AP Calculus BC
2.1 – 2.3 Quiz Review
Quiz Format:
1. 5 AP-Style Multiple Choice Problems (about 10 minutes)
2. 6 Free Response Problems (about 35 minutes)
Keep in mind the test will target all of the following skills:
- Remembering: Recall or remember previous information
- Understanding: Comprehend or explain the meaning of ideas and concepts
- Applying: Use and apply a concept in a new situation
- Analyzing: Distinguish/separate concepts into parts to understand a larger structure
- Evaluating: Justify a stand or decision about the value of ideas or materials
- Creating: Create a build a new structure or pattern from diverse elements
1.) Which of the following values is the average rate of f (x)  x  1 over the interval (0, 3)?
a) -3
b) -1
c) -1/3
d) 1/3
e) 3
2.) Which of the following is an equation for the tangent line to f ( x )  9  x 2 when x =2?
1
4
a) y  x 
9
2
b) y = -4x + 13
c) y = -4x – 3
d) y = 4x – 3
e) y = 4x + 13
3.) Use the limit definition of a derivative to find the instantaneous rate of change of f (x)  2x  x 2 ; at x  2 .
4.) Let f (x)  x  1 . Which of following statements about f are true?
I. f is continuous at x = -1.
II. f is differentiable at x = -1
III. f has a corner at x = -1
a) I only
b) II only
c) III only
d) I and III only
e) I and II only
5.) Find the derivative of the following
a) y  4x 2  8x  1
b) y  10 x 7
2
3
c) y  5 cos(x )  sin(x )
d) y 
2
5x 3
 5x  7 
6.) Suppose the position of a fly can be given by the function s(t )  4x 2  10 cosx  in f feet, over the interval
[0, 4π] in time t seconds.
a) Find the average rate of the fly over the interval.
b) Find the rate at which fly is traveling at t 
3
4
7.) An astronaut standing on the moon throws a rock upward. The height of the rock is:
27
𝑠(𝑡) = − 10 𝑡 2 + 27𝑡 + 6, where 𝑠 is measured in feet and 𝑡 is measured in seconds.
a. Find expressions for the velocity and acceleration of the rock.
b. Find the time when the rock is at its highest point by finding the time when the velocity is zero. What is the
height of the rock at this time?
8.) Find lim
[2(𝑥+ℎ)2 −1]−[2𝑥2 −1]
ℎ
ℎ→0
(Hint: There is a simple way to do this)
9.) What are the equations of the tangent and normal line to 𝑦 = 4𝑥 3 − 𝑥 2 + 𝑥 at 𝑥 = 1?
10.) Find the derivative of the following:
a. 𝑦 = 𝑥 + cos 𝑥 + 4𝑥 5
b. 𝑦 = 3𝑥 2 sin 𝑥
2𝑥 5
c. 𝑦 = 𝑥+2
d. 𝑦 = tan 𝑥 + sec 𝑥 − 𝑥 2
e. 𝑓 (3) (𝑥) =? When, 𝑓(𝑥) = 3𝑥 3 − 4𝑥 2 + 16𝑥 − 4
11.) A ball is thrown straight down from the top of a 600-foot building with an initial velocity of −30 ft/sec.
Use the position function 𝑠(𝑡) = −16𝑡 2 + 𝑣0 𝑡 + 𝑠0 to answer the following questions:
a. Determine the position and velocity functions for the ball.
b. Determine the average velocity on the interval [1,3].
c. Find the instantaneous velocities when 𝑡 = 1 and 𝑡 = 3.
d. Find the time required for the ball to reach the ground.
e. Find the velocity of the ball at impact.
12.) Given the graph of 𝑓(𝑥), draw the graph of 𝑓′(𝑥).
http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_matching.html (use this website for more
practice)
13.) Given the graph of 𝑓 ′′ (𝑥) = 0, draw possible graphs of 𝑓′(𝑥) and 𝑓(𝑥)? **Note: there is more than
possible solution**
14.) Draw two different cases where the derivative does not exist.
15.) Find the horizontal tangents of the curve 𝑦 = 5𝑥 3 − 3𝑥 5
Additional Problems
1. Finding Derivative: PG 115: #15-18, 21, 51
PG 126: #2, 6, 9, 11, 39, 54
2. Higher Order Derivatives: PG 128: #99-102
3. Equation of Tangent Line: PG 127: #64, 66
PG 104: #34
PG 127: #73, 75
4. Graphing: PG 128: #111-114
PG 116: #71-72
PG 128: #109-110
5. Differentiability: PG 106: #81-86