Homework Homework Assignment #21 Review Sections 3.1 – 3.11 Page 207, Exercises: 1 – 121 (EOO), skip 73, 77 Chapter 3 Test next time Rogawski Calculus

Homework

Homework Assignment #21 Review Sections 3.1 – 3.11 Page 207, Exercises: 1 – 121 (EOO), skip 73,

77 Chapter 3 Test next time

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 207 1. Compute the average ROC of f (x) over [0, 2]. What is the graphical interpretation of this average ROC?


7 13

2 0 represents the slope

of the secant line between

0,1 and 2,7 .

avg

avg

ROC

ROC

Homework, Page 207 Compute f ′ (a) using the limit definition and find an equation of the tangent line to the graph of f (x) at x = a.

5.


2 , 1f x x x a

0

2 2

0

2

0

2

0 0

2

1 1lim

1 1 1 1lim

1 2 1 0lim

2lim lim 2 1 2 1 1

1 1 1 0 1 1

h

h

h

h h

f h ff a

h

h h

h

h h h

h

h h hh

h

f y x

Homework, Page 207 Compute dy/dx using the limit definition.

9.


24y x

0

2 2

0

2 2 2

0

2

0 0

lim

4 4lim

4 2 4lim

2lim lim 2 2

2

h

h

h

h h

f x h f xdy

dx h

x h x

h

x xh h x

h

xh hx h x

h

dyx

dx

Homework, Page 207 Express the limit as a derivative.

13.


0

1 1limh

h

h

0 0

0

1 11 1lim lim , 1

1 1lim , 1

h h

h

f h fh dx a

h h dx

h dx a

h dx

Homework, Page 207 17. Find f (4) and f ′(4) if the tangent line to the graph of f (x) at x = 4 has an equation y = 3x – 14.


3 14 3 4 2 2 3 4

4 2, 4 3

y x x y x

f f

Homework, Page 207 21. A girl’s height h(t) (in cm) is measured at time t (years) for 0 ≤ t ≤ 14:

52, 75.1, 87.5, 96.7, 104.5, 111.8, 118.7, 125.2,

131.5, 137.5, 143.3, 149.2, 155.3, 160.8, 164.7

(a) What is the girl’s average rate of growth over the 14-yr period?

(b) Is the average growth rate larger over the first half or second half of this period?

164.7 52 8.05 cm/yr

14 0avg h t

1

2

125.2 52 10.547 cm/yr

7 0164.7 125.2

5.643 cm/yr14 7

The average growth rate is greater over the first seven years.

avg h t

avg h t

Homework, Page 207 21. Estimate h′(t) (in cm/yr ) for t = 3, 8.

104.5 87.53 8.5 cm/yr

4 2137.5 125.2

8 6.15 cm/yr9 7

h

h


Homework, Page 207 25.

By Theorem 1, 2 ln 2 2 .x xd

dx

1

Which of the following is equal 2

2 ln 2 2

1 2 2

ln 2

x

x x

x x

d

dx

a b

c x d


Homework, Page 207 Compute the derivative.

29. 3

24y x

3 512 2

52

34 6

2

6

dyx x

dx

dyx

dx



33. 3 2

4 9

ty

t

2 2

2 2

2

3 2, 33 2

4 9, 44 9

4 9 3 3 2 4

4 9

12 27 12 8 19

4 9 4 9

19

4 9

u t uty

v t vt

t tdy vu uv

dx v t

t t

t t

dy

dx t



33. 3 2

4 9

ty

t

2 2

2 2

2

3 2, 33 2

4 9, 44 9

4 9 3 3 2 4

4 9

12 27 12 8 19

4 9 4 9

19

4 9

u t uty

v t vt

t tdy vu uv

dx v t

t t

t t

dy

dx t



37. 3 41 4y x x

3 23 4

4 3

3 3 4 2

3 3 4 2

1 , 3 1 11 4

4 , 4 4 1

1 4 4 4 3 1

4 1 4 3 4 1

u x u xy x x

v x v x

dyuv vu x x x x

dx

dyx x x x

dx



41.

1

1 2y

x x

11 2

1 2 2

3 312 2 2

3 11 22 2

3 22

11 2

1 2

1 , 1 1 1 1

1 12 , 2 1 22 2

11 2 2 12

1 1

1 22 1 2

y x xx x

u x u x x

v x v x x

dyuv vu x x x x

dx

dy

dx x xx x



45. 2sin 1y x

12 2 2

1 12 22 2

12 2 2

2

2

sin 1 sin , 1

1 1 2 12

cos cos 1 1

cos 11

y x u u x

u x x x x

dyu u x x x

dx

dy xx

dx x



49.

, 1csc 9 1 ,

csc 9 1 , 9csc 9 1 cot 9 1

9csc 9 1 cot 9 1 csc 9 1

csc 9 1 1 9cot 9 1

u z uy z z

v z v z z

dyuv vu z z z z

dx

dyz z

dx

csc 9 1y z z



53.

cos cos cos , sincos cos cos ,

cos , sin

sin sin cos cos sin cos sin

sin cos cos sin cos sin

u v u v vy

v v

dyu u

dx

dy

dx

cos cos cosy


Homework, Page 207 Find the derivative.

