2f @ = 15x2 y 4 + 4x3 y + xexy @x@y @x = 30xy 4 + 12x2 y + exy + yexy ,
fyy =
@ 2f @ = 15x2 y 4 + 4x3 y + xexy @y 2 @y = 60x2 y 3 + 4x3 + x2 exy .
◆
◆ fxy = fyx
z = f (x, y) f
(x0 , y0 )
f (x, y) f (x0 , y0 ) (x, y)
(x0 , y0 ) f
f (x, y)
(x0 , y0 )
f (x0 , y0 )
(x, y)
(x0 , y0 ) fy (x, y)
fx (x, y) R
(x0 , y0 )
fx (x0 , y0 ) = 0,
(x, y) (x0 , y0 )
R
fy (x0 , y0 ) = 0.
f
R
(x0 , y0 )
(x0 , y0 )
z = f (x, y)
D (x0 , y0 )
D z = f (x, y) fx (x0 , y0 ) = fy (x0 , y0 ) = 0
(x0 , y0 ) D = fxx (x0 , y0 )fyy (x0 , y0 )
[fxy (x0 , y0 )]2 .
•
D>0
fxx (x0 , y0 ) > 0
f
(x0 , y0 )
•
D>0
fxx (x0 , y0 ) < 0
f
(x0 , y0 )
•
D 0,
f
( 1, 0)
( 1, 2) D = 6( 1) · (6(2)
6)
02 =
f
36 < 0.
( 1, 2)
(1, 0) D = 6(1) · (6(0) f
(1, 2)
6)
02 =
36 < 0. (1, 0)
fxx ( 1, 0) =
6 < 0.
(1, 2) D = 6(1) · (6(1)
6)
f
fxx (1, 0) = 6 > 0.
(1, 2) f
( 1, 2)
02 = 36 > 0,
( 1, 0) (1, 0)
(1, 2)
f (x, y) = 2xy f (x, y) = 2x2 ey + 5y 3 f (1, 0) fxx fyy
fx fxy
ln x :
f (x, y) = 2x3 y
@2z @x2
@z @y
@2z @x@y
@2z @y@x
P (x, y) = x y x
x2
xy
x2
y 2 + 6x + 4y + 2 y3
24x + 16y + 1 1.5y 2 + 80x + 90y
0.5x
0.3y
pl = 300
0.5x
0.5y,
pl x
y
px = 600 x
0.4x
py = 400
y
y C(x, y) = 60,000 + 30x + 40y. R(x, y)
R(x, y) Rx (65, 35) Ry (65, 35) D
2,000
@2z @y 2
pb = 200
pb
x2
f (x, y) = x3 + 3xy
fy fyx
z = 3exy + x2 y 3 : @z @x
f (x, y) =
y3
P (x, y)
0.2y,
z = f (x, y) = x3 + 2yex
y2
f (0, 2) fx fy fxx fxy fyx fyy
f (x, y) = 4xy
p b = 200
1.5x
0.3y
p l = 300
0.1x
0.8y,
pb
pl
x3
y2
D
x y
R(x, y)
Rx (60, 25)
Ry (60, 25)
y 10 5
( 1, 1) 2
6a + 3 2t + 3
!4
!2
h 6= 0
2x + 2h + 3
2
4
2
4
x
!5 !10 !15 y
!20
15
x
f (x)
y = 5x + 3
10
y 20
5 !4
2
!2
x
4
10
!5 !4
!10
!2
x
!10
( 1, 1) 50a2
2 2a
!20
y=
2
2x2 + 4xh + 2h2
y = 2(x
2) y = 2x 4 3 3 14 y 1= (x 3) y = x+ 5 5 5 y 1 = 2x y = 2x + 1
2
4x + 2h h 6= 0 x
F (x)
y 20 15 10 5
2 !4
x 6=
x
1.5x + 3
!2
2
!5 !10
4
y
7=0
y
2=
y
2=0
y=7 3(x
1)
y=
3x + 5
y=2
x
4 S = 4x + 680 m=4
1
m=4
$4
G(x)
b = $680 10 5 !4
2
!2 !5
4
V =
x
m=
3000t + 31, 000
3000
!10
m=
3, 000
$3, 000 b = $31, 000 T ⇡ 10.3
9 4 m = 9/4
m=
9/4
b = 32 0
C(x) = 128 + 0.8x
= 32
m = $0.8 b = $128 R(x) = 2.5x x ⇡ 76
0.3x 1200 1000 800 600 400 200 0
P (x) = 1.7x
p x
x = D(p) = 500
10p
pE = $15.27 xE = 43.65 0
200
400
600
x 800 1000
1, 500
600
p = $100
p = $50
pE = $80 xE = 1, 200
m = 0.3
x 4000
R(x) = 1.2x 0 x 1, 000 1200 1000 800 600 400 200 0
3000 2000 1000 0
0
200
400
P (x) = 0.9x
100
150
200
600 (0, 1) y
