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Sullivan AP˙Sullivan˙Chapter01 September October 8, 20, 2016 2016 17:416:7
1 Limits and Continuity
(b) Investigate lim cos π by using a table and evaluating the
x→0 x2 function f (x) = cos π x 2 at
x =− 2 3 , − 2 5 , − 2 7 , − 2 9 ,..., 2 9 , 2 7 , 2 5 , 2 3 .
(c) Compare the results from (a) and (b). What do you conclude
about the limit? Why do you think this happens? What is your
view about using a table to draw a conclusion about limits?
(d) Use technology to graph f . Begin with the x-window
[−2π, 2π] and the y-window [−1, 1]. If you were finding
lim f (x) using a graph, what would you conclude? Zoom in
x→0
on the graph. Describe what you see. (Hint: Be sure your
calculator is set to the radian mode.)
PAGE
x − 8
85 57. (a) Use a table to investigate lim .
x→2 2
(b) How close must x be to 2, so that f (x) is within 0.1 of the
limit?
(c) How close must x be to 2, so that f (x) is within 0.01 of the
limit?
58. (a) Use a table to investigate lim(5 − 2x).
x→2
(b) How close must x be to 2, so that f (x) is within 0.1 of the
limit?
(c) How close must x be to 2, so that f (x) is within 0.01 of the
limit?
59. First-Class Mail As of April
2016, the U.S. Postal Service
charged $0.47 postage for
first-class letters weighing up to
and including 1 ounce, plus a flat
fee of $0.21 for each additional
or partial ounce up to and
including 3.5 ounces. First-class
letter rates do not apply to letters
weighing more than 3.5 ounces.
Source: U.S. Postal Service Notice 123
(a) Find a function C that models the first-class postage charged,
in dollars, for a letter weighing w ounces. Assume w>0.
(b) What is the domain of C?
(c) Graph the function C.
(d) Use the graph to investigate lim C(w) and lim C(w). Do
w→2− w→2 +
1.1 Limits of Functions Using
Numerical these suggest and that Graphical lim C(w) exists?
w→2
(e) Techniques Use the graph to investigate lim C(w).
w→0 +
1.2(f) Limits Use the ofgraph Functions to investigate Using lim
Properties of Limits
C(w).
w→3.5 −
60. 1.3 First-Class Continuity Mail As of April 2016, the U.S. Postal Service
charged $0.94 postage for first-class large envelope weighing up to
1.4 Limits and Continuity of
and including 1 ounce, plus a flat fee of $0.21 for each additional
or partial Trigonometric, ounce up toExponential, and includingand
13 ounces. First-class rates do
notLogarithmic apply to largeFunctions
envelopes weighing more than 13 ounces.
1.5 Source: Infinite U.S. Limits; Postal Service LimitsNotice at Infinity; 123
Asymptotes
(a) Find a function C that models the first-class postage charged,
1.6 The in dollars, ε-δ Definition for a largeofenvelope a Limitweighing w ounces. Assume
Chapter w>0. Review
(b)
Chapter
What is
Project
the domain of C?
Kathryn Sidenstricker /Dreamstime.com
(c) Graph the function C.
(d) Use the graph to investigate lim C(w) and lim C(w). Do
w→1− w→1 +
these suggest that lim C(w) exists?
w→1
(e) Use the graph to investigate lim C(w) and lim C(w).
w→12− w→12 +
Do these suggest that lim C(w) exists?
w→12
(f) Use the graph to investigate lim C(w).
w→0 +
(g) Use the graph to investigate lim C(w).
w→13 −
61. Correlating Student Success to Study Time Professor Smith
claims that a student’s final exam score is a function of the time t
(in hours) that the student studies. He claims that the closer to
seven hours one studies, the closer to 100% the student scores
on the final. He claims that studying significantly less than seven
hours may cause one to be underprepared for the test, while
studying significantly more than seven hours may cause
“burnout.”
(a) Write Professor Smith’s claim symbolically as a limit.
(b) Write Professor Smith’s claim using the ε-δ definition
of limit.
Source: Submitted by the students of Millikin University.
