t 1980 1985 1990 1995 2000
129.9 187.3 271.9 409.3 568.6 C共t兲
98845_02_ch02_p182-191.qk_98845_02_ch02_p182-191 8/16/11 9:58 AM Page 191
192 CHAPTER 2 DERIVATIVES
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39. 40.
41. If , find .
42. If , find .
43. Find if .
44. Find if .
45–46 Find the limit.
45. 46.
47–48 Find an equation of the tangent to the curve at the given point.
47. , 48. ,
49–50 Find equations of the tangent line and normal line to the curve at the given point.
49. ,
50. ,
51. (a) If , find .
(b) Find equations of the tangent lines to the curve at the points and .
; (c) Illustrate part (b) by graphing the curve and tangent lines on the same screen.
; (d) Check to see that your answer to part (a) is reasonable by comparing the graphs of and .
y苷 t4 1
t4 1 y苷 sin共cos x兲
y苷 1
sin共x sin x兲 y苷 tan s1 x
xy4 x2y苷 x 3y y苷 sec共1 x2兲 x2cos y sin 2y 苷 xy y苷 sec 2
1 tan 2
y苷 1兾s3x sx y苷 共1 x1兲1
y苷ssin sx sin共xy兲 苷 x2 y
y苷 cot共3x2 5兲 y苷 共x 兲4
x44
y苷 sx cos sx y苷 sin mx
x y苷 tan2共sin 兲 x tan y苷 y 1
y苷 共x 1兲共x 4兲 共x 2兲共x 3兲 y苷 s5x tan x
y苷 sin
(
tan s1 x3)
y苷 sin2(
cosssin x)
f共2兲 f共t兲 苷 s4t 1
t共兾6兲 t共兲 苷 sin
x6 y6苷 1 y
f共x兲 苷 1兾共2 x兲 f共n兲共x兲
limt l 0
t3 tan3 2t
x l 0lim sec x 1 sin x
共0, 1兲 y苷 x2 1
x2 1 共兾6, 1兲
y苷 4 sin2x
共2, 1兲 x2 4xy y2苷 13 y苷 s1 4 sin x 共0, 1兲
f共x兲 f共x兲 苷 xs5 x
共4, 4兲 共1, 2兲
y苷 xs5 x
f f
52. (a) If , , find and .
; (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of , , and .
53. At what points on the curve , ,
is the tangent line horizontal?
54. Find the points on the ellipse where the tangent line has slope 1.
55. Find a parabola that passes through the point and whose tangent lines at and have slopes 6 and , respectively.
56. How many tangent lines to the curve ) pass through the point ? At which points do these tangent lines touch the curve?
57. If , show that
58. (a) By differentiating the double-angle formula
obtain the double-angle formula for the sine function.
(b) By differentiating the addition formula
obtain the addition formula for the cosine function.
59. Suppose that and , where
, , , , and .
Find (a) and (b) .
60. If and are the functions whose graphs are shown, let
, , and .
Find (a) , (b) , and (c) .
61–68 Find in terms of .
61. 62.
63. 64.
65. 66.
67. 68.
f f f
y苷 sin x cos x 0 x 2
x2 2y2苷 1
f f
f共x兲 苷 4x tan x
y苷 ax2 bx c
共1, 4兲 x苷 1 x苷 5
2
y苷 x兾共x 1 共1, 2兲
f共x兲 苷 共x a兲共x b兲共x c兲 f共x兲
f共x兲 苷 1
x a 1
x b 1 x c
cos 2x苷 cos2x sin2x
sin共x a兲 苷 sin x cos a cos x sin a
h共x兲 苷 f 共x兲t共x兲 F共x兲 苷 f 共t共x兲兲
f共2兲 苷 3 t共2兲 苷 5 t共2兲 苷 4 f 共2兲 苷 2 f共5兲 苷 11 h共2兲 F共2兲
f t
P共x兲 苷 f 共x兲t共x兲 Q共x兲 苷 f 共x兲兾t共x兲 C共x兲 苷 f 共t共x兲兲 P共2兲 Q共2兲 C共2兲
0 g
f y
1 x 1
f t
f共x兲 苷 x2t共x兲 f共x兲 苷 t共x2兲 f共x兲 苷 关t共x兲兴2 f共x兲 苷 xat共xb兲
f共x兲 苷 t共t共x兲兲 f共x兲 苷 sin共t共x兲兲
f共x兲 苷 t共sin x兲 f共x兲 苷 t
(
tan sx)
98845_02_ch02_p192-196.qk_98845_02_ch02_p192-196 8/16/11 9:58 AM Page 192
CHAPTER 2 REVIEW 193 69–71 Find in terms of and .
69. 70.
71.
72. A particle moves along a horizontal line so that its coor-dinate at time is , , where and are positive constants.
(a) Find the velocity and acceleration functions.
(b) Show that the particle always moves in the positive direction.
73. A particle moves on a vertical line so that its coordinate at
time is , .
(a) Find the velocity and acceleration functions.
(b) When is the particle moving upward and when is it moving downward?
(c) Find the distance that the particle travels in the time interval .
; (d) Graph the position, velocity, and acceleration functions for .
(e) When is the particle speeding up? When is it slowing down?
74. The volume of a right circular cone is , where is the radius of the base and is the height.
(a) Find the rate of change of the volume with respect to the height if the radius is constant.
(b) Find the rate of change of the volume with respect to the radius if the height is constant.
