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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
156 Chapter 1 • Limits and Continuity
Answers to Chapter 1 AP®
Review Problems
1. B
2. B
3. C
4. C
5. C
6. B
7. C
8. C
9. D
10. B
11. D
TRM Full Solutions for the
Chapter 1 Project
CHAPTER 1 AP® REVIEW PROBLEMS
u 1. Which line is an asymptote to the graph of f (x) = e 2x ?
u 2.
(A) x = 0 (B) y = 0 (C) y = 2 (D) y = x
3x 3 + 4x
lim
x→∞ 5 − 2x 4 =
(A) − 3 2
(B) 0
(C)
3
5
(D)
3
2
u 3. If f (x) = 5x 3 − 1, then lim x→0
f (x) − f (0)
x 3 =
(A) 0 (B) 1 (C) 5 (D) The limit does not exist.
θ 2
u 4. lim θ→0 1 − cos θ =
(A) 0 (B) 1 (C) 2 (D) 4
u 5. The table gives values of three functions:
x –0.15 –0.1 –0.05 0 0.05 0.1 0.15
f(x) 0.075 0.05 0.025 –4 0.025 0.05 0.075
g(x) –8.3 –8.2 –8.1 undefined –7.9 –7.8 –7.7
h(x) 1.997 1.99 1.9975 1 1.005 1.02 1.045
For which of these functions does the table suggest that the
limit as x approaches 0 exists?
(A) f only (B) h only
(C) f and g only (D) f and h only
u 6. If a function f is continuous on the closed interval [1, 4] and if
f (1) = 6 and f (4) = –1, then which of the following must be
true?
(A) f (c) = 0 for some number c in the open interval (–1, 6).
(B) f (c) = 1 for some number c in the open interval (1, 4).
(C) f (c) = 1 for some number c in the open interval (–1, 6).
(D) f (c) = –2 for any number c in the open interval (1, 4).
u 7. Which are the equations of the asymptotes of the graph of the
x
function f (x) =
x(x 2 − 9) ?
(A) x = –3, x = 0, x = 3, y = 0
(B) x = –3, x = 0, x = 3, y = 1
(C) x = –3, x = 3, y = 0
(D) x = –3, x = 3, y = 1
u 8. Find the value of k that makes the function
{
x
f (x) =
2 + 2 if x ≤−1
kx + 4 if x > −1
continuous for all real numbers.
(A) –3 (B) –1 (C) 1 (D) 3
u 9. An odd function f is continuous for all real numbers.
If lim f (x) =−2, then which of the statements
x→∞
must be true?
I. f has no vertical asymptotes.
II. lim f (x) = 0
x→0
III. The horizontal asymptotes of the graph of f are y = –2
and y = 2.
(A) I only (B) III only
(C) I and III only (D) I, II, and III
u 10. lim x→0
sin(2x)
tan(3x) =
(A) 0
(B)
2
3
(C) 1
(D)
u 11. If a function f is continuous for all real numbers, and if
f (x) = x3 + 8
when x = −2, then f (–2) =
x + 2
(A) 0 (B) 4 (C) 8 (D) 12
3
2
CHAPTER 1 PROJECT
Pollution in Clear Lake
AP Photo
The Toxic Waste Disposal
Company (TWDC) specializes in
the disposal of a particularly
dangerous pollutant, Agent Yellow
(AY). Unfortunately, instead of
safely disposing of this pollutant,
the company simply dumped AY
in (formerly) Clear Lake.
Fortunately, they have been caught and are now defending
themselves in court.
The facts below are not in dispute. As a result of TWDC’s
activity, the current concentration of AY in Clear Lake is now
10 ppm (parts per million). Clear Lake is part of a chain of rivers
and lakes. Fresh water flows into Clear Lake and the
contaminated water flows downstream from it. The Department
of Environmental Protection estimates that the level of
contamination in Clear Lake will fall by 20% each year. These
facts can be modeled as
p(0) = 10 p(t + 1) = 0.80p(t)
where p = p(t), measured in ppm, is the concentration of
pollutants in the lake at time t, in years.
1. Explain how the above equations model the facts.
2. Create a table showing the values of t for t = 0, 1, 2,...,20.
3. Show that p(t) = 10(0.8) t .
4. Use technology to graph p = p(t).
5. What is lim p(t)?
t→∞
Lawyers for TWDC looked at the results in 1–5 above and argued
that their client has not done any real damage. They concluded
that Clear Lake would eventually return to its former clear and
unpolluted state. They even called in a mathematician, who wrote
the following on a blackboard:
lim p(t) = 0
t→∞
and explained that this bit of mathematics means, descriptively,
that after many years the concentration of AY will, indeed, be
close to zero.
156
Chapter 1 • Limits and Continuity
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