200
400
20
x 800 1000
600
15 10 5
$420 1200 1000 800 600 400 200 0
50
x 800 1000
600
P 400 200 0 !200 !400 !600 !800
0
667
!4
2
4
2
4
!2
x
!5 !10
(1, 1) y 20 15 0
200
400
600
10
x 800 1000
C(x) = 25, 000 + 8x R(x) = 16x P (x) = 8x 25, 000 3125
5 !4
!2
!10
( 1.5, 5.5) y
y
20
80 000
15
60 000
10
40 000
5
20 000 0
x
!5
0
x 1000 2000 3000 4000 5000
!4
!2
2 !5 !10
4
x
p
128
x
(1.5, 5.5)
f (x)
y 4
y 10
2
5 !4
2
!2
x
4
!4
2
!2
!5
x
4
!2
!10 !4
!15 !20
2
p
6 2+
p
F (x) = 6
x x
⇢
x 2 x+2
x< x
2 2
F (x) 4
1
2
y 1 0
2
4
6
8
10
x
!4
x
2
!2
!1
!2
!2
!4
!3 !4
x 6= 3
x
f (x)
8
4
6
2
4 2 !10
5
!5
10
!6
x
!4
2
!2
4
6
x
!2
!2
x 6=
1 y
y 3
!4
x
4
f (x)
2
3
1
2 2
!2
4
1
x
!1 !4
x
1
!1
!3
!2
x
t=2
1
pE = 2.05 xE = 4.15 pE = 8.64 xE = 6.64
x 6= 1, 3 x 6= 2
2
!2
!2
x= x=1
3
t ⇡ 6.9
4
x
x
f (x)
P (5) = 10, 272.2
y 30 25 20 15 10 5 !3
x
!2
P (5) = 3, 571.02 P (5) = 8, 549.73
!1
f (x)
1
2
3
1
2
3
x
y 30 25 20 15 10 5 !3
x
!2
!1
f (x)
x
y 3 2
log 1000 = 3
1
!4
2
!2
4
ln b = k 2
loga J = h
x
3 =9
ek = 10
aG = H
!1
x x
f (x)
y 10
f (x)
y
5
3 2
!10
5
!5
10
x
!5
1
!10 !4
2
!2 !1
4
x
x
g(x)
y 10 5
!10
5
!5 !5 !10
10 5
P (5) = 6, 388.89
!10
5
!5 !5
P (5) = 6, 788.45
!10
10
x
10
x
ln 8 ln 5 ln 100 t= 3 ln 1.2 t= 0.02 3 x = e x ln 3 ln K log3 K = ln 3 ln H loga H = ln a
t = ln 100 ln M ln a ln 1.4 t= 4 10b = eb ln 10 t=
ah = eh ln a log2 7 =
0
t=
ln 7 ln 2
( 1, 1)
50a2
2 2a
2x + 4xh + 2h
A(5) = $6, 261.61 ln 2 T = = 15.4 0.045 ln 1.6 T = = 10.44 0.045 $29, 845.3 ln 8/3 k= = 0.0577 = 5.77% 17 r < 22% r
2
4x + 2h h 6= 0
2
y 10 8 6 4 2
3 ln 1.5 t= = 15.6 0.026 ln 1.5 t= = 5.33 0.076 ln 1.25 k= = 0.056 = 5.6% 4 ln 2 T = = 12.38 0.056 ln 2 k= = 0.139 = 13.9% 5 ln 2 70/6 = 11.67 T = = 11.55 0.06 A(t) = 5, 000e0.045t
2
2
!4
2
!2 !2 !4
m=
x
4
1/3 y 10 8 6 4 2
!4
!2 !2 !4
2
y = 3x + 3 V = m =
x
4
y=
3x + 5
4, 000t + 57, 000 4, 000
m =
4, 000
$4, 000
b = $57, 000 T ⇡ 14.25 15x 0 x 1, 000
22%
R(x) = 25x 0 x 1, 000
r = 5.1% P (t) = 600e[(ln 1.5)/1.5]t
6, 000 0 x 1, 000
P (x) = 10x
P (5) = 2, 318 ln 2 T = = 2.564 (ln 1.5)/1.5 t=
ln(40/3) = 9.58 (ln 1.5)/1.5
m = $10
P (t) = 17e(ln(91/17)/45)t P (66) = 199 T (t) = 55e t=
+ 75
2 ln(11/8) ⇡ 3.17 ln(11/9) T (t) = 110e
t=
(ln(11/9)/2)t
(ln(22/13)/30)t
30 ln(22/5) ⇡ 84.5 ln(22/13)
+ 75
25 000 20 000 15 000 10 000 5000 0
0
x 200 400 600 800 1000
C(x) = 128 + 0.8x m = $0.8 b = $128
x
( 1.5, 5.5)
4000
y
3000
20 15
2000
10
1000
5 !4
