Sullivan Microsite DigiSample

Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.1 • Assess Your Understanding 89 87
In Problems (b) Investigate 13–16, use limtechnology cos π by using to complete a tablethe andtable evaluating and the In Problems (c) Graph 21–28, the function use the graph C. to investigate lim f (x). If the limit
x→0 x2 x→c
investigate the limit.
function f (x) = cos π does
x 2 at
(d)
not exist,
Use the
explain
graph
why.
to investigate lim C(w) and lim C(w). Do
w→1− w→1 +
PAGE
2 − 2e x
82 13. lim x→0 x
x =− 2 3 , − 2 5 , − 2 7 , − 2 9 ,..., 2 9 , 2 7 , 2 5 , 2 3 .
21. these suggest that lim C(w) w→1 22. exists?
y (e) Use the graph to investigate
y
x approaches 0 x approaches 0
lim C(w) and lim C(w).
w→12− w→12
(c) Compare the resultsfrom the (a) left and (b). What from do youthe conclude right
+
about the limit? Why
−−−−−−−−−−→
do you think this happens?
←−−−−−−−−−−
Do these suggest that lim C(w) exists?
What is your 3
w→12 y f (x)
viewx about using −0.2 a table −0.1to −0.01 draw a conclusion → 0 ← about 0.01limits?
0.1 0.2
y f (x)
2
(f) Use the graph to investigate lim C(w).
w→0 +
(d) f (x) Use = 2 technology − 2
2ex
1
to graph f . Begin with the x-window
1 (g) Use the graph to investigate lim
x
C(w).
w→13
[−2π, 2π] and the y-window [−1, 1]. If you were finding
− (c, 1)
(c, 1)
lim ln xf (x) using a graph, what would you conclude? Zoom in
c
x
61. Correlating c Student Success x to Study Time Professor Smith
14. lim x→0
x→1 on x −the 1 graph. Describe what you see. (Hint: Be sure your
claims that a student’s final exam score is a function of the time t
calculator is set to the radian mode.)
(in hours) that the student studies. He claims that the closer to
x approaches 1 x approaches 1
PAGE
x − 8
23. seven hours one studies, the closer 24. to 100% the student scores
85 57. (a) Use a table to investigate from the lim left . from the right y
x→2 2
on the final. He claims that studying y
−−−−−−−−−−→ ←−−−−−−−−−−
significantly less than seven
(b) How x close must 0.9 x0.99 be to0.999 2, so that → f 1(x) ←is within 1.001 0.11.01 of the1.1
hours may cause one to be underprepared for the test, while
limit?
studying significantly more than seven hours may cause
(c) f (x) How =
ln close x
3 (c, 3) 3
must x be to 2, so that f (x) is within 0.01 of the 2 “burnout.”
2
x − 1
limit?
y f (x)
y f (x)
1 (a) Write Professor Smith’s claim1
symbolically as a limit.
58. (a) Use a table to investigate lim(5 − 2x).
PAGE
1 − cos x
(c, 1)
x→2 (b) Write Professor Smith’s claim using the ε-δ definition
82 15. lim , where x is measured in radians
(b) x→0
How
x
close must x be to 2, so that f (x) is within 0.1 of the
ofclimit.
x
c
x
limit? x approaches 0 x approaches 0
Source: Submitted by the students of Millikin University.
(c) How close must x from be to 2, thesoleft
that f (x) is within from the 0.01right
of the
PAGE
limit? −−−−−−−−−−→ ←−−−−−−−−−− 84 25. 62. The definition of the slope of the26.
tangent line to the graph of
59. First-Class
x (in radians)
Mail
−0.2
As of
−0.1
April
−0.01 → 0 ← 0.01 0.1 0.2
y f (x) − f (c)
y = f (x) at the point (c, f (c)) is m tan = lim
.
x→c
2016, the U.S. Postal Service
x − c
f (x) = 1 − cos x
y
2
charged $0.47 x postage for
Another way y f to(x)
express this slope 3is to define a new variable
1
first-class letters weighing up to
h = x − c. Rewrite the slope of the tangent (c, 2) line my tan using f (x)
sin x
h and c.
