348
CHAPTER 4
APPLICATIONS OF DIFFERENTIATION
the acceleration must be negative and we have
a共t兲 苷
dv
苷 ⫺32
dt
Taking antiderivatives, we have
v共t兲 苷 ⫺32t ⫹ C
To determine C we use the given information that v共0兲 苷 48. This gives 48 苷 0 ⫹ C , so
v共t兲 苷 ⫺32t ⫹ 48
The maximum height is reached when v共t兲 苷 0, that is, after 1.5 s. Since s⬘共t兲 苷 v共t兲, we
antidifferentiate again and obtain
s共t兲 苷 ⫺16t 2 ⫹ 48t ⫹ D
Using the fact that s共0兲 苷 432, we have 432 苷 0 ⫹ D and so
Figure 5 shows the position function of the ball
in Example 7. The graph corroborates the conclusions we reached: The ball reaches its maximum height after 1.5 s and hits the ground
after 6.9 s.
500
s共t兲 苷 ⫺16t 2 ⫹ 48t ⫹ 432
The expression for s共t兲 is valid until the ball hits the ground. This happens when s共t兲 苷 0,
that is, when
⫺16t 2 ⫹ 48t ⫹ 432 苷 0
t 2 ⫺ 3t ⫺ 27 苷 0
or, equivalently,
Using the quadratic formula to solve this equation, we get
3 ⫾ 3s13
2
t苷
8
0
We reject the solution with the minus sign since it gives a negative value for t. Therefore
the ball hits the ground after 3(1 ⫹ s13 )兾2 ⬇ 6.9 s.
FIGURE 5
4.9
Exercises
1–22 Find the most general antiderivative of the function.
19. f 共x兲 苷 5e x ⫺ 3 cosh x
(Check your answer by differentiation.)
1. f 共x兲 苷 x ⫺ 3
21. f 共x兲 苷
2. f 共x兲 苷 2 x 2 ⫺ 2x ⫹ 6
1
3. f 共x兲 苷 2 ⫹ 4 x 2 ⫺ 5 x 3
4. f 共x兲 苷 8x 9 ⫺ 3x 6 ⫹ 12x 3
5. f 共x兲 苷 共x ⫹ 1兲共2 x ⫺ 1兲
6. f 共x兲 苷 x 共2 ⫺ x兲 2
7. f 共x兲 苷 7x 2兾5 ⫹ 8x ⫺4兾5
8. f 共x兲 苷 x 3.4 ⫺ 2x s2⫺1
1
3
4
9. f 共x兲 苷 s2
3
11. f 共x兲 苷 3sx ⫺ 2 s
x
13. f 共x兲 苷
1
2
⫺
5
x
15. t共t兲 苷
1⫹t⫹t
st
17. h共␪ 兲 苷 2 sin ␪ ⫺ sec ␪
18. f 共t兲 苷 sin t ⫹ 2 sinh t
Graphing calculator or computer required
F共0兲 苷 4
F共1兲 苷 0
2
16. r共␪ 兲 苷 sec ␪ tan ␪ ⫺ 2e␪
2
;
24. f 共x兲 苷 4 ⫺ 3共1 ⫹ x 2 兲⫺1,
3t ⫺ t ⫹ 6t
t4
2 ⫹ x2
1 ⫹ x2
tion. Check your answer by comparing the graphs of f and F .
3
12. f 共x兲 苷 s
x 2 ⫹ x sx
3
22. f 共x兲 苷
; 23–24 Find the antiderivative F of f that satisfies the given condi23. f 共x兲 苷 5x 4 ⫺ 2x 5,
14. f 共t兲 苷
2
3
10. f 共x兲 苷 e 2
4
20. f 共x兲 苷 2sx ⫹ 6 cos x
x ⫺ x ⫹ 2x
x4
5
25– 48 Find f .
