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Sullivan AP˙Sullivan˙Chapter01 October September 8, 2016 20, 2016 17:414:45
Section 1.1 1.2 • Assess Your Understanding 89 99
(b) Investigate lim cos π by using a table and evaluating the
(c) Graph the function C.
x→0 x2 function f (x) = cos π x 2 at
In Section P.1, we defined the (d) difference Use the graph quotient to investigate of a function lim C(w) f at xand as lim C(w). Do
w→1− w→1 +
f
x =− 2 3 , − 2 5 , − 2 7 , − 2 9 ,..., 2 9 , 2 7 , 2 5 , 2 these (x + suggest h) − f that (x) lim
3 .
w→1
h C(w) = 0 exists?
h
(e) Use the graph to investigate lim C(w) and lim C(w).
w→12− CALC
w→12
(c) Compare the results from (a) and (b). What do you conclude
+ EXAMPLE 16 Finding theDo Limit theseof suggest a Difference that lim C(w) Quotient exists?
about the limit? Why do you think this CLIPhappens? What is your
w→12
view about using a table to draw a conclusion about limits? (f) Use the graph to investigate lim
(a) For f (x) = 2x 2 f (x C(w). + h) − f (x)
− 3x + 1, find the difference quotient w→0 + , h = 0.
(d) Use technology to graph f . Begin with the x-window
(g) Use the graph to investigate lim C(w). h
w→13
[−2π, 2π] and the y-window [−1, 1]. If
(b)
you were
Find
finding
the limit as h approaches 0 of the difference quotient − of f (x) = 2x 2 − 3x + 1.
lim f (x) using a graph, what would you conclude? Zoom in 61. Correlating Student Success to Study Time Professor Smith
x→0 Solution (a) To find the difference quotient of f, we begin with f (x + h).
on the 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.) f (x + h) = 2(x + h) 2 (in− hours) 3(x + that h) the + 1 student = 2(xstudies. 2 + 2xh He+ claims h 2 ) −that 3x the − 3h closer + 1to
PAGE
85 57. (a) Use a table to investigate lim
2
.
(b) How close must x be to 2, so that f (x) is within 0.1 of the
limit?
Now
(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 f (x + 0.1h) of− thef (x)
limit?
h
(c) How close must x be to 2, so that f (x) is within 0.01 of the
x→2
x − 8
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
Summary or partial ounce up to and
including 3.5 ounces. First-class
Twoletter Basic rates Limits do not apply to letters
weighing more than 3.5 ounces.
• lim A = A, where A is a constant.
x→c Source: U.S. Postal Service Notice 123
• lim x = c
x→c
(a) Find a function C that models the first-class postage charged,
in dollars, for a letter weighing w ounces. Assume w>0.
Properties (b) What ofisLimits
the domain of C?
If f and (c) gGraph are functions the function for which C. lim f (x) and lim g(x) both exist,
x→c x→c
(d) Use the graph to investigate lim C(w) and lim C(w). Do
and k is a constant, then
w→2− w→2 +
• Limit
these
of a
suggest
Sum or
that
a Difference:
lim C(w) exists?
w→2
lim (e) [ Use f (x) the ± g(x)] graph = to investigate lim f (x) ± lim
g(x) C(w).
x→c x→c w→0 x→c +
• Limit (f) Use of the a Product: graph tolim
investigate [ f (x) · g(x)] lim = C(w). lim f (x) · lim g(x)
x→c w→3.5 − x→c x→c
• Limit of a Constant Times a Function: lim[kg(x)] = k lim g(x)
60. First-Class Mail As of April 2016, the x→c U.S. Postal Service x→c
charged $0.94 postage for first-class large envelope weighing up to
and including 1 ounce, plus a flat fee of $0.21 for each additional
or partial ounce up to and including 13 ounces. First-class rates do
not apply to large envelopes weighing more than 13 ounces.
1.2Source: Assess
U.S.
Your
Postal Service
Understanding
Notice 123
(a) Find a function C that models the first-class postage charged,
Concepts and Vocabulary
in dollars, for a large envelope weighing w ounces. Assume
1. (a) lim w>0. (−3) = ; (b) lim π =
x→4 x→0
(b) What is the domain of C?
2. If lim f (x) = 3, then lim[ f (x)] 5 = .
x→c x→c
3. If lim x→c
f (x) = 64, then lim x→c
3 √ f (x) = .
The limit of the difference quotient
tends to appear as a topic for a
multiple-choice question on the exam.
In the next chapter, students learn
to recognize this as the definition of
the derivative. Once this connection
is made, the limit of the difference
quotient can be found by simply taking
the derivative of the function. This will
be a great time saver.
