332460_010R.qxd
11/1/04
2:29 PM
Page 91
REVIEW EXERCISES
Review Exercises for Chapter 1
In Exercises 1 and 2, determine whether the problem can be
solved using precalculus or if calculus is required. If the problem
can be solved using precalculus, solve it. If the problem seems to
require calculus, explain your reasoning. Use a graphical or
numerical approach to estimate the solution.
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
19. lim
x 3 125
x5
20. lim
x2 4
x3 8
x→5
x→2
1 cos x
sin x
1. Find the distance between the points 1, 1 and 3, 9 along the
curve y x 2.
21. lim
2. Find the distance between the points 1, 1 and 3, 9 along the
line y 4x 3.
22. lim
4x
tan x
23. lim
sin6 x 12
x
In Exercises 3 and 4, complete the table and use the result to
estimate the limit. Use a graphing utility to graph the function
to confirm your result.
x→0
x→ 4
x→0
[Hint: sin sin cos cos sin ]
24. lim
x→0
0.1
x
0.01 0.001
0.001
0.01
In Exercises 25 and 26, evaluate the limit given lim f x 34
x→c
and lim gx 23.
x→c
4x 2 2
x→0
x
4x 2 2 4. lim
x
x→0
3. lim
25. lim f xgx
x→c
26. lim f x 2gx
x→c
In Exercises 5 and 6, use the graph to determine each limit.
x 2 2x
x
cos x 1
x
[Hint: cos cos cos sin sin ]
0.1
f x
5. hx 91
6. gx y
y
3x
x2
Numerical, Graphical, and Analytic Analysis
and 28, consider
In Exercises 27
lim f x.
x→1 (a) Complete the table to estimate the limit.
8
h
2
x
−2
−2
2
g
(b) Use a graphing utility to graph the function and use the
graph to estimate the limit.
4
4
x
−4
−4
−4
4
8
(c) Rationalize the numerator to find the exact value of the
limit analytically.
x
(a) lim hx (b) lim hx
x→0
(a) lim gx (b) lim gx
x→1
x→2
7. lim 3 x
x→1
9. lim x 2 3
x→2
8. lim x
10. lim 9
11. lim t 2
12. lim 3 y 1
t2
13. lim 2
t→2 t 4
t2 9
14. lim
t→3 t 3
15. lim
x 2
x4
1x 1 1
17. lim
x
x→0
x→4
y→4
16. lim
x→0
18.
4 x 2
x
11 s 1
lim
s→0
1.001
1.0001
2x 1 3
x1
3 x
1
x1
Hint: a3 b3 a ba 2 ab b2
x→5
In Exercises 11–24, find the limit (if it exists).
t→4
27. f x 28. f x x→9
1.01
f x
x→0
In Exercises 7–10, find the limit L. Then use the - definition
to prove that the limit is L.
1.1
s
Free-Falling Object In Exercises 29 and 30, use the position
function st 4.9t 2 200, which gives the height (in meters)
of an object that has fallen from a height of 200 meters. The
velocity at time t a seconds is given by
lim
t→a
sa st
.
at
29. Find the velocity of the object when t 4.
30. At what velocity will the object impact the ground?
332460_010R.qxd
92
11/1/04
2:30 PM
CHAPTER 1
Page 92
Limits and Their Properties
In Exercises 31– 36, find the limit (if it exists). If the limit does
not exist, explain why.
31. lim
x→3
51. Let f x x 3
x2 4
. Find each limit (if possible).
x2
(a) lim f x
x→2
x3
(b) lim f x
32. lim x 1
x→2
x→4
x 22, x ≤ 2
2 x, x > 2
1 x, x ≤ 1
34. lim gx, where gx x 1,
x > 1
t 1, t < 1
35. lim ht, where ht t 1, t ≥ 1
s 4s 2, s ≤ 2
36. lim f s, where f s s 4s 6,
s > 2
33. lim f x, where f x x→2
(c) lim f x
x→2
52. Let f x xx 1 .
(a) Find the domain of f.
(b) Find lim f x.
x→1
x→0
3
(c) Find lim f x.
1
2
t→1
x→1
2
2
s→2
In Exercises 53–56, find the vertical asymptotes (if any) of the
graphs of the function.