25. f ⬙共x兲 苷 20x 3 ⫺ 12x 2 ⫹ 6x
26. f ⬙共x兲 苷 x 6 ⫺ 4x 4 ⫹ x ⫹ 1
27. f ⬙共x兲 苷 3 x 2兾3
2
1. Homework Hints available at stewartcalculus.com
28. f ⬙共x兲 苷 6x ⫹ sin x
SECTION 4.9
30. f ⵮共t兲 苷 e t ⫹ t ⫺4
29. f ⵮共t兲 苷 cos t
31. f ⬘共x兲 苷 1 ⫹ 3sx ,
33. f ⬘共t兲 苷 4兾共1 ⫹ t 2 兲,
34. f ⬘共t兲 苷 t ⫹ 1兾t ,
√
f 共1兲 苷 0
36. f ⬘共x兲 苷 共x ⫺ 1兲兾x,
0
f 共1兲 苷 6
⫺␲兾2 ⬍ t ⬍ ␲兾2, f 共␲兾3兲 苷 4
35. f ⬘共t兲 苷 2 cos t ⫹ sec 2 t,
f is continuous and f 共0兲 苷 ⫺1.
f 共1兲 苷 1, f 共⫺1兲 苷 ⫺1
y
f ( 12 ) 苷 1
38. f ⬘共x兲 苷 4兾s1 ⫺ x 2 ,
39. f ⬙共x兲 苷 ⫺2 ⫹ 12x ⫺ 12x 2,
2
f 共0兲 苷 4, f ⬘共0兲 苷 12
42. f ⬙共t兲 苷 3兾st ,
f 共0兲 苷 3,
f 共4兲 苷 20,
43. f ⬙共x兲 苷 4 ⫹ 6x ⫹ 24x 2,
44. f ⬙共x兲 苷 x 3 ⫹ sinh x,
45. f ⬙共x兲 苷 2 ⫹ cos x,
f ⬘共0兲 苷 4
f 共0兲 苷 3, f 共1兲 苷 10
47. f ⬙共x兲 苷 x ⫺2,
f 共1兲 苷 0,
x ⬎ 0,
x
2
(b) Starting with the graph in part (a), sketch a rough graph of
the antiderivative F that satisfies F共0兲 苷 1.
(c) Use the rules of this section to find an expression for F共x兲.
(d) Graph F using the expression in part (c). Compare with
your sketch in part (b).
f 共␲兾2兲 苷 0
f 共0兲 苷 0, f 共␲ 兲 苷 0
1
; 56. (a) Use a graphing device to graph f 共x兲 苷 2x ⫺ 3 sx .
f 共0兲 苷 1, f 共2兲 苷 2.6
46. f ⬙共t兲 苷 2e t ⫹ 3 sin t,
48. f ⵮共x兲 苷 cos x,
0
_1
f ⬘共4兲 苷 7
f 共0兲 苷 ⫺1,
y=fª(x)
1
40. f ⬙共x兲 苷 8x 3 ⫹ 5, f 共1兲 苷 0, f ⬘共1兲 苷 8
41. f ⬙共␪ 兲 苷 sin ␪ ⫹ cos ␪,
t
55. The graph of f ⬘ is shown in the figure. Sketch the graph of f if
f 共1兲 苷 12 , f 共⫺1兲 苷 0
2
37. f ⬘共x兲 苷 x ⫺1兾3,
figure. Sketch the graph of a position function.
f 共⫺1兲 苷 2
t ⬎ 0,
3
349
54. The graph of the velocity function of a particle is shown in the
f 共4兲 苷 25
32. f ⬘共x兲 苷 5x 4 ⫺ 3x 2 ⫹ 4,
ANTIDERIVATIVES
f 共2兲 苷 0
f 共0兲 苷 1, f ⬘共0兲 苷 2, f ⬙共0兲 苷 3
; 57–58 Draw a graph of f and use it to make a rough sketch of the
antiderivative that passes through the origin.
49. Given that the graph of f passes through the point 共1, 6兲
57. f 共x兲 苷
and that the slope of its tangent line at 共x, f 共x兲兲 is 2x ⫹ 1,
find f 共2兲.
50. Find a function f such that f ⬘共x兲 苷 x 3 and the line x ⫹ y 苷 0
sin x
,
1 ⫹ x2
⫺2␲ 艋 x 艋 2␲
58. f 共x兲 苷 sx 4 ⫺ 2 x 2 ⫹ 2 ⫺ 2,
⫺3 艋 x 艋 3
is tangent to the graph of f .
51–52 The graph of a function f is shown. Which graph is an anti-
59–64 A particle is moving with the given data. Find the position
derivative of f and why?
of the particle.