2
!2
4
0
x
0
t = ln 100
(1.5, 5.5)
t=
y
56x
10 5 2
!2
4
!5 !15 !20
x=
p
x=2
x= p 6 2+ 6
x 6= 1, 3 x 6= x
x=
2, 3
x=3
3
x=2
2 1
x
ln 100 3 = e(6 ln 5)x
k=
1
p
4 2
!4
2
!2 !2
D(50) = 1, 500
2.5(ln 10)
22% (ln 1.8/1.5)t
P (5) = 3, 547 ln 2 T = = 1.77 ln 1.8/1.5 ln 16 t= = 7.08 ln 1.8/1.5
g(x)
!6
ln 120 ln 10 ln 1.2 t= 0.02 21/3 = e(ln 2)/3
ln 8/3 = 0.0577 = 5.77% 17 r < 22% P (t) = 500e
S(50) = 600
p = $100 p = $50
200
t=
r
x
150
ah = eh ln a 10 2.5 = e ln 1.5 t= = 11.3 0.036 ln 1.25 k= = 0.056 = 5.6% 4 A(t) = 8, 000e0.051t A(5) = $10, 323.7 ln 2 T = = 13.6 0.051 ln 7/4 t= = 11 0.051 $29, 845.3
x
!10
x 6=
100
!5 !10
!4
50
pE = $80 xE = 1, 200
4
6
x
T (t) = 58e
(ln(58/48)/2)t
2 ln(58/43) t= ⇡ 3.16 ln(58/48)
1
1
+ 72
1
e3
3x2
y
lim f (x)
20
x! 1
15 10
lim f (x) = 2 = f ( 2)
5 !4
!2
x! 2
2
4
x
!5
g( 1)
!10
lim g(x) = 1 = g(3)
x!3
y
3
10
lim f (x)
x!1
5
!4
2
!2
4
3
x
3
3 3 lim f (x) =
x!1
!5 !10
2x 1/x
2
y
1
3.0 2.5 2.0 1.5 1.0 0.5 !4 !2
1
2
4
6
8
10
x
6x + 3h h 6= 0 1 + h h 6= 0
2x
y 4
5 h 6= 0
2 5
!5
x
!2
3 h 6= 0 x(x + h)
!4
4x 2h h 6= 0 x2 (x + h)2
y 4 2 5
!5 !2
x
0 h 6= 0 4x
!4 !6
1
3x2 2x
2h h 6= 0 3xh
h2 h 6= 0
2 + h h 6= 0
3 = f (1)
128 2 2, 0.5 x2 y = 2x + 4 y = 0.5x
64
2 y 10
96
5
!4
48
!2
2
4
2
4
x
!5 !10
2x
4, 0,
2
y = 4x + 4 y = 0 y =
2x + 1 10 5
!4
!2
x
!5 !10
4x
4, 0,
8
y = 4x + 4 y = 2 y =
8x + 10 y 10 5
!4
2
!2
4
x
!5 !10
3x2 y=
3x
3, 0,
3
2 y=0 y=
3x + 2
10 5
!4
2
!2
4
x
!5 !10
6x + 3h h 6= 0
6x
2x
2
y=
6x
6,
2, 0
1 y=
2x + 3 y = 2 y 15
2x 1
1 + h h 6= 0
5 h 6= 0
2x
1
10 5
5 !4
3 h 6= 0 x(x + h)
3 x2
2
!2 !5
4
x
x1 x4 x7
x2 x6
x3
x8 v(t) = 32t 0 t 4 v(3) = 96
v(t) = 32t a(t) = 32 0 t 4 s(3) = 112 v(3) = a(3) =
112 96 32
v(t) = 32t + 80 0 t 5 v(3) = 16
p 5 21 x2 5 4 x3 5 7
f (x) = 6x5 + 3x4 f 0 (x) = 30x4 + 12x3
3 p 0.24x 4 8 x5 6x2 20x + 6ex 3 2 p 3ex x2 2 x 5 3 10 + p + p 3 7 2 x 3 x5 2x 6 3 x3 5ex exe 1
(0, 1) (5, 25) ( 2, 21) (0, 0) ( 2, 19)
f 0 (x) = 9x2 · (2x2 + x) + 3x3 · (4x + 1) = 18x4 + 9x3 + 12x4 + 3x3 = 30x4 + 12x3
9 p 5 5 x2 1 p x3 x 7e 2
f (x) = x1/2 · x1/3 = x5/6 5 5 f 0 (x) = x 1/6 = p 6 66x 1 1 f 0 (x) = x 1/2 · x1/3 + x1/2 · x 2/3 2 3 5 = 12 x 1/6 + 13 x 1/6 = 56 x 1/6 = 6 p 6x
1.1
f (x) = x4 f 0 (x) = 4x3 6x5 · x2 x6 · 2x 4x7 = = 4x3 x4 x4 1 3 f (x) = 3 2x3 f 0 (x) = 4 6x2 x x (2x 16x7 ) · x5 (x2 2x8 ) · 5x4 f 0 (x) = x10 6 12 3x 6x 3 = = 4 6x2 x10 x f (x) = x 1 x 6= 1 f 0 (x) = 1 x 6= 1 2x · (x + 1) (x2 1) · 1 f 0 (x) = (x + 1)2 2 x + 2x + 1 = 2 = 1 x 6= 1 x + 2x + 1 (14x6 + 20x3 ) · (7x2 6x + 1) +(2x7 + 5x4 3) · (14x 6) 3 8 ( p ) · (ex + x2 + 4) x2 2 x p +(3 x + x8 ) · (ex + 2x) f 0 (x) =