57.

ln , , 1

1 1 1

1

1

x x x

x x

x x

x

x

f x x e u x e v u e

dy du e ef x

du dx x e x e

ef x

x e

ln xf x x e



61. 12 tg t t e

2

121 1

2

1 1 122

1

, 2, 1

,

12 2 1

2 1

t

t t

t t t

t

u t u tg t t e

v e v et

g t uv vu t e e t e tt

g t e t



65. 2sin xf x e

2

2 2

2

sin 2

sin sin

sin

, sin , 2sin cos

2sin cos 2 sin cos

2 sin cos

x u

u x x

x

f x e e u x u x x

f x e u e x x e x x

f x e x x



69. 1tanG s s

1

2 2

1tan , ,

21 1 1 1

1 2 2 11

1

2 1

G s s u s us

G x uu s s ss

G xs s


Homework, Page 207 Use the table of values to calculate the derivative of the given function at x = 2.

81.

f xR x

g x


x f (x) g (x) f ′ (x) g ′ (x)

2 5 4 –3 9

4 3 2 –2 3

2 2

4 3 5 2 2 1

16 84

1

8

g x f x f x g xR x

g x

R x

Homework, Page 207 Let f (x) = x3 – 2x2 + x + 1.

85. Find the points on the graph where the tangent line has a slope of 10

3 2 2

2 2

2 1 3 4 1

3 4 1 10 3 4 9 0

1.189,2.523

f x x x x f x x x

x x x x

x


Homework, Page 207 Let f (x) = x3 – 2x2 + x + 1.

89. (a) Show that there is a unique value of a such that f (x) has the same slope at both a and a + 1.

(b) Plot f (x) together with the tangent lines at x = a and x = a + 1 and confirm the answer in part (a).

3 2 2

22

2 2

2 1 3 4 1

3 4 1 3 1 4 1 1

13 4 1 3 6 3 4 4 1 0 6 1

6

f x x x x f x x x

a a a a

a a a a a a a


Homework, Page 207 Calculate y″.

93. 2 3y x

2 2

3

2 3, 2 3, 2

1, 01 1 2 1

12 3,12 2 2 3 2 3

2 3

12 3 0

2 3

2 3

1

2 3

y x u x u

u u

y uv x vu x x

x

xvu uv xy

v x

yx


Homework, Page 207 97. In Figure 5, label f, f ′, and f ″.


Homework, Page 207 Compute dy/dx.

101. 3 3 4x y

3 3 2 2 2 2

2 2

2 2

4 3 3 0 3 3

3

3

dy dyx y x y y x

dx dx

dy x dy x

dx dxy y


Homework, Page 207 Compute dy/dx.

105. 2 23 2x y x y

22 2 2 2

2 2 2 2 2 2

3 2 3 2

2 3 2 2 2 0

4 2 2 2 0 2

2

x y x y x y x y

x xy y x y x xy y

dy dy dyx x y y y x y x

dx dx dx

dy y x

dx y x


Homework, Page 207 109. Find the points on the graph of x3 – y3 = 3xy – 3 where the tangent line is horizontal.

3 3 2 2

2 2 2 2

22 2

2

33 2 2 6 3

3 3 3 3 3 3

0 0

3 3 3 3 0

1.559,0.925

dy dyx y xy x y x y

dx dxdy dy dy

x y x y x y x ydx dx dx

dy x y dyx y y x

dx dxx y

x x x x x x

x


Homework, Page 207 113. Water pours into the tank in Figure 7 at a rate of 20 m3/min. How fast is the water level rising when the eater level is h = 4m?

1 11 2

21

121.5

2 8 22 24 1.52 1.5

10 24 7.52 2

1 1 115 20 / min

15 15 8 6

1 ft min6

l l ll l lV wh l h V wh

hhl h

V wh h h

dV dh dh dVh ft

dt dt dt h dt

dh

dt


Homework, Page 207 117. (a) Side x of the triangle in Figure 9 is increasing at 2 cm/s and side y is increasing at 3 cm/s. Assume that θ decreases in such a way that the area of the triangle has a constant value of 4 cm2. How fast is θ decreasing when x = 4, y = 4?


Homework, Page 207 117. (a)

1

1 1sin cos sin sin 0

2 2

1 1 14 4 4 sin sin sin

2 2 2 6

sin sincos sin sin

cos

1 14 3 4 2

102 2

8 334 4

2

dA d dy dxA xy xy x y

dt dt dt dt

dy dxx yd dy dx d dt dtxy x y

dt dt dt dt xy

d

dt

0.722 /rad s


Homework, Page 207 117. (b) How fast is the distance between P and Q changing when x = 2, y = 3?


11 1 4sin 4 2 3 sin sin . . .

2 2 3Triangle with area 4 with sides 2 and 3 does not exist.

A xy D N E

A x y

Homework, Page 197Use logarithmic differentiation to find the derivative .


1sin121.

ln

xe xy

x

1 1

2

1 1 2

1

1 2

ln ln ln sin ln ln ln ln sin ln ln

1 11 1 111 1

sin ln lnsin 1

sin 1 11

ln lnsin 1

x

x

y e x x x e x x

dy x xy dx x x x xx x

dy e x

dx x x xx x

Documents

Homework Homework Assignment #21 Review Sections 3.1 – 3.11 Page 207, Exercises: 1 – 121 (EOO), skip 73, 77 Chapter 3 Test next time Rogawski Calculus