2
x
4x
4 x2
In Exercises 37– 46, determine the intervals on which the function is continuous.
53. gx 1 37. f x x 3
55. f x 3x 2 x 2
38. f x x1
In Exercises 57–68, find the one-sided limit.
3x 2 x 2 , x 1
x1
39. f x 0,
x1
57.
59.
1
x 2 2
3
43. f x x1
x
45. f x csc
2
41. f x 42. f x 44. f x x x 1
x1
2x 2
46. f x tan 2x
f x xcx3,6,
x ≤ 2
x > 2
48. Determine the values of b and c such that the function is
continuous on the entire real line.
f x x 1,
x 2 bx c,
1 < x < 3
x2 ≥ 1
49. Use the Intermediate Value Theorem to show that
f x 2x 3 3 has a zero in the interval 1, 2.
50. Delivery Charges The cost of sending an overnight package
from New York to Atlanta is $9.80 for the first pound and $2.50
for each additional pound or fraction thereof. Use the greatest
integer function to create a model for the cost C of overnight
delivery of a package weighing x pounds. Use a graphing
utility to graph the function and discuss its continuity.
56. f x csc x
2x 2 x 1
x2
58.
lim
x1
x3 1
60.
x→1 61. lim
x→1
lim
x→ 12 x1
x4 1
lim x 2 2x 1
x1
62.
64. lim
x→1
x
2x 1
lim
x→1 x 2 2x 1
x1
1
63. lim x 3
x
x→0
x→2
1
3 x2 4
65. lim
sin 4x
5x
66. lim
sec x
x
67. lim
csc 2x
x
68. lim
cos 2 x
x
x→0
x→0
47. Determine the value of c such that the function is continuous on
the entire real line.
8
x 10 2
lim x→2
5 x, x ≤ 2
40. f x 2x 3, x > 2
54. hx x→0
x→0
69. Environment A utility company burns coal to generate electricity. The cost C in dollars of removing p% of the air
pollutants in the stack emissions is
C
80,000p
,
100 p
0 ≤ p < 100.
Find the cost of removing (a) 15%, (b) 50%, and (c) 90% of
the pollutants. (d) Find the limit of C as p → 100.
70. The function f is defined as shown.
f x tan 2x
,
x
(a) Find lim
x→0
x0
tan 2x
(if it exists).
x
(b) Can the function f be defined at x 0 such that it is
continuous at x 0?
332460_010R.qxd
11/1/04
2:30 PM
Page 93
P.S.
P.S.
Problem Solving
93
Problem Solving
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
1. Let Px, y be a point on the parabola y x 2 in the first quadrant. Consider the triangle PAO formed by P, A0, 1, and the
origin O0, 0, and the triangle PBO formed by P, B1, 0, and
the origin.
y
3. (a) Find the area of a regular hexagon inscribed in a circle of
radius 1. How close is this area to that of the circle?
(b) Find the area An of an n-sided regular polygon inscribed in
a circle of radius 1. Write your answer as a function of n.
(c) Complete the table.
P
A
n
1
6
12
24
48
96
An
B
O
x
1
(d) What number does An approach as n gets larger and larger?
y
(a) Write the perimeter of each triangle in terms of x.
6
(b) Let rx be the ratio of the perimeters of the two triangles,
P(3, 4)
1
Perimeter PAO
rx .
Perimeter PBO
2
−6
Complete the table.
4
x
2
1
0.1
0.01
Perimeter PAO
Q
x
2
6
−6
Figure for 3
Perimeter PBO
−2 O
Figure for 4
4. Let P3, 4 be a point on the circle x 2 y 2 25.
r x
(a) What is the slope of the line joining P and O0, 0?
(b) Find an equation of the tangent line to the circle at P.
(c) Calculate lim rx.
x→0
2. Let Px, y be a point on the parabola y x 2 in the first quadrant. Consider the triangle PAO formed by P, A0, 1, and the
origin O0, 0, and the triangle PBO formed by P, B1, 0, and
the origin.
y
(c) Let Qx, y be another point on the circle in the first quadrant.
Find the slope mx of the line joining P and Q in terms of x.
(d) Calculate lim mx. How does this number relate to your
x→3
answer in part (b)?