2
16. and lim including , 1where ounce, xplus is measured a flat in radians
x→0 1 + tan x
63. If f (2) c = 6, can you conclude x anything about lim f (x)? Explain
fee of $0.21 for each additional
1
x→2
your reasoning.
or partial ounce up to and x approaches 0 x approaches 0 2
64. If lim f (x) = 6, can you conclude anything about f (2)? Explain
including 3.5 ounces. First-class
c
x
from the left from the right
x→2
letter rates do not apply−−−−−−−−−−→
to letters
←−−−−−−−−−− your reasoning.
weighing x (in radians) more than−0.2 3.5 ounces. −0.1 −0.01 → 0 ← 0.01 0.1 0.2
65. The graph of f (x) = x − 3 is a straight line with a point punched
Source: U.S. Postal Service Notice 123
3 − x
f (x) =
sin x
27. 28.
y out.
y
1 + tan x
(a) Find a function C that models the first-class postage charged, (a) What straight line and what point?
in dollars, for a letter weighing w ounces. Assume w>0. 3
3
In Problems 17–20, use the graph to investigate
(b) Use the graph of f to investigate the one-sided limits of f as
(a) lim
(b) What
f (x),
is
(b)
the
lim
domain
f (x),
of
(c)
C?
lim
x approaches 3.
(c) Graph the function C.
f (x).
(c, 2)
y f (x)
2
y f (x)
2 (c, 2)
x→2− x→2 + x→2
1 (c) Does the graph suggest that lim1
f (x) exists? If so, what is it?
(d) Use the graph to investigate lim C(w) and lim C(w). Do
x→3
17. 18.
w→2− w→2 +
y
y
66. (a) Use c a table to investigate x lim(1 + x)
these suggest that lim C(w) exists?
1/x . c
x
x→0
4
w→2
(e) Use the graph to investigate lim 4 C(w).
(b) Use graphing technology to graph g(x) = (1 + x) 1/x .
w→0 + (c) What do (a) and (b) suggest about lim(1 + x) 1/x ?
(f) Use the graph to investigate lim C(w).
x→0
2 y f (x)
w→3.5 2 − y f (x)
CAS In Problems (d) Find29–36, lim(1 use + x) a 1/x graph . to investigate lim f (x) at the
(2, 2)
x→c x→0
60. First-Class Mail As of April 2016, the U.S. Postal Service number c.
charged $0.94 postage for first-class large envelope weighing up to
2x + 5 if x ≤ 2
2 4 x
2 4 x 29. f (x) =
at c = 2
and including 1 ounce, plus a flat fee of $0.21 for each additional
4x + 1 if x > 2
or partial ounce up to and including 13 ounces. First-class rates do Challenge Problems 2x + 1 if x ≤ 0
19. not apply to large envelopes weighing 20. more than 13 ounces. 30. For Problems f (x) = 67–70, investigate eachatof c = the0
2x if x > 0 following limits.
y
Source: U.S. Postal Service Notice 123
y
⎧ { 1 if x is an integer
6
8
⎨ 3x −f (x) 1 = if x < 1
(a) (2, Find 6) a function C that models the first-class postage y charged, f (x) PAGE
0 if x is not an integer
in dollars, y for f a(x)
large envelope6
84 31. f (x) = 4 if x = 1 at c = 1
weighing w ounces. Assume
⎩
4x if x > 1
3 w>0.
4
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
(b) What is the domain of C? 2
⎨ x + 2 if x < 2
32. f (x) = 4 if x = 2 at c = 2
⎩
2 4 x
2 4 x
x 2 if x > 2
Kathryn Sidenstricker /Dreamstime.com
− e
13. lim 2 2 x
=−2. For table values
x→0
x
see TSM.
x
14. lim ln − =
→ x 1 1 . For table values see
x 1
TSM.
− x
15. lim 1 cos = 0. For table values
x→0
x
see TSM.
x
16. lim
sin = 0. For table values
x→0
1+
tanx
see TSM.
17. (a) 2
(b) 2
(c) 2
18. (a) 4
(b) 4
(c) 4
19. (a) 3
(b) 6
(c) Limit does not exist.
20. (a) 4
(b) 2
(c) Limit does not exist.
21. 1
22. 1
23. 1
24. Limit does not exist.
25. Limit does not exist, because the two
one-sided limits are not equal.
26. Limit does not exist.
27. Limit does not exist, because the two
one-sided limits are not equal.
28. Limit does not exist.
29. 9
30. Limit does not exist.
31. Limit does not exist.
32. 4
Section 1.1 • Assess Your Understanding
87
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