Upload
theodora-brown
View
229
Download
0
Embed Size (px)
Citation preview
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?
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Compute the derivative.
29. 3
24y x
3 512 2
52
34 6
2
6
dyx x
dx
dyx
dx
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Compute the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Compute the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Compute the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Compute the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Compute the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Compute the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Compute the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Find the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Find the derivative.
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 Find the derivative.
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
Rogawski CalculusCopyright © 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.
81.
f xR x
g x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 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.
(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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 207 97. In Figure 5, label f, f ′, and f ″.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 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.
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
Rogawski CalculusCopyright © 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?
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
Rogawski CalculusCopyright © 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 CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 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?
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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 .
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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