3ex
y = 13x + 6 y=x 1
2
f (x) = x9 f 0 (x) = 9x8 f 0 (x) = 7x6 · x2 + x7 · 2x = 7x8 + 2x8 = 9x8
3 p 2 x
16x 6 5 3 p + p 2 x 2 x3 2520x2
2
v(t) = 32t + 80 a(t) = 32 0 t 5
32
14x6
96
2
32
y=x (0, 2)
(2, 1.8) (1, 6) 2 (4, 10 ) 3 (2, 13)
2
4x3 + 3x2 + 10 (x3 + 5)2
(x, 7)
2ex (x 2) x3 2(1 x) ex
dy 5 = dx 7 7ex (2x 1) (2x + 1)2 p p 4 (2x 3)(2 4 x + 3) (x2 3x + 2)/(2 x3 ) p (2 4 x + 3)2 2 + 3x2 ex + x3 ex (2x3 4x2 + 12ex 20x p 3 2 ( p + 3 x)ex x2 2 x
3)ex
35x2 + 15 (7x2 + 3)2 p +4.5(x + 1)/ x 2 3x(2 x)ex + 4x (3ex + 2)2 3(x 2)ex 4 + x3 3ex (x3 5x2 + 4x + 4) (x2 + 4)2 18x 28 (x2 + 4x + 2)ex 54 (3x + 1)3 y = 5x + 2 8 20 y = x+ 9 9 8 20 y= x+ 9 9
3
7(2x + 5x
2
6
0.04e (27x 4
4x + (3x2 + 4x3 )e4x+1 2(10x2 + 6x
3)(10x2 + 26x + 3)e2x+5 p + 3 2x + 1ex
x
p
3e 2x + 1 6(x + 1)ex p 2x + 1
3x3 + 2)
2/3
· (5x4
9x2 )
2
2(1 + 2x2 )ex
2
24(x2 + 1)2 (7x2 + 1) y = 2x + 1 y = 2x + 3 y = 2x + 3
y=4
y=2
(3x + 1)2 p x2 1 2
+x
8)(x + 1)6 (3x
2)
3x2
4x (6x + 5)e p x2 2x2 + 1 p x2 + 1 + p x2 + 1 x2 + 1 3 2 f (x) = (x + 2x)(x + 1) 1 0 f (x) = (3x2 + 2)(x2 + 1) 1 2x(x3 + 2x)(x2 + 1) 2 (3x2 + 2)(x2 + 1) (x3 + 2x)(2x) (x2 + 1)2 f (x) = (x2 2x)e x f (x) = (4x x2 2)e x (2x 2)ex (x2 2x)ex (ex )2 2 4x x 2 = ex 0
5(3x2 + ex )4 (6x + ex )
6xex + 10(2x + 1)4
0.02x
2
4
42(2x + 1)
y = 8x(x2 + 1)3 · (3ex + 2) 1 3(x2 + 1)4 · (3ex + 2) 2 · ex
3) (6x + 10x)
2
◆2 5x + 3 93 · 7x 2 (7x 2)2 r 1 3x 5 19 · 2 2x + 3 (3x 5)2
1 (x5 3 0
3(3x2 + ex ) p 2x3 + 2ex 0.3e0.06x 2(6x + 1)e3x
✓
3x2 + 1 x
3e2x + 2ex + 1
1 x>0 e3x
2
+2x+1
5
f (u) = u g(x) = 2x + 3 p f (u) = u g(x) = ex + 2 f (u) = eu g(x) = 2x2 + x dy du = 3u2 = 3ex du dx dy = 3u2 · 3ex = 9(3ex + 2)2 ex dx dy du = eu = 6x du dx 2 dy = eu · 6x = 6xe3x +4 dx dy 1 du = = 4x3 du u dx dy 1 4x3 = · 4x3 = 4 dx u x +1 3 x 2+ +e x 20x 2x2 + 3 5 4 5x + 3 4x + 1 1 x
1 4x ln x + 1
(1/3, 98/27)
6(ln x + 4ex
3x2 )5 · (
1 + 4ex x
6x)
5 3 10 +p + p 3 7 x 3 x5
2
3(ln x) x 9 (3x + 1)2 2 ln x + 3 y = 5x
x
20x 3 7e 2/x 3 8 ( p )(ex + x2 + 4) x2 2 x p +(3 x + x8 )(ex + 2x)
3(ln 6x + 2) p 2 x
y0 =
(2, 1)
3
x2 4x 6 (x 2)2 5ex + exe 1 10x + 0.28e x2 + 1
5
1 x + (ln 3 + 3 1 y = x + ln 4 4 1 y = x + (ln 5 5 y=
1 ) 3
7(2x3 + 5x2 5xex
2
3)6 (6x2 + 10x)
/2
14x(x2 + 1)6 (3x