62. The definition of the slope of the tangent line to the graph of
f (x) − f (c)
y = f (x) at the point (c, f (c)) is m tan = lim
.
x→c x − c
Another way to express this slope is to define a new variable
h = x − c. Rewrite the slope of the tangent line m tan using h and c.
63. If f (2) = 6, can you conclude anything about lim f (x)? Explain
x→2
your reasoning.
64. If lim f (x) = 6, can you conclude anything about f (2)? Explain
x→2
your reasoning.
65. The graph of f (x) = x − 3 is a straight line with a point punched
3 − x
out.
(a) What straight line and what point?
(b) Use the graph of f to investigate the one-sided limits of f as
x approaches 3.
AP Photo
(c) Does the graph suggest that lim f (x) exists? If so, what is it?
x→3
Oil Spills and Dispersant Chemicals
66. (a) Use a table to investigate lim(1 + x) 1/x .
x→0
On April 20, 2010, the Deepwater (b) Horizon Use graphing drillingtechnology rig exploded toand graph initiated g(x) = the (1 worst + x) 1/x marine . oil
spill in recent history. Oil gushed (c) from What thedo well (a) for and three (b) suggest monthsabout and released lim(1 + millions x) 1/x ? of gallons
of crude oil into the Gulf of Mexico. One technique used to help clean x→0
up during and after the
CAS
spill was the use of the chemical (d) dispersant Find lim(1 + x) 1/x .
x→0
Corexit. Oil dispersants allow the oil particles to
spread more freely in the water, thus allowing the oil to biodegrade more quickly. Their use is
debated, however, because some of their ingredients are carcinogens. Further, the use of oil
dispersants can increase toxic Challenge hydrocarbon Problems levels affecting sea life. Over time, the pollution
caused by the oil spill and the dispersants will eventually diminish and sea life will return, more
or less, to its previous condition.
For Problems
In the short
67–70,
term,
investigate
however,
each of the following limits.
{
pollution raises serious questions
about the health of the local sea life and the safety of fish 1 and if shellfish x an for integer human consumption.
f (x) =
Explore a hypothetical situation of pollution in a lake in the Chapter 0 1 Project if x isonnot p. 156. an integer
67. lim f (x) 68. lim f (x) 69. lim f (x) 70. lim f (x)
x→2 x→1/2 x→3 x→0
77
PD Chapter 1 Overview
Watch the chapter overview video for
expert advice on teaching the content in
this chapter, anticipating likely problem
spots, and guidance on staying on pace.
Teaching Tip
Limits are foundational to the study of
calculus. This chapter opens with a
practical application of limits by modeling
the level of contamination in Clear Lake.
Students will have an opportunity to
apply what they have learned through
this chapter by completing the Chapter 1
Project, Pollution in Clear Lake, at the end
of this unit of study.
TRM Chapter 1 Bell Ringers
Each day, the students review and
evaluate 5 trigonometric functions and 2
logarithmic functions without the use of
a calculator. Each problem is presented
on a PowerPoint slide, and the problems
transition on a timer, allowing you time to
take attendance or check homework while
your students work the problems. At the
end of the Bell Ringer file is a 20-question
quiz that can be given on the day that you
review the chapter.
WEB SITE
Mathisfun: The Math Is Fun Web site is a
helpful resource for the student who would
benefit from learning about limits in simple
terms with lots of illustrations. A link to this
resource is available on the Additional
Chapter 1 Resources document, available
for download.
Section 1.1 • Assess Your Understanding 89
Where to Find the
u Teacher Resources?
t
All of the Teacher Resource Materials listed in
the blue pages for this chapter and referenced
through the PD and TRM icons may be
found by clicking on the links in the Teacher’s
e-Book (TE-book), logging in to LaunchPad
(password required) highschool.bfwpub.com/
launchpad/apsullivan2e, or opening the
Teacher’s Resource Flash Drive (TRFD).
TRM AP® Calc AB Exam Prep Flashcards
You may want to give your students the AP ® Calc
AB Exam Prep Flashcards now so that they may
use them throughout the year. Have them separate
the Chapter 1 cards by referring to the chapter
number in the bottom corner.
Chapter 1 • Limits and Continuity
77
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