Sullivan Microsite DigiSample

Sullivan AP˙Sullivan˙Chapter01 October8, 8, 2016 17:4
Sullivan
TRM Alternate Examples Section 1.1
You can find the Alternate Examples for
this section in PDF format in the Teacher’s
Resource Materials.
TRM AP® Calc Skill Builders
Section 1.1
You can find the AP ® Calc Skill Builders for
this section in PDF format in the Teacher’s
Resource Materials.
WEB SITE
Many Web sites provide help for teaching
and learning calculus. Some provide
advice and concept development videos
or applets, while others offer step-by-step
support in working calculus problems.
Three very useful sites:
• Khan Academy is a well-known source
of good tutorials, and the College Board
has partnered with Khan to provide
instructional videos on released AP ®
Exam problems. https://www.
khanacademy.org/math/calculus-home/
ap-calc-topic
• Lin McMullin’s Teaching Calculus blog
offers an extensive set of resources for
teachers and students, including
instructional videos for most topics,
suggestions for addressing the MPACs,
and other helpful information.
https://teachingcalculus.com/
• The Wolfram|Alpha site offers help with
solving many types of problems by “doing
dynamic computations based on a vast
collection of built-in data, algorithms, and
methods.” It offers much helpful guidance
to students when they are stuck and
looking for quick specific help.
https://www.wolframalpha.com/
Links to these resources, and others,
are found in the Additional Chapter 1
Resources document, available for download.
88 78 Chapter 1 • Limits and Continuity
T
2x 2 if x < 1
33. f (x) =
3x 2 at c = 1
he concept of a limit 53. is Slope central of a to Tangent calculus. Line To understand For f (x) = calculus, 1
− 1 if x > 1
2 x2 − 1: it is essential to
know what it means for a function to have a limit, and then how to find a limit of a
x 3 function. Chapter 1 explains (a) what Finda the limit slope is, shows m
if x < −1
sec ofhow the secant to findline a limit containing of a function, the and
34. f (x) =
x 2 at c =−1 demonstrates how to prove thatpoints limitsP exist = (2, using f (2)) the anddefinition Q = (2 + of h, limit. f (2 + h)).
− 1 if x > −1
We begin the chapter using (b) Use numerical the result and from graphical (a) to complete approaches the following to explore table: the idea
x 2 if x ≤ 0
35. f (x) =
at c = 0 of a limit. Although these methods seem to work well, there are instances in which they
2x + 1 if x > 0
⎧
fail to identify the correct limit. h −0.5 −0.1 −0.001 0.001 0.1 0.5
⎨ x 2 if x < 1
In Section 1.2, we provide analytic m sec techniques for finding limits. Some of the proofs
36. f (x) = 2 if x = 1 at c = 1
⎩
of these techniques are found in Section 1.6, others in Appendix B. A limit found by
−3x + 2 if x > 1
correctly applying these analytic (c) Investigate techniques theis limit precise; of thethere slopeis ofno thedoubt secantthat lineit found is correct. in (a)
as h → 0.
In Sections 1.3–1.5, we continue to study limits and some ways that they are used.
(d) What is the slope of the tangent line to the graph of f at the
Applications and Extensions
For example, we use limits to define continuity, an important property of a function.
point P = (2, f (2))?
In Problems 37–40, sketch a graph of a function with the Section given 1.6 provides a(e) precise On the definition same set of ofaxes, limit, graph the so-called f and the tangent ε-δ (epsilon-delta)
line to f at
properties. Answers will vary.
definition, which we use to showP = when (2, f a(2)).
limit does, and does not, exist. ■
37. lim f (x) = 3; lim f (x) = 3; lim f (x) = 1;
x→2 x→3− x→3 +
f (2) = 3; f (3) = 1
54. Slope of a Tangent Line For f (x) = x 2 − 1:
1.1 Limits of Functions Using Numerical
38. lim f (x) = 0; lim f (x) =−2; lim f (x) =−2;
(a) Find the slope m sec of the secant line containing the
x→−1 x→2− x→2 + and Graphical Techniques
points P = (−1, f (−1)) and Q = (−1 + h, f (−1 + h)).
f (−1) is not defined; f (2) =−2
39. lim f (x) = 4; lim f (x) =−1; lim f
OBJECTIVES
(x) = 0;
When you(b) finish Usethis the result section, from you (a) to should complete be able the following to: table:
x→1 x→0− x→0 +
AP®
f
EXAM
(0) =−1;
INSIGHT
f (1)
Limits
= 2
is the first 1 Discuss the slope of a tangent line to a graph (p. 78)
h −0.1 −0.01 −0.001 −0.0001 0.0001 0.001 0.01 0.1
Big Idea in the AP® Calculus curriculum.
40. lim f (x) = 2; lim f (x) = 0; lim 2 Investigate a limit using a table (p. 80)
f (x) = 1;
m sec
x→2 x→−1 x→1 3 Investigate a limit using a graph (p. 82)
f (−1) = 1; f (2) = 3
(c) Investigate the limit of the slope of the secant line found
Calculus can be used to solve certain (a) asfundamental h → 0. questions in geometry. Two of these
In Problems 41–50, use either a graph or a table toquestions investigate are:
(d) What is the slope of the tangent line to the graph of f at the
each limit.
point P = (−1, f (−1))?
|x − 5|
|x − 5|
41. lim
42. lim
43.