75. The mass of part of a wire is kilograms, where is measured in meters from one end of the wire. Find the linear density of the wire when m.
76. The cost, in dollars, of producing units of a certain com-modity is
(a) Find the marginal cost function.
(b) Find and explain its meaning.
(c) Compare with the cost of producing the 101st item.
77. The volume of a cube is increasing at a rate of 10 . How fast is the surface area increasing when the length of an edge is 30 cm?
78. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of , how fast is the water level rising when the water is 5 cm deep?
79. A balloon is rising at a constant speed of . A boy is cycling along a straight road at a speed of . When he passes under the balloon, it is 15 m above him. How fast is the distance between the boy and the balloon increasing 3 s later?
h f t
h共x兲 苷 f共x兲t共x兲
f共x兲 t共x兲 h共x兲 苷冑tf共x兲共x兲
h共x兲 苷 f 共t共sin 4x兲兲
t x苷 sb2 c2t2 t 0 b c
t y苷 t3 12t 3 t 0
0 t 3 0 t 3
V苷13r2h
r h
x
(
1 sx)
x
x苷 4 x
C共x兲 苷 920 2x 0.02x2 0.00007x3
C共100兲 C共100兲
cm3兾min
2 cm3兾s
2 m兾s 5 m兾s
80. A waterskier skis over the ramp shown in the figure at a speed of . How fast is she rising as she leaves the ramp?
81. The angle of elevation of the sun is decreasing at a rate of . How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is ?
; 82. (a) Find the linear approximation to near 3.
(b) Illustrate part (a) by graphing and the linear approximation.
(c) For what values of is the linear approximation accurate to within 0.1?
83. (a) Find the linearization of at . State the corresponding linear approximation and use it to give an approximate value for .
; (b) Determine the values of for which the linear approx-imation given in part (a) is accurate to within 0.1.
84. Evaluate if , , and .
85. A window has the shape of a square surmounted by a semi -circle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in com-puting the area of the window.
86–88 Express the limit as a derivative and evaluate.
86. 87.
88.
89. Evaluate .
90. Suppose is a differentiable function such that
and . Show that .
91. Find if it is known that
92. Show that the length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant.
1 m 5 m
0.25 rad兾h
兾6
f共x兲 苷 s25 x2 f
x
f共x兲 苷 s31 3x a苷 0 s31.03
x
dy y苷 x3 2x2 1 x 苷 2 dx苷 0.2
lim
x l1
x17 1
x 1 lim
h l 0
s416 h 2 h
l 兾3lim
cos 0.5
兾3
lim
x l 0
s1 tan x s1 sin x x3
f f共t共x兲兲 苷 x
f共x兲 苷 1 关 f 共x兲兴2 t共x兲 苷 1兾共1 x2兲 f共x兲
d
dx 关 f 共2x兲兴 苷 x2
x2兾3 y2兾3苷 a2兾3 10 m兾s 98845_02_ch02_p192-196.qk_98845_02_ch02_p192-196 8/16/11 9:58 AM Page 193
Before you look at the example, cover up the solution and try it yourself first.
How many lines are tangent to both of the parabolas and
? Find the coordinates of the points at which these tangents touch the parabolas.
SOLUTION
To gain insight into this problem, it is essential to draw a diagram. So we sketch the parabolas (which is the standard parabola shifted 1 unit upward) and (which is obtained by reflecting the first parabola about the
x-axis). If we try to draw a line tangent to both parabolas, we soon discover that there areonly two possibilities, as illustrated in Figure 1.
Let P be a point at which one of these tangents touches the upper parabola and let a be its x-coordinate. (The choice of notation for the unknown is important. Of course we could have used b or c or or instead of
a. However, it’s not advisable to use x inplace of a because that x could be confused with the variable x in the equation of the parabola.) Then, since P lies on the parabola , its y-coordinate must be
Because of the symmetry shown in Figure 1, the coordinates of the point Q where the tangent touches the lower parabola must be .
To use the given information that the line is a tangent, we equate the slope of the line
PQ to the slope of the tangent line at P. We haveIf , then the slope of the tangent line at P is . Thus the condi-tion that we need to use is that
Solving this equation, we get , so and . Therefore the points are (1, 2) and ( 1, 2). By symmetry, the two remaining points are (1, 2) and (1, 2).
1. Find points and on the parabola so that the triangle formed by the -axis and the tangent lines at and is an equilateral triangle (see the figure).
; 2. Find the point where the curves and are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent.
3. Show that the tangent lines to the parabola at any two points with -coordinates and must intersect at a point whose -coordinate is halfway between and .
4. Show that
5. If , find the value of .
y
苷 1 x
2 EXAMPLE 1y
苷 1 x
2y
苷 x
2 y苷 1 x
2y
苷 1 x
2x1
x0
y
苷 1 x
21 a
2.
共a, 共1 a
2兲兲
mPQ
苷 1 a
2共1 a
2兲
a
共a兲 苷 1 a
2 a f共a兲 苷 2a
f共x兲 苷 1 x
21 a
2 a苷 2a
a
苷 1
a2苷 1
1 a
2苷 2a
2ABC y苷 1 x2
Q P
Q P x
y苷 3共x2 x兲 y苷 x3 3x 4
y苷 ax2 bx c
p x
q p x
q
d
dx冉1 cot xsin2x 1 tan xcos2x 冊苷 cos 2x
f共兾4兲 f共x兲 苷 lim
t l x
sec t sec x t x