1 ) 5
0.04x
2
3
2)2 + 6(x2 + 1)7 (3x
2)
x
(3x + 3 x 3x)e r dy 1 7x 3 22 = · dx 2 5x + 1 (7x 3)2
2(10x2 + 26x + 3)(10x2 + 6x p 3 ln x 3 2x + 1 p + x 2x + 1
3)e2x+5
840x3 1 4 x2
3
3 3/8 dy du 1 = 3u2 + eu = p du dx 2 x p
1
dy 3x + e x p = dx 2 x dy 1 du 1 = p = 2x + du x 2 u dx 2x + 1/x dy = p dx 2 x2 + ln x v(t) = 32t + 80 a(t) = 32 t = 2.5
lim f (x)
x! 1
lim f (x) = 2 = f ( 2)
x! 2
lim f (x)
x!1
1 2
3
12
10x
2
$93.96
x2
128 $112.45
64
v(t) = v(2) =
32t 0 t 4 64
$1202, 06 $400.69 y = 13x + 6
y=x
1
y=x
2
2
M C(x) = C 0 (x) = 42.4x 36
1/5
M C(77) = $17.79 36
P (x) = 275x 0.05x2 +300e 0.02x 3, 000
P (x) = 70x 100 ln(x + 5)
0.2x2 1, 500
95
5 110 130
x = 0.5 y = 3.25 x = h y = 6h + h2 y = 0.61 f 0 (2) x = 0.6 8.6 8.61
M P (1, 500) = 0.3 > 0 P p = D(x) = 840 R(x) = 840x
0
y = 1.5 f (3) x = 1.5 y= y= y=
0.23077 f 0 (1) x = 0
0.075 f (1) x =
2x
2x2
0.3 p = D(x) = 80
0.08
0
0.009975 f (0) x =
R(x) = 80x
0.01
0.2x
0.2x2
y = 0.2624 f 0 (0) x = 0.3 y = 0.12048 f 0 (1) x = 0.12
f f
( 1, 1.5) ( 4, 1)
g
0.1109 g
(1.5, 4)
( 2.5, 1) ( 4, 2.5)
g
(1, 2.5) (2.5, 4)
( 1, 1) f0 > 0 0
f 0 h0 < 0
x = 2.5 ( 1, 1.5)
( 4, 1)
0
h =0
(1.5, 4)
x=1 5
x= (1.5, 2)
x=
( 1.5, 2.5)
( 4, 2.5)
2 2
x=
x=2
( 1.5, 2.5)
f
x=0 2
x=
3 0
x=2
( 4, 0)
f
3
(0, 4)
f
(0, 1) f
0
f A
( 4, 0)
f0
AU
(0, 4) f 00
2
( 4, 0)
f 00
f0
(0, 4)
-x
0
f
+ x = 2.5
f p ( 1/ 3, 5/9)
p (1/ 3, 5/9)
h
(3, 3) (0.59, 0.19)
2.5
(3.41, 0.38)
( 3, 2) ( 4, 3)
A AU
g
g0
(0.5, 4)
+
-x
0
( 2, 0.5)
x = 1, 5
( 3, 1) ( 2, 3) ( 1, 1.5) (0.5, 2) (3, 0.5) ( 2, 3) ( 3, 1) (4, 4)
A AU
h
(0.5, 2) (0.5, 2)
( 1, 1.5) (1.5, 1) (3, 0.5) ( 1, 1.5)
( 4, 3) ( 3, 1) ( 2, 12) ( 1, 2)
( 2, 2)
h0
(3, 4)
(2, 1)
f
( 1, 0)
+
A AU
AU 0
K0
2
0
+
(0, 3/2)
s A
s0
(1, 1) (1, 1)
(0, 1)
0
A AU
AU
f
-x
x = 0, 3/4
(3/2, 27/16)
(3/2, 27/16)
( 1, 0)
-x
x = 0, 2
( 1, 0)
f
(0, 0)
0
K A
(0, 4)
(3/2, 1)
0
(1.5, 3)
( 2, 12) (2, 20)
(0, 0)
5
(2, 20)
f
(0, 1)
+
1
0
3/4
0
0 x=
2, 0, 2
-x +
A U A
A U A
f
2
f0
0
+
fA
AU
0
2
0
0
- x f0
+
x=2
2.5
-x
0
+
g
(0, 4)
g
(2, 0)
g A
U A
g
2
(0, 2)
+
h
g0 2 0
h0
A AU +
1
5
0
0
(2, 16) ( 1, 2)
f f
(2.5, 0.25)
A AU
f
x = 2, 0, 2 f 00 ( 2) = 32 > 0 f x = 2 f 00 (0) = 16 < 0 f x = 0 f 00 (2) = 32 > 0 f x=2
f0
+
2
2
0
0
s
-x +
(2, 5)
s
(0, 1) s
s
(2, 1)
( 2, 2)
x = 0, 3/4 s00 (3/4) = 9/4 > 0 x = 3/4
(2.5, 1) ( 1, 2.5)
+
( 2, 16)
f
x=0 0 K
f
(5, 1)
(1, 5)
h
x = 0, 2
2
( 1, 1)
h
00
f
+
(1, 8 13 )
h