Kathryn Sidenstricker /Dreamstime.com
AP® Exam Tip
6 Find the Limit of a Difference Quotient
seven hours one studies, the closer to 100% the student scores
= 2x 2 + 4xh on+ the2h final. 2 − He 3x claims − 3h + that 1 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.”
f (x +h)− f (x) = (2x 2 +4xh+2h 2 −3x −3h +1)−(2x 2 −3x +1) = 4xh+2h 2 −3h
(a) Write Professor Smith’s claim symbolically as a limit.
Then, the difference quotient is
(b) Write Professor Smith’s claim using the ε-δ definition
of limit.
= 4xh + 2h2 − 3h h(4x + 2h − 3)
= = 4x + 2h − 3, h = 0
h
h
Source: Submitted by the students of Millikin University.
62. The definition of the slope of the tangent line to the graph of
f (x + h) − f (x)
(b) lim
= lim(4x + 2h − 3) = 4x + 0 − 3 = 4x −f (x) 3 − f (c) ■
h→0 h y = h→0 f (x) at the point (c, f (c)) is m tan = lim
.
x→c x − c
Another way to express this slope is toNOW defineWORK a new variable Problem 71.
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 [ ] n
• Limit your reasoning. of a Power: lim
65. The graph of f (x) = x [ − f (x)] n 3 = lim f (x) x→c x→c
where n ≥ 2 is an integer is a straight line with a point punched
3 − x
•
out.
√ √
n
Limit of a Root: lim f (x) = n lim f (x)
x→c x→c
provided (a) Whatf straight (x) >0line if nand ≥ 2what is even point?
(b) Use the graph of f to investigate
[
the one-sided
] m/n limits of f as
• Limit of [f(x)]
x approaches m/n : lim[ f (x)]
3.
m/n = lim f (x) x→c x→c
provided (c) Does[ the f (x)] graph m/n is suggest defined that forlim
positive f (x) integers exists? If mso, andwhat n is it?
[ x→3 ] f (x)
lim
66. • f (x)
x→c
Limit (a) Use of a table Quotient: to investigate lim lim(1 = + x) 1/x .
x→c g(x) x→0 lim g(x) x→c
provided
(b) Use graphing
lim g(x)
technology
= 0
to graph g(x) = (1 + x) 1/x .
(c) What x→c do (a) and (b) suggest about lim(1 + x) 1/x ?
• Limit of a Polynomial Function: lim P(x) x→0 = P(c)
CAS (d) Find lim(1 + x) 1/x .
x→c
• Limit of a x→0 Rational Function: lim R(x) = R(c)
x→c
if c is in the domain of R
Challenge Problems
For Problems 67–70, investigate each of the following limits.
{ 1 if x is an integer
f (x) =
4. (a) lim x = 0 ; (b) if xlim
is not an integer
x→−1 x→e
67. 5. (a) lim f (x) 68. lim f (x) 69. lim f (x) x→2 lim (x − 2) = ; (b) lim (3 + x) =
x→0
x→1/2
x→1/2
x→3
70. lim x→0
6. (a) lim (−3x) = x→2
; (b) lim (3x) =
x→0
7. True or False If p is a polynomial function,
then lim p(x) = p(5).
x→5
Alternate Example
Finding the Limit of a Difference
Quotient
For fx ( ) = x + 1 find the limit of the
difference quotient as h approaches 0 + .
Solution
fx+ h −f x
lim ( ) ( )
+
→0
h
h
=
=
=
=
=
lim
+
h→0
lim
+
h→0
lim
+
h→0
lim
+
h→0
lim
+
h→0
( x+ h+ 1) − ( x + 1)
h
x+ h−
x
h
( x+ h−
x ) ( x+ h+
x )
⋅
h
( x+ h+
x )
x+ h−x
h( x+ h+
x)
h
h( x+ h+
x)
1
= lim
+
h→0
x+ h+
x
1
=
x+ 0 + x
1
=
2 x
Must-Do Problems for
Exam Readiness
AB: 35, 37, 39, 41, 43, 47, 48, 51–65
odd, 67, 71, 73–79 odd, AP ® Practice
Problems
BC: 10, 19, 37, 39, 43, 59, 75, and all AP ®
Practice Problems (especially 5)
TRM Full Solutions to Section
1.2 Problems and AP® Practice
Problems
Answers to Section 1.2
Problems
1. (a) −3
(b) π
2. 243
3. 4
4. (a) −1
(b) e
5. (a) −2
(b) 7 2
6. (a) −6
(b) 0
7. True
Section 1.2 • Assess Your Understanding
99
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