51.
y
52.
f
59. v共t兲 苷 sin t ⫺ cos t,
f
b
a
y
60. v共t兲 苷 1.5 st ,
a
x
x
b
c
c
61. a共t兲 苷 2t ⫹ 1,
s共4兲 苷 10
s共0兲 苷 3, v 共0兲 苷 ⫺2
62. a共t兲 苷 3 cos t ⫺ 2 sin t,
63. a共t兲 苷 10 sin t ⫹ 3 cos t,
64. a共t兲 苷 t ⫺ 4t ⫹ 6,
2
53. The graph of a function is shown in the figure. Make a rough
sketch of an antiderivative F, given that F共0兲 苷 1.
y
y=ƒ
0
1
x
s共0兲 苷 0
s共0兲 苷 0, v 共0兲 苷 4
s共0兲 苷 0,
s共2␲兲 苷 12
s共0兲 苷 0, s共1兲 苷 20
65. A stone is dropped from the upper observation deck (the Space
Deck) of the CN Tower, 450 m above the ground.
(a) Find the distance of the stone above ground level at
time t.
(b) How long does it take the stone to reach the ground?
(c) With what velocity does it strike the ground?
(d) If the stone is thrown downward with a speed of 5 m兾s,
how long does it take to reach the ground?
350
CHAPTER 4
APPLICATIONS OF DIFFERENTIATION
66. Show that for motion in a straight line with constant acceleration a, initial velocity v 0 , and initial displacement s 0 , the dis-
placement after time t is
a苷
s 苷 at 2 ⫹ v 0 t ⫹ s 0
1
2
67. An object is projected upward with initial velocity v 0 meters
per second from a point s0 meters above the ground. Show that
68. Two balls are thrown upward from the edge of the cliff in
Example 7. The first is thrown with a speed of 48 ft兾s and the
other is thrown a second later with a speed of 24 ft兾s. Do the
balls ever pass each other?
69. A stone was dropped off a cliff and hit the ground with a speed
of 120 ft兾s. What is the height of the cliff?
70. If a diver of mass m stands at the end of a diving board with
length L and linear density ␳, then the board takes on the shape
of a curve y 苷 f 共x兲, where
EI y ⬙ 苷 mt共L ⫺ x兲 ⫹ ␳ t共L ⫺ x兲
1
2
再
9 ⫺ 0.9t
0
if 0 艋 t 艋 10
if t ⬎ 10
If the raindrop is initially 500 m above the ground, how long
does it take to fall?
74. A car is traveling at 50 mi兾h when the brakes are fully applied,
关v共t兲兴 2 苷 v02 ⫺ 19.6关s共t兲 ⫺ s0 兴
2
E and I are positive constants that depend on the material
of the board and t 共⬍ 0兲 is the acceleration due to gravity.
(a) Find an expression for the shape of the curve.
(b) Use f 共L兲 to estimate the distance below the horizontal at
the end of the board.
y
0
downward acceleration is
x
producing a constant deceleration of 22 ft兾s2. What is the distance traveled before the car comes to a stop?
75. What constant acceleration is required to increase the speed of
a car from 30 mi兾h to 50 mi兾h in 5 s?
76. A car braked with a constant deceleration of 16 ft兾s2, pro-
ducing skid marks measuring 200 ft before coming to a stop.
How fast was the car traveling when the brakes were first
applied?
77. A car is traveling at 100 km兾h when the driver sees an accident
80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup?
78. A model rocket is fired vertically upward from rest. Its acceler-
ation for the first three seconds is a共t兲 苷 60t, at which time the
fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (downward) velocity slows linearly to ⫺18 ft兾s in 5 s. The rocket
then “floats” to the ground at that rate.
(a) Determine the position function s and the velocity function v (for all times t). Sketch the graphs of s and v.
(b) At what time does the rocket reach its maximum height,
and what is that height?
(c) At what time does the rocket land?
79. A high-speed bullet train accelerates and decelerates at the rate
71. A company estimates that the marginal cost ( in dollars per
item) of producing x items is 1.92 ⫺ 0.002x. If the cost of producing one item is $562, find the cost of producing 100 items.
72. The linear density of a rod of length 1 m is given by
␳ 共x兲 苷 1兾sx , in grams per centimeter, where x is measured in
centimeters from one end of the rod. Find the mass of the rod.
73. Since raindrops grow as they fall, their surface area increases
and therefore the resistance to their falling increases. A
raindrop has an initial downward velocity of 10 m兾s and its
of 4 ft兾s2. Its maximum cruising speed is 90 mi兾h.
(a) What is the maximum distance the train can travel if it
accelerates from rest until it reaches its cruising speed and
then runs at that speed for 15 minutes?
(b) Suppose that the train starts from rest and must come to
a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions?
(c) Find the minimum time that the train takes to travel
between two consecutive stations that are 45 miles apart.
(d) The trip from one station to the next takes 37.5 minutes.
How far apart are the stations?