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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
Copyright © 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 1
3
20
represents the slope
ROCavg 
ROCavg
of the secant line between
 0,1 and  2, 7  .
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
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.
2
f
x

x
 x, a  1


5.
f 1  h   f 1
f   a   lim
h 0
h
1  h 

 lim
2
 

 1  h   1  1
2
h 0
h
1  2h  h 2  1  h  0
 lim
h 0
h
2h  h 2  h
 lim
 lim  2  h  1  2  1  1
h 0
h 0
h
f 1  1  1  0  y  1 x  1
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute dy/dx using the limit definition.
9. y  4  x 2
f  x  h  f  x
dy
 lim
dx h0
h

 lim
 
4   x  h   4  x2
2

h 0
h
4  x 2  2 xh  h 2  4  x 2
 lim
h 0
h
2 xh  h 2
 lim
 lim  2 x  h   2 x
h 0
h 0
h
dy
 2 x
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Express the limit as a derivative.
13. lim 1  h  1
h 0
h
f 1  h   f 1 d
1  h 1
lim
 lim

x, a  1
h 0
h 0
h
h
dx
1  h 1 d
lim

x, a  1
h 0
h
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
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.
y  3 x  14  3  x  4   2  y  2  3  x  4 
f  4   2, f   4   3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
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?
164.7  52
avg h  t  
 8.05 cm/yr
14  0
(b) Is the average growth rate larger over the first half or second
half of this period?
125.2  52

avg h  t 1 
 10.547 cm/yr
70
164.7  125.2

avg h  t 2 
 5.643 cm/yr
14  7
The average growth rate is greater over the first seven years.
Homework, Page 207
21. Estimate h′(t) (in cm/yr ) for t = 3, 8.
104.5  87.5
h  3  
 8.5 cm/yr
42
137.5  125.2
h  8  
 6.15 cm/yr
97
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
25. Which of the following is equal
a
c
 b   ln 2  2 x
2x
x2
d x
2
dx
x 1
d 
1 x
2
ln 2
d x
By Theorem 1,
2   ln 2  2 x.
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute the derivative.
3
29. y  4 x 2
dy
5
 3   3 2 1
 4   x
 6 x 2
dx
 2
dy
5
 6 x 2
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute the derivative.
3t  2
33. y 
4t  9
u  3t  2, u   3
3t  2
y

4t  9 v  4t  9, v  4
dy vu   uv  4t  9  3   3t  2  4 


2
2
dx
v
 4t  9 

12t  27  12t  8
 4t  9 
2

19
 4t  9 
2
dy
19

dx  4t  9 2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute the derivative.
3t  2
33. y 
4t  9
u  3t  2, u   3
3t  2
y

4t  9 v  4t  9, v  4
dy vu   uv  4t  9  3   3t  2  4 


2
2
dx
v
 4t  9 

12t  27  12t  8
 4t  9 
2

19
 4t  9 
2
dy
19

dx  4t  9 2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute the derivative.
3
4
37. y   x  1  x  4 
y   x  1  x  4 
3
4
u   x  13 , u   3  x  12 1


4
3

v   x  4  , v  4  x  4  1



dy
3
3
4
2


 uv  vu   x  1 4  x  4    x  4  3  x  1
dx

dy
3
3
4
2
 4  x  1  x  4   3  x  4   x  1
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute the derivative.
1
41. y 
1  x  2  x
y
1
1  x 
2 x
 1  x 
1
2  x
1
2
u  1  x 1 , u   11  x 2  1  1  x 2


1
3
3
2
2
1
1
v   2  x  , v   2  2  x   1  2  2  x  2
dy
1
3
1
2
2
2
1


 uv  vu  1  x 
 2  x    2  x  1  x 
2
dx
dy
1
1


3
dx 2 1  x  2  x  2 1  x 2 2  x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute the derivative.
45. y  sin x 2  1
 
u   1  x  1  2 x   x  x  1
2
dy

 cos u u   cos x  1  x  x  1
dx

y  sin x  1  sin u, u  x  1
2
2
2
1
2
dy

dx
x
x2  1
2
1
2
2
1
2
2
1
2



cos x 2  1
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute the derivative.
49. y  z csc  9 z  1
u  z , u   1
y  z csc  9 z  1 , 
v  csc  9 z  1 , v  9 csc  9 z  1 cot  9 z  1
dy
 uv  vu   z  9 csc  9 z  1 cot  9 z  1   csc  9 z  1
dx
dy
 csc  9 z  1 1  9 cot  9 z  1 
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute the derivative.
53. y  cos  cos  cos   
u  cos  cos    cos v, u    sin v v
y  cos  cos  cos    , 
v  cos  , v   sin 
dy
  sin u u    sin  cos  cos      sin  cos      sin  
dx
dy
  sin  cos  cos    sin  cos   sin 
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Find the derivative.
x
f
x

ln
x

e


57.




f  x   ln x  e x , u  x  e x v, u   1  e x
dy du
1 1 ex 1 ex
f  x 


x
du dx x  e
1
x  ex
1 ex
f  x 
x  ex
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Find the derivative.
1
2 t
61. g  t   t e
u  t 2 , u   2t
1
2 t 
g t   t e , 
1
1 1t
t
v  e , v   2 e
t