x = 1, 5 h00 (1) = 4 < 0 h x=1 00 h (5) = 4 > 0 h x=5
s00 (0) = 0 s
0
-x
(5, 2 13 )
x = 2.5 1 0 f x=2
K (2) =
2
h
+ x=2
00
+
0
-x
+
g 00 (2.5) =
A AU
g
x=2
f
(2, 1)
-x
g0
h
( 1, 0)
g
(0, 2) ( 1, 0)
(2, 1)
s
(1, 2) ( 1, 2)
s s A
0 s0
0
+
p
A AU
s
p ( 3, 1) p (0, 3)
1
0
0
s +0
1
0
+
f
f
p
f0
0
f
A U A 3 +
g
0
p
0
0
- x
g
0
(2, 1)
+ f A
A AU
AU
( 2, 18) (0, 2) ( 1, 2) (0, 2) ( 2, 0) (2, 1)
(2, 18) g
(0, 2) ( 1, 0)
f 3
0 f0
2
0
+
-x
0
g
( 3, 2)
g
A U A
g
2
A AU 0
g0 + 0
2
0
+
( 3, 1) ( 1, 3)
g
- x
g
0 gA
h h
AU
(1.5, 3.69) (1.5, 1) ( 1, 1.5)
h h
h A
3 g0
h0
0
1.5
0
0
+
( 1, 1)
-x + hA
A AU
AU
f f
(1, 1)
3
(1, 1) ( 1, 1)
f f
-x
h h
A U A
U A
-x
h0
fA
U A
f0
2.5
-x
1
+
- x
(2, 0.54) (0, 0)
f
A AU
(1, 1)
A AU
(0, 4) 3, 13)
p ( 3, 0) p ( 1, 3)
f
-x
0
(
p ( 3, 13) f
( 1, 1) ( 1, 1)
s
2
f f
s
A U A
U A
f
(2.5, 0.25) f
y
y
8
10 8
6
!4
6
4
4
2
2
2
!2
4
6
x
!4
2
!2
4
x
!2
!2
!4
g
(0, 4)
(2, 0)
g
(
(1, 2)
p
f 3, 13)
(0, 4)
p ( 3, 13)
f (1, 9)
y
( 1, 9) y
10 5 5
!4
2
!2
4
!3
x
!2
1
!1
2
3
x
!5
!5 !10 !10 !15
(1, 8 13 )
h h
(5, 2 13 ) (3, 3)
g (2, 18)
( 2, 18) (0, 2)
g
p ( 2/ 3, 10.9)
y 10
p (2/ 3, 10.9) y
8
20
6
15
4 2 !4
10 2
!2
4
6
8
x 5
!2 !4 !4
f
( 2, 16) f
(2, 16) (0, 0)
2
!2
h
(0, 2)
(0, 2)
y
10
4
2
!2
4
2
x !2
!10
1
!1 !2
!20 !4
(2, 5) s
(1.5, 3.69)
(1, 3)
20
s
x
h
y
!4
4
(0, 1) (1, 3)
f f
(1, 1)
2
3
x
y
h h
4
( 3, 2) ( 3, 2) y
2
!1.0
8 0.5
!0.5
1.0
1.5
2.0
6
x
4
!2
2 !4 !6
s ( 1, 2) s p ( 1/ 2, 1.24) (0, 0)
!2 !4
f
p (1/ 2, 1.24)
(2.5, 0.25)
f f f
4
f f
2 1
!1
2
g g g
!4
g (0, 0)
(2, 0.54)
g g
(3.41, 0.38)
g
2.0
h
1.5
h h
1.0
h
0.5 2
!2
4
h h
x
!0.5
h
!1.0
f
f
( 0.5, 0.184) f
f ( 1.5, 0.1)
f f
g g
f f
( 3, 2) f y
(1, 8 13 ) (5, 2 13 ) ( 1, 1) (5, 1) (1, 5) (3, 3) (3, 1) ( 1, 3) ( 2, 16) (2, 16) ( 1, 2) (2, 1) ( 2, 2) (0, 0) (0, 1) ( 1, 0) (2, 5) (0, 1)
s s s s
2
!2
(0, 4) (2, 0) ( 1, 0) (2, 1) (0, 2) (1, 2) (1, 1) ( 1, 1)
s
4
!4
( 1, 1)
s
6
!6
(2.5, 1) ( 1, 2.5)
x
!2
f f (0.59, 0.19)
2
!2
(1, 2) (0, 0)
y
!2
!4
2
x
s
(0, 2) ( 1, 0) ( 1, 1) (1, 1)
(2, 1) (1, 3)
x
f f
p ( 3, 13) f
f f (1, 9) f
(
p
(0, 4) 3, 13)
f (0.59, 0.19) f (3.41, 1) f (0.59, 3.41)
p p ( 3, 0) ( 3, 1) p p ( 1, 3) (0, 3) ( 1, 9) ( 1, 1) ( 1, 1)
f
(1, 1)
g
( 3, 2)
g ( 3, 1) ( 1, 3)
g g (2, 18) g g g p ( 2/ 3, 10.9) g g
( 2, 18) (0, 2) ( 1, 2) (0, 2) ( 2, 0) (2, 1)
g g g ( 3, 1)
p (2/ 3, 10.9) p p ( 2/ 3, 2/ 3) p ( 1, 2/ 3)
p (2/ 3, 1)
(3.41, 0.38) ( 1, 0.59)
( 1, 3)
h h h h
( 1, 1)
h
