Study guide for exam 2 for AP Calculus AB Fall 2015
Format of the exam:
Expect:
6 multiple-choice questions to be done without a calculator. One of these questions will be worth 5 points, and the
others will be worth 7 points.
6 multiple-choice questions to be done with a calculator. All will be worth 7 points.
2 Free-Response questions worth 9 points each. You can use a calculator on these questions
Topics to Review:
AP Calculus Unit 2 Review Sheet
Section Topics
2.1
2.1
Find the one sided and two sided limit of a function graphically and analytically.
The limit by direct substitution.
Limits algebraically.
Factoring and canceling.
Multiplying by the conjugate.
Simplifying with common denominators
sin 𝑥
cos 𝑥−1
lim
=1
lim
=0
𝑥→0
2.2
𝑥
𝑥→0
𝑥
Properties of limits.
The limit as 𝑥 → ∓∞, or horizontal asymptote.
Limits of Rational Functions as 𝑥 → ∞
If the function is a rational function, divide out the variable of the highest degree on all terms
and observe the following:
1
if the function reduces to 𝑛 then function goes to zero at infinity.
𝑥
𝑥𝑛
if the function reduces to then function goes to ±∞ at infinity.
1
if the function reduces to a constant, then the function goes to that value at infinity.
End behavior model.
lim 𝑓(𝑥) = ±∞, or vertical asymptote.
2.2
𝑥→𝑎
Behavior of 𝑓(𝑥) left and right of asymptote.
Sign Charts
Finding continuity and discontinuity.
A function is continuous on an interior point c of its domain iff lim 𝑓(𝑥) = 𝑓(𝑐)
𝑥→𝑐
2.3
2.3
Types of discontinuities:
 jump
 infinite
 oscillating
 removable
“Fixing” removable discontinuities.
Intermediate Value Theorem.
The Intermediate Value Theorem for Continuous Functions
A function 𝑦 = 𝑓(𝑥) that is continuous on a closed interval [a, b] takes on every
value between 𝑓(𝑎)and𝑓(𝑏). In other words, if 𝑦𝑜 is between 𝑓(𝑎)and𝑓(𝑏), then 𝑦𝑜 = 𝑓(𝑐)
for some c in [a, b].
2.4
AROC - Slope of the Secant – Average Rate of Change - graphically, algebraically.
𝑓(𝑏) − 𝑓(𝑎)
𝑚=
𝑏−𝑎
Instantaneous Rate of Change – IROC – slope of the tangent
Derive the definition lim
h 0
2.4
f ( a  h)  f ( a )
h
.
Use it to find the slope of the tangent.
Find IROC (SOT) algebraically and determine tangent and normal line to a curve.
Find IROC at a junction point of a piecewise function.
Sample questions: Note that some of these questions come directly from AP exams:
Sample Multiple-choice questions no calculator:
1. The graph of the function f is shown below. Which of the following statements is false?
(a) lim 𝑓(𝑥) exists.
𝑥→2
(b) lim 𝑓(𝑥) exists.
𝑥→3
(c) lim 𝑓(𝑥) exists.
𝑥→4
(d) lim 𝑓(𝑥) exists.
𝑥→5
(e) The function f is continuous at x = 3.
2. The line y = 5 is a horizontal asymptote to the graph of which of the following functions?
(a) 𝑦 =
sin(5𝑥)
𝑥
(b) y = 5x
1
(c) 𝑦 = 𝑥−5
5𝑥
(d) 𝑦 = 1−𝑥
(e) 𝑦 =
3.
4.
5.
20𝑥 2 −𝑥
1+4𝑥 2
6.
Further review (that is homework questions to do)
2.1: pg. 68: #66-70; 2.2: pg. 77: #61, 63, and 64 and also the quick quiz questions 1-3 at the bottom of the page; 2.3:
pg. 86: #58, 59; 2.4: the quick quiz questions 1-3 at the bottom of the page.
Sample Multiple-choice questions with calculator:
Note that the first of these questions come almost directly from AP exams:
7. Let f be a function that is continuous on the closed interval [2, 4] with f(2) = 10 and f(4) =20. Which of the
following is guaranteed by the Intermediate Value theorem?
a. f(x) = 13 has at least one solution in the open interval (2, 4)
b. f(3) =15
c. f attains a maximum on the open interval (2, 4)
d. The slope of the tangent line must equal 5 for at least one x-value in the open interval (2, 4)
e. The slope of the tangent line is positive for all x in the open interval (2, 4)
8. Let m and b be real numbers and let the function f be defined by:
−2𝑏𝑥 2 + 3 𝑓𝑜𝑟 𝑥 ≤ 1
𝑚𝑥 + 𝑏 𝑓𝑜𝑟 𝑥 > 1
This function is continuous at x = 1, and there is a slope of the tangent line at x =1.What are the
values of m and b?
𝑓(𝑥) = {
(a) m = -3, b = 12
(b) m = 3, b = -12
(c) m = 12, b = -3
(d) m = -12, b = 3
(e) none of the above
(𝑥−6)2
9. Which of the following is true about function f if 𝑓(𝑥) = 2𝑥 2 −11𝑥−6
i. The graph of f has a vertical asymptote at x = 6
ii. The graph has a horizontal asymptote at y = ½
iii. The graph is continuous as x = 6
(a) i only
(b) ii only
(c) iii only
(d) ii and iii only
(e) i, ii, and iii
Further review (that is homework questions to do)
2.2: pg. 76: #62; 2.3: pg. 86: #56; 2.4: pg. 94: #40
Sample free response questions (these are also homework questions to do):
Pg. 77 the quick quiz question 4 at the bottom of the page, parts a, b and c only. Pg. 94 the quick
quiz question 4 at the bottom of the page. Pg. 97 54 and 55 (Note: even though it says in the text
not to use a calculator, you can use one for these questions after all).
Answers to the sample questions:
1. (c) lim 𝑓(𝑥) exists.
𝑥→4
2. (e) 𝑦 =
20𝑥 2 −𝑥
1+4𝑥 2
3. (E) 5
1
4. (B) 6
5.
6.
7.
8.
9.
(A) y = 0
(B) ¼
a. f(x) = 13 has at least one solution in the open interval (2, 4)
(c) m = 12, b = -3
(b) ii only