1
1 1t  1t
2
g   t   uv  vu  t   2 e   e 2t  e t  2t  1
 t

g t   e
1
t
 2t  1
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Find the derivative.
sin 2 x
65. f  x   e
f  x  e
sin 2 x
 eu , u  sin 2 x, u   2sin x cos x
f   x   e u  e
u
f   x   2e
sin 2 x
sin 2 x
2sin x cos x  2e
sin 2 x
sin x cos x
sin x cos x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Find the derivative.
1
s
69. G  s   tan
 
G  s   tan
1
 s ,u 
1
G  x   2
u 
u 1
s , u 
1
2 s
1
1
1

2
s  1 2 s 2  s  1 s
 
1
G  x  
2  s  1 s
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Use the table of values to calculate the derivative of the given
function at x = 2.
x
2
4
81. R  x  
R  x  
f (x)
5
3
g (x)
4
2
f ′ (x)
–3
–2
g ′ (x)
9
3
f  x
g  x
g  x f  x  f  x g x
 g  x 
2

4  3  5  2 
 4
2
2
1


16
8
1
R  x   
8
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
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
f  x   x3  2 x 2  x  1  f   x   3x 2  4 x  1
3 x 2  4 x  1  10  3 x 2  4 x  9  0
x  1.189, 2.523
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
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.
f  x   x3  2 x 2  x  1  f   x   3x 2  4 x  1
3a 2  4a  1  3  a  1  4  a  1  1
2
1
3a  4a  1  3a  6a  3  4a  4  1  0  6a  1  a 
6
(b) Plot f (x) together with the tangent lines at x = a and x =
a + 1 and confirm the answer in part (a).
2
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Calculate y″.
y  2x  3
93.
y  2 x  3, u  2 x  3, u   2
u  1, u   0
1
1
2
1



y 
u 


1
2 u
2 2x  3 1
2x  3
v  2 x  3, v  2 x  3

1
2x  3 0 
vu   uv
2x  3
y 

2
v2
2x  3

y 

1
2x  3


3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
97.
In Figure 5, label f, f ′, and f ″.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute dy/dx.
x3  y 3  4
101.
x3  y 3  4  3x 2  3 y 2
dy
dy
 0  3y2
 3x 2
dx
dx
dy 3 x 2
dy x 2
 2
 2
dx 3 y
dx y
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
Compute dy/dx.
2
2
105. x  y  3x  2 y
x  y  3x  2 y   x  y   3x 2  2 y 2
2
2
2
x 2  2 xy  y 2  3x 2  2 y 2  2 x 2  2 xy  y 2  0
dy
dy
dy
4x  2x  2 y  2 y
 0   y  x   y  2x
dx
dx
dx
dy y  2 x

dx
yx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
109. Find the points on the graph of x3 – y3 = 3xy – 3 where the
tangent line is horizontal.
dy
dy
x  y  3 xy  3  3 x  3 y
 3x  3 y
dx
dx
dy
2 dy
2
2 dy
x y
 x  y  x y
 x2  y
dx
dx
dx
dy x 2  y
dy
2
2



0

x

y

0

y

x
dx x  y 2
dx
3
3
2
2

 
x  x
3
2 3

 
 3x x 2  3  x 6  3x3  3  0
x  1.559, 0.925
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
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?
l1   l1  l 
l1  l2
l 12
V
wh 
  l  1.5h  V 
wh
2
h
8
2
2  24   1.5h
2l1  1.5h
V
wh 
10  h  24  7.5h 2
2
2
dV
dh
dh
1 dV
1
1
 15h



20

  ft / min
dt
dt
dt 15h dt 15  8 
6
dh 1 ft

dt 6 min
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
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?
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
117. (a)
1
dA 1 
d
dy
dx 
xy sin  
  xy cos 
 x sin 
 y sin    0
2
dt 2 
dt
dt
dt 
1
1
1 
4   4  4  sin   sin      sin 1 
2
2
2 6
dy
dx
x sin 
 y sin 
d
dy
dx  d

dt
dt
xy cos 
   x sin 
 y sin   

dt
dt
dt 
dt
xy cos 

1
1
4     3   4     2 

d
10
2
2




 0.722rad / s
dt
 3
8 3
 4  4   
 2 
A
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 207
117. (b) How fast is the distance between P and Q changing when
x = 2, y = 3?
1
1
4
xy sin   4   2  3 sin     sin 1  D.N .E.
2
2
3
Triangle with area A  4 with sides x  2 and y  3 does not exist.
A
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 197
Use logarithmic differentiation to find the derivative .
e x sin 1 x
121. y 
ln x
ln y  ln  e x   ln  sin 1 x   ln  ln x   x ln e  ln  sin 1 x   ln  ln x 
1
1
2
1 dy
1
1
1

x
x
 1

 1

1

1
2
y dx
sin x ln x
x ln x
sin  x  1  x
dy e x sin 1 x 
1
1 
1 




1
2
dx
ln x  sin  x  1  x
x ln x 


Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company