h h h h h (0, 2) h h
( 3, 2)
( 3, 1) ( 1, 3)
(1.5, 3.69) (1.5, 1) ( 1, 1.5)
y 12 10 8
(1, 3) ( 1, 0) (0, 1)
6
(1, 1)
4 2
f f
(1, 1) f
f f f
!8
!6
f
!4
!2
4
x
!5
(2, 0.54) (0, 0) (0, 2) ( 1, 0)
2
x
5
(1, 1) s p p ( 1/ 2, 1.24) (0, 0) (1/ 2, 1.24) p s ( 1, 1/ 2) p (0, 1/ 2) s p p ( 1/ 2, 0) (1/ 2, 1)
f
8
y
( 1, 1) ( 1, 1)
f
6
10
(1, 2) ( 1, 2)
f
4
!2
( 1, 1)
s
s
2
!2
(1, 1) ( 1, 1)
s s
!4
!10 y 10 5
5
!5 !5
(2, 1)
!10
x
y 8
y 8
6
6
4
4
2
2 !4
2
!2
!4
x
4
2
!2
x
4
!2 !4
!2 !4 y y 3
4
2 2
1 2
!2
4
6
x 2
!2
4
6
x
!1 !2 !2 !3 y y
3
12
2
10
1
8 6 !6
4
!4
!2
2 !4
2
4
6
2
4
6
2
4
6
x
!1 !2 2
!2
4
6
8
x !3
!2 y
y 8
6
6
4
4
2
2 !6
!4
2
!2
x
!6
!4
!2
!2
x
!2
!4 y
30
10
20 5
10 5
!5
x
!6
!4
!2
x
!10 !5
!20 !30
!10
P (x) = 70x
0.2x2
1, 500 0 x 400
y
x ⇡ 52
4000
9x 0 x 100
M P (x) = 464
2000 0
100
200
300
400
x 400
!2000 200 !4000
0
20
x = 175 M P (x) = 70 y
40
60
80
100
40
60
80
100
80
100
x
!200
0.4x 0 x 400
!400
100
x ⇡ 52
50 0
100
200
300
400
x y
!50
20 000
!100
15 000
x = 175
10 000
y
5000
40 000 0 30 000
0
20
x
20 000 y
10 000 0
0
100
200
300
400
x
500 400 300
y
200
140 120
100
100
0
80
0
20
40
60
x
60 40
x ⇡ 52
20 0
0
100
200
300
400
x
x
x = 175 P (x) = 464x 0 x 100
4.5x
2
100
4, 000
80 60
8000
40
6000
20
4000 2000 0 !2000 !4000
xE ⇡ 62 pE ⇡ 16
0 20
40
60
80
100
x
0
5
10
xE ⇡ 7 p E ⇡ 9
15
20
25
30
p
x
y
14
4
12 10
2
8 6
!4
2
!2
4
x
!2
2 0
4
0
2
4
6
8
10
12
p
14
!4
x 6= 0 K x=2
x=
K y
4 x=4
x=1 x=2
x=0 y=0
20
x=1 15 10
y=2
y=0
y=2
y=3
5
!4
y=
1 x 6= 0
f
x 6= 0
x=0 y=0
2
!2
s
4
x=0 y=1 s
f
y 10
y
8 4
6
2
!4
4 2
!2
2
x
4
!2
!4
2
!2
4
x
!2 !4
x 6= 2 g
x=2
x 6= 0
y=0
x=0 y=x
g f
y
(
p
p 2, 2 2)
p p ( 2, 2 2) f
4 2
10 2
!2
4
6
x 5
!2 !4 !4
x= h
1
x 6=
2
!2 !5
1 y=0 h
!10
x
4
x
x 6= 2
x=2
10
y=2
g
5
g y !4
15
!10
2
!2
10
!5
5
!10 5
!5
10
x
u(x) =
!5
1 x 6= x 2 x 6= 2, 2
2 x=2 y=0
!10
u x 6=
x
4
u y
2, 2
3
x=
2 x=2
2
y=0
1
u
(0, 1/2) u !6
!4
2
!2
4
6
x
!1 y
!2
4
!3
( 1, 1)
2
!4
2
!2
4
x
y=0 f
!2
p (0, 2) ( 1/ 3, 3/2)
f
p (1/ 3, 3/2)
y
!4 4
x 6= 0 h
3
x=0 y=2
2
h
1
y !4
10
2
!2
4
x
!1 5
!4
x 6= 2
!2
4
x
y=
1, 1 x=
1 x=1
1
g
(0, 0) g
!5 y !10 4
x 6= 0
2
x=0 y=x K p p ( 2, 2 2) K
(
p
p 2, 2 2)
!4
2
!2 !2 !4
4
x
x 6= 2
x=2
( 1, 2)
g g
y = x/2 + 1 h (4.24, 4.24) h
( 0.24, 0.3)
(2, 1)
( 1, 2) (2, 1)
g g y 4
10
2 5 2
!2 !4
2
!2
4
6
8
x
4
x
6
!2 !4
!5
x 6=
!10
s(x) = x
2 x 6=
x 6=
s
1 y x=
2
2
(0, 2)
1 y=0
h s
( 1, 1)
h
y
h
2
( 1, 1) ( 1, 1)
h x
5
!5
