Sullivan Microsite TE SAMPLE

Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
88 Chapter 1 • Limits and Continuity
33.
2x 2 if x < 1
f (x) =
3x 2 − 1 if x > 1
at c = 1
34. f (x) =
x 2 − 1 if x > −1
x 3 if x < −1
at c =−1
35. f (x) =
at c = 0
2x + 1 if x > 0
⎧
⎨ x 2 if x < 1
x 2 if x ≤ 0
36. f (x) = 2 if x = 1 at c = 1
⎩
−3x + 2 if x > 1
Calculus AB Pacing Guide (continued)
Applications and Extensions
In Problems 37–40, sketch a graph of a function with the given
properties. Answers will vary.
37. lim f (x) = 3;
x→2
lim f (x) = 3;
x→3− lim f (x) = 1;
x→3 +
f (2) = 3; f (3) = 1
38. lim f (x) = 0;
x→−1 lim f (x) =−2;
x→2− lim f (x) =−2;
x→2 +
f (−1) is not defined; f (2) =−2
39. lim f (x) = 4;
x→1
lim f (x) =−1;
x→0− lim f (x) = 0;
x→0 +
f (0) =−1; f (1) = 2
f (x) = 1;
x→1
In Problems 41–50, use either a graph or a table to investigate
each limit.
|x − 5|
|x − 5|
41. lim
42. lim
43. lim
x→5 + x − 5
x→5 − x − 5
x→ 12
2x
−
44. lim
x→ 12
2x 45. lim
+
x→ 23
2x 46. lim
−
x→ 23
2x
+
47. lim |x|−x 48. lim |x|−x
x→2 + x→2 −
3 3
49. lim x−x 50. lim x−x
x→2 + x→2 −
51. Slope of a Tangent Line For f (x) = 3x 2 :
(a) Find the slope of the secant line containing the points (2, 12)
and (3, 27).
(b) Find the slope of the secant line containing the points (2, 12)
and (x, f (x)), x = 2.
(c) Create a table to investigate the slope of the tangent line to the
graph of f at 2 using the result from (b).
(d) On the same set of axes, graph f , the tangent line to the graph
of f at the point (2, 12), and the secant line from (a).
(a) Find the slope of the secant line containing the points (2, 8)
and (3, 27).
(b) Find the slope of the secant line containing the points (2, 8)
and (x, f (x)), x = 2.
(c) Create a table to investigate the slope of the tangent line to the
graph of f at 2 using the result from (b).
(d) On the same set of axes, graph f , the tangent line to the graph
of f at the point (2, 8), and the secant line from (a).
53. Slope of a Tangent Line For f (x) = 1 2 x2 − 1:
Day Topic Sullivan/Miranda Chapter Objectives Suggested Assignment
8 Section 1.4
9 Section 1.5
10 Section 1.5
3. Determine where the trigonometric functions are continuous
4. Determine where an exponential or a logarithmic function is
continuous
1. Investigate infinite limits
2. Find the vertical asymptotes of a graph
3. Investigate limits 40. at lim f (x) = 2; lim f (x) = 0; lim
x→2
infinity
x→−1
4. Find the horizontal asymptotes f (−1) = 1; f (2) of = a 3graph
5. Find the asymptotes of the graph of a rational function
(a) Find the slope m sec of the secant line containing the
points P = (2, f (2)) and Q = (2 + h, f (2 + h)).
(b) Use the result from (a) to complete the following table:
h −0.5 −0.1 −0.001 0.001 0.1 0.5
m sec
(c) Investigate the limit of the slope of the secant line found in (a)
as h → 0.
(d) What is the slope of the tangent line to the graph of f at the
point P = (2, f (2))?
(e) On the same set of axes, graph f and the tangent line to f at
P = (2, f (2)).
54. Slope of a Tangent Line For f (x) = x 2 − 1:
All AP ® Practice Problems
(a) Find the slope m sec of the secant line containing the
points P = (−1, f (−1)) and Q = (−1 + h, f (−1 + h)).
(b) Use the result from (a) to complete the following table:
2–26, 27–41 odd
h −0.1 −0.01 −0.001 −0.0001 0.0001 0.001 0.01 0.1
m sec
43–59 odd, 67–71 odd, All AP ® Practice
Problems
(c) Investigate the limit of the slope of the secant line found
in (a) as h → 0.
(d) What is the slope of the tangent line to the graph of f at the
point P = (−1, f (−1))?
(e) On the same set of axes, graph f and the tangent line to f
at P = (−1, f (−1)).
11 Review Chapter 1 Review Exercises Chapter 1 AP ® Review Problems
12 Test Chapter 1 Test AP ® Practice Exam, Big Idea 1: Limits
Calculus BC Pacing Guide
PAGE
85 55. (a) Investigate lim cos π by using a table and evaluating the
x→0 x
function f (x) = cos π x at
Day Topic Sullivan/Miranda Chapter Objectives Suggested Assignment
1 Section 1.1
2 Section 1.2
3 Section 1.2
4 Section 1.3
5 Section 1.4
6 Section 1.5
1. Discuss the slope of the tangent line to the graph
2. Investigate a limit using a table
3. Investigate a limit using a graph
1. Find the limit of a sum, a difference, and a product
2. Find the limit of a power and the limit of a root
3. Find the limit of a polynomial
52. Slope of a Tangent Line For f (x) = x 3 :
4. Find the limit of a quotient
5. Find the limit of an average rate of change
6. Find the limit of a difference quotient
1. Determine whether a function is continuous at a number
2. Determine intervals on which a function is continuous
3. Use properties of continuity
4. Use the Intermediate Value Theorem
1. Use the Squeeze Theorem to find a limit
2. Find limits involving trigonometric functions
3. Determine where the trigonometric functions are continuous
4. Determine where an exponential or a logarithmic function is
continuous
1. Investigate infinite limits
2. Find the vertical asymptotes of a graph
3. Investigate limits at infinity
4. Find the horizontal asymptotes of a graph
5. Find the asymptotes of the graph of a rational function
x =− 1 2 , − 1 4 , − 1 8 , − 1 10 , − 1 12 ,..., 1
12 , 1 10 , 1 8 , 1 4 , 1 2 .
(b) Investigate lim cos π by using a table and evaluating the
x→0 x
function f (x) = cos π x at
9, 11, 13, 15, 17, 19, 31, 37, 39, 43, 47,
51, 55, 59, 63–66, AP ® Practice Problems
3–8
x =−1, − 1 3 , − 1 5 , − 1 7 , − 1 9 ,..., 1 9 , 1 7 , 1 5 , 1 3 , 1.
(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.)
10, 13, 19, 35, 37, 39, 43, 47, 51, 55, 59
63, 66, 74, 75, 83, 93, 96
All AP ® Practice Problems (especially 5)
56. (a) Investigate lim cos π by using a table and evaluating the
x→0 x2 function f (x) = cos π at x =−0.1, −0.01, −0.001,
x2 −0.0001, 0.0001, 0.001, 0.01, 0.1.
13–18, 24, 29, 45, 51–56, 59, 62, 63,
73, 79
All AP ® Practice Problems
11,15,19, 23, 24, 25, 35, 38, 43, 45, 55,
70
All AP ® Practice Problems
17–26, 30–35, 40, 45, 50, 53, 55, 60, 63,
70, 73, 74, 75, 78, 86
All AP ® Practice Problems
7 Review Chapter 1 Review Exercises Chapter 1 AP ® Review Problems
8 Test Chapter 1 Test AP ® Practice Exam, Big Idea 1: Limits
1-6 Chapter 1 • Limits and Continuity
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