• Given
lim
a
x→5 + x − 5
x→5 − x − 5
x→ 12
2x
function f and(e) a point On theP same on its set graph, of axes, what graph is the f and slope the tangent of the line
to f
tangent −
to the graph of f at atP? P = See (−1, Figure f (−1)). 1.
• Given a nonnegative function f whose domain
PAGE
44. lim
x→ 12
2x 45. lim
+
x→ 23
2x 46. lim
−
x→ 23
2x
+
85 55. (a) Investigate lim cos π is the closed interval [a, b], what is
the area of the region enclosed by the by using a table evaluating the
x→0
graph of x f , the x-axis, and the vertical lines
x = a and x = b? See Figure 2.
function f (x) = cos π
47. lim |x|−x 48. lim |x|−x
x at
x→2 + x→2 −
x =− 1
3 3 2 , − 1 4 , − 1 8 , − 1 10 , − 1 12 ,..., 1
12 , 1 10 , 1 8 , 1 4 , 1 y
y f (x)
y
2 .
49. lim x−x 50. lim x−x
x→2 + x→2 − l T
51. Slope of a Tangent Line For f (x) = 3x 2 (b) Investigate lim cos π y f (x)
by using a table and evaluating the
:
Tangent x→0 x
P
line
(a) Find the slope of the secant line containing the points (2, 12)
function f (x) = cos π x at
and (3, 27).
(b) Find the slope of the secant line containing the points (2, 12)
x =−1, − 1 3 , − 1 5 , − 1 7 , − 1 9 ,..., 1 9 , 1 7 , 1 5 , 1 x
a
3 , 1. b x
and (x, f (x)), x = 2.
(c) Compare the results from (a) and (b). What do you conclude
(c) Create a table to investigate the slope of the DFtangent Figureline 1 to the
Figure 2
about the limit? Why do you think this happens? What is
graph of f at 2 using the result from (b).
your view about using a table to draw a conclusion about
(d) On the same set of axes, graph f , the tangent line to the graph
These questions, traditionally limits? called the tangent problem and the area problem,
of f at the point (2, 12), and the secant line from (a).
were solved by Gottfried Wilhelm von Leibniz and Sir Isaac Newton during the late
(d) Use technology to graph f . Begin with the x-window
52. Slope of a Tangent Line For f (x) = x 3 : seventeenth and early eighteenth [−2π, centuries. 2π] and The thesolutions y-window to[−1, the1]. twoIfseemingly you were finding different
problems are both based on the idea
(a) Find the slope of the secant line containing the points (2, 8)
lim f of (x) a limit. using Their a graph, solutions what would not you onlyconclude? are related Zoom to each in
x→0
other, but are also applicable to many other problems in science and geometry. Here,
and (3, 27).
on the graph. Describe what you see. (Hint: Be sure your
we begin to discuss the tangent problem. The discussion of the area problem begins in
(b) Find the slope of the secant line containing the points (2, 8)
calculator is set to the radian mode.)
Chapter 5.
and (x, f (x)), x = 2.
56. (a) Investigate lim cos π by using a table and evaluating the
(c) Create a table to investigate the slope of the tangent line to the
x→0 x2 graph of f at 2 using the result from (b). 1 Discuss the Slope offunction a Tangent f (x) = Line cos π to at x a=−0.1, Graph −0.01, −0.001,
(d) On the same set of axes, graph f , the tangent line to the graph
x2 NEED TO REVIEW? The slope of a line
Notice that the line
is discussed of finatAppendix the point A.3, (2, 8), p. A-18.
T in Figure 1 just touches the graph of f at the point P. This unique
and the secant line from (a).
−0.0001, 0.0001, 0.001, 0.01, 0.1.
line is the tangent line to the graph of f at P. But how is the tangent line defined?
∑ Mathematical Practices Tip
MPAC 4: Connecting Multiple Representations
Because limits can be presented numerically,
graphically, and algebraically, you will have many
opportunities to reinforce MPAC 4. Students
may find that one method is more useful or
appealing than another for a given function, but
it is important that they have experience with
multiple approaches. Show students how to solve
the same problem by creating a table, examining
a graph, and manipulating the problem using
algebra. By constructing one representational form
from another, you will help to solidify the concept
in the students’ minds as they see that the various
methods all lead to the same result.
Teaching Tip
In the AB pacing guide, two days have been
allocated to Section 1.1 and two days have been
allocated to Section 1.2. If you teach limits with all
three techniques together (tabular, graphical, and
analytical), then you may spend 4 days working
on limits and still adhere to the schedule. The
suggested pacing guide will allow you to cover all
concepts on the AP ® Calculus Exam and still have
approximately 3 weeks for review before the test.
78
Chapter 1 • Limits and Continuity
TE_Sullivan_Chapter01_PART 0.indd 7
11/01/17 9:51 am