( 1, 1)
h
y
!2 4 !4
2
!6 !4
x 6= 0
x=0 y=0
( 1, 0)
f
!4
x 6= 0
(0, 1)
f
( 1, 0) (0, 1)
K K
y
K
4
( 1, 0)
K
2
!4
(0, 1)
y
2
!2
4
20
x
15
!2
10
!4
x 6= 2 y
x=2 y=0
g
x=0 y=0
K
(0, 1) ( 1, 0)
f
x
4
!2
f f
2
!2
5
(0, 1/2) !4
!2
x 6= 0
2
4
x
x=0 y=1
(2, 1) ( 1, 2)
g g
y
s
15
( 1, 0) (0, 1)
s s
10 5
s ( 1, 0)
s
(0, 1)
!10
y
5
!5
10
!5
8
!10
6
x= 2 x=2 y=0
2
!4
2
!2
4
x
x 6= 2, 2 (0, 1/2)
y
4
10
x
u
!2
(0, 1/2) u ( 1, 2) ( 2, 0) (0, 2) (2, 1)
u u
x 6= 0
u
x=0
( 1, 2) ( 2, 2)
u u
y=x
(2, 1)
y
f ( f
p
p
2, 2 2) f
4
p p ( 2, 2 2) p ( 1, 2)
2
p ( 2, 1) f p p ( 2, 0) (0, 2)
!4
2
!2
4
x
!2
f !4
(0, 1) ( 1, 0)
f f
x 6= 0 x
10
h
5
( 1, 0)
h !4
2
!2
4
x
h h
y
!10
10
x 6= 2 (0, 2.5) ( 2.5, 0)
5
x=2 y=2
!4
g g
(0, 1)
(0, 1) ( 1, 0)
h !5
g
( 5/2, 0)
x=0 y=2
2
!2 !5
( 1, 2)
(2, 1)
!10
4
x
x 6= 0
u x=0
(2, 1)
u
( 1, 2)
u
y=x
( 2, 2)
y
K ( K
p
p 2, 2 2) K p p ( 2, 2 2) ( 1,
p
3 2
p
1
2)
( 2, 1) K p p ( 2, 0) (0, 2)
!6
!4
2
!2
4
6
x
!1
K
!2
(0, 1) ( 1, 0)
K K
!3
10
( 1, 1)
y
(0, 2)
5
y=0 !4
2
!2
4
x
f
(0, 2) ( 1, 0)
f
!5
(0, 1)
f !10
f 2 x 6=
s(x) = x
2
x 6= 2 (0, 2) (2, 0)
p ( 1/ 3, 3/2)
f
p (1/ 3, 1) f p p ( 1/ 3, 1/ 3)
s ( 1, 2)
s s
p (1/ 3, 3/2) p ( 1, 1/ 3)
y
( 2, 1)
4 3
s y
2
2 1 5
!5
x !4
!4
y !6
y
x
x 6=
1, 1
(0, 0)
x=
1
x 6= 2 x 2 x 6= 2, 2 (0, 1/2)
1 x=1
y= g
1 (0, 0)
g
x=2 y=0
(0, 1) g
(1, 1)
( 1, 1)
( 1, 0)
g
u u ( 2, 2)
4
!1
!2
u(x) =
2
!2
( 1, 2) (2, 1)
g
( 1, 1) g
( 1, 1)
(1, 1)
y 10 4
8
2
!4
6 2
!2
x
4
4
!2
2
!4
0
x 6= 2 y
(0, 1/4)
0
1
2
3
4
20
30
40
p
pE = 20 xE = 3 x
x=2
10 8
y = x/2 + 1
6
h
4
( 0.24, 0.3) h
2
(4.24, 4.24)
0
h ( 1, 0.24)
(4.24, 1)
h
( 0.24, 2)
0
10
p
(2, 4.24)
h (2, 1)
h
g(2.5) = 2.5 g( 2.5) = 2.5
( 1, 2)
h
g(2) = 2.3 g( 2) = 2
10
0.5
5
!4
2
!2
4
6
8
x
[ 1, 1]
0.5 [ 1, 1] g( 2) = 2 f ( 1.5) =
!5
f (1.5) = 2 f (0) = 1
!10
f ( 3) = P (x) = 70x AP (x) = 70
0.2x2
0.2x
1, 500 0 x 400
1, 500/x
y=
2 13
0.2x + 70
40 20 100
!20
200
300
400
x
2
6
x=0 0.25 x = 2.5
3
x= 1 x=2
!80
3
!100
6 23 2 13
pE ⇡ 1.5 xE ⇡ 4.4
x=1 x=5
0.5 x=2 0.5 x=
!40 !60
3
0.54 x=2 0 x=0
y
0
f (0) = 1 f ( 1.5) = 8 13
1 x 400
1.3
x=2 x=5
3
29.6 x= 0 x=0
2
P (x) = 464x
4.5x2
4, 000
0 x 100
0.5 x=2 0 x=0
52
10.98 x=4 0 x= 3 5 4
x = 1, 4 x=2
2
x=0 x=
79 0.54 0.37
3, 3
x=2 x=1
0
x=0 x=
0.5
2
9.9 x=3 3 x= 2 4 5
x
x·y
x= 2 x= 1
8 13
y
x=1 A(x) = 10x
0.54
x=2
0.5
x2
0 x 10 x⇤ = 5
x=2
⇥ 0.25
x = 2.5
2x + 2y
x 7
0
x=4
y
x=0 0.4 0.5
0
x= 4
P (x) = 2x + 128/x
x= 1 x= 2
0