5. Let P5, 12 be a point on the circle x 2 y 2 169.
y
P
A
1
15
B
O
5
x
1
−15
−5 O
(a) Write the area of each triangle in terms of x.
(a) What is the slope of the line joining P and O0, 0?
Area PBO
.
Area PAO
(b) Find an equation of the tangent line to the circle at P.
Complete the table.
4
x
Area PAO
Area PBO
ax
2
1
0.1
0.01
(c) Let Qx, y be another point on the circle in the fourth quadrant. Find the slope mx of the line joining P and Q in terms
of x.
(d) Calculate lim mx. How does this number relate to your
x→5
answer in part (b)?
6. Find the values of the constants a and b such that
lim
(c) Calculate lim ax.
x→0
Q 15
P(5, −12)
(b) Let ax be the ratio of the areas of the two triangles,
ax x
5
x→0
a bx 3
x
3.
332460_010R.qxd
94
11/1/04
2:30 PM
CHAPTER 1
Page 94
Limits and Their Properties
7. Consider the function f x 3 x13 2
x1
12. To escape Earth’s gravitational field, a rocket must be launched
with an initial velocity called the escape velocity. A rocket
launched from the surface of Earth has velocity v (in miles per
second) given by
.
(a) Find the domain of f.
(b) Use a graphing utility to graph the function.
f x.
(c) Calculate lim
x→27
v
(d) Calculate lim f x.
x→1
8. Determine all values of the constant a such that the following
function is continuous for all real numbers.
ax ,
f x tan x
a 2 2,
v
2GM
r
2
0
x < 0
9. Consider the graphs of the four functions g1, g2, g3, and g4.
y
v
g2
2
1
x
3
1
y
2
3
v
y
3
3
g3
2
x
2
x
3
v
10,600
r
2
0
6.99.
13. For positive numbers a < b, the pulse function is defined as
1
1
2.17.
2
0
Find the escape velocity for this planet. Is the mass of this
planet larger or smaller than that of Earth? (Assume that the
mean density of this planet is the same as that of Earth.)
g4
2
1
v
1920
r
Find the escape velocity for the moon.
x
2
1
2
3
For each given condition of the function f, which of the graphs
could be the graph of f ?
0,
Pa,bx Hx a Hx b 1,
0,
where Hx (a) lim f x 3
x→2
1,0,
x < a
a ≤ x < b
x ≥ b
x ≥ 0
is the Heaviside function.
x < 0
(a) Sketch the graph of the pulse function.
(b) f is continuous at 2.
(b) Find the following limits:
(c) lim f x 3
x→2
(i)
1
10. Sketch the graph of the function f x .
x
lim Pa,bx
x→a
(iii) lim Pa,bx
x→b
(ii)
lim Pa,bx
x→a
(iv) lim Pa,bx
x→b
(a) Evaluate f , f 3, and f 1.
(c) Discuss the continuity of the pulse function.
(b) Evaluate the limits lim f x, lim f x, lim f x, and
x→1
x→1
x→0
lim f x.
(d) Why is
1
4
x→0
Ux (c) Discuss the continuity of the function.
11. Sketch the graph of the function f x x x.
1
(a) Evaluate f 1, f 0, f 2 , and f 2.7.
(b) Evaluate the limits lim f x, lim f x, and lim1 f x.
x→1
48
(c) A rocket launched from the surface of a planet has velocity
v (in miles per second) given by
1
1
2
0
(b) A rocket launched from the surface of the moon has
velocity v (in miles per second) given by
3
g1
v
192,000
r
where v0 is the initial velocity, r is the distance from the rocket to
the center of Earth, G is the gravitational constant, M is the mass
of Earth, and R is the radius of Earth (approximately 4000 miles).
y
2
2GM
R
(a) Find the value of v0 for which you obtain an infinite limit
for r as v tends to zero. This value of v0 is the escape
velocity for Earth.
x ≥ 0
3
x→1
(c) Discuss the continuity of the function.
x→ 2
1
P x
b a a,b
called the unit pulse function?
lim f x L, then
14. Let a be a nonzero constant. Prove that if x→0
lim f ax L. Show by means of an example that a must be
x→0
nonzero.