AP Calculus
Name: ________________________
Review Unit 3
SHOW ALL WORK!
This review is required and will count as part of your test grade. On the test, you will be required to show all of your work. The
test will be divided into calculator and no calculator parts. This review is not comprehensive. Please look back over your notes,
your homework, and your quizzes to help you study for the test.
I. Chain Rule: Find the derivative of each function using shortcuts. Be sure to use correct notation.
1.
f ( x)  ( x 2  3) 29
2.
y  2  3x 2
3.
y  cos 2 (3x  2)
4.
y  (csc x  cot x) 1
5.
g (t )  sin(3t 4 )
6.
f (t )  sin 3 (5t 2 )
Use your calculator to differentiate each function at the given value of x, round to 3 decimal places. Put your calculator in
radian mode. Find the equation of the line tangent to the curve at the given point.
7.
y  cot(2 x) , x 

8.
12
y  x2  9 , x  5
II. Derivatives with Tables and Graphs:
Let
h( x)  f ( x) g ( x) , j ( x) 
9.
h' (2)
10.
j ' (0)
11.
f ( x)
, k ( x)  f ( g ( x)) .
g ( x)
k ' (2)
Given
p ( x )  f ( x)  g ( x ) , q ( x ) 
12.
p' (1)
13.
q' (4)
14.
h' (1)
Find each of the following using the information in the table.
x
-2
f(x)
1
f’(x)
-1
g(x)
0
g’(x)
4
-1
0
-2
1
1
0
-1
2
-2
1
1
2
-2
-1
2
2
3
-1
2
-2
f ( x)
, h( x)  f ( g ( x)) .
g ( x)
Find each of the following using the graph shown.
III. Interpreting derivatives
15. Let
y (t )
t (years)
y (people)
be the population of a town since 1995. The table shows the population recorded every two years.
0
2500
2
2912
a) Estimate the value of
4
3360
6
3815
8
4330
10
4875
y' (7) , and explain its meaning.
b) A model for the population is given by
Show your computation.
P(t )  (2t  50) 2 .
Use the model and your calculator to find
P' (7)
IV. Particle Motion
For problems 16-21, use the following information:
A particle moves along a line so that its position at any time t  0 is given by the function
s(t )  t 2  3t  2 , where s is
measured in meters and t is measured in seconds.
Find each of the following:
16. Displacement on [0,5]
17. Average velocity on [0, 5]
18. Instantaneous velocity when t = 4.
19. Acceleration when t = 4.
20. At what values of t does the particle move right?
Explain why.
21. At what values of t does the particle change direction?
Explain why.
The graph shows the velocity of a particle moving along the x-axis.
22. For what times is the particle moving backward
(i.e. in the negative direction)? Justify your answer.
23. For what times is the acceleration of the particle negative?
Justify your answer.
24. For what times is the speed of the particle decreasing?
Justify your answer.
V. Implicit Differentiation
23. Using the implicit relation
a. Find
dy
dx
y  x ( x  7) ,
2
2
by implicit differentiation.
24. Consider the curve
a. Find
dy
dx
x 2  4 xy  y 2  12 .
in terms of x and y.
b. Find the coordinates of all horizontal tangent lines.
b. Find an equation for the tangent line to the curve at
(-3, 6)
c. Find any points where the tangent line is vertical (if it
exists).
c. Find the equation of the tangent line at the point
(-4, 14)
VI. Related Rates:
120 mm2/sec. What is the rate of increase in
4


36 cubic millimeters? V  r 3 , S  4r 2 
3


25. The surface area of a spherical balloon is increasing at a constant rate of
the radius of the balloon at the instant when the volume is
26. A woman 1.6 m tall is walking away from a lamp post 10 m tall. If the woman is walking at a speed of 1.2 m/sec, how fast is
her shadow increasing when she is 15m from the lamp post?
27. On Halloween night Linus VanPelt and Charlie Brown have been trying to catch the Great Pumpkin as it rises from the
pumpkin patch to bring all good calculus students presents and candy, so they can keep all the goodies for themselves. Linus and
Charlie spot the Great Pumpkin just starting to rise at a distance of 80 feet away. The angle of elevation of the Great Pumpkin is
changing at the rate of  radians per second.
36
a) How fast is the distance off the ground of the Great Pumpkin changing when the Great Pumpkin is 6 feet off the ground?
b) How fast is the distance from Linus and Charlie to the Great Pumpkin changing when the Great Pumpkin is 6 feet off the
ground?
28. A dingy is pulled toward a dock by a rope from the bow through a ring on the dock 6 feet above the bow as
shown in the figure. The rope is hauled in at a the rate of 2 feet per second.
(a) How fast is the boat approaching the dock when 10 feet of rope are out?
(b) At what rate is the angle

changing at that moment?
29. Sand falls from a conveyor belt at the rate of 10 cubic meters per minute onto the top of a conical pile. The height of the
pile is always three-eighths of the base diameter. How fast are the height and radius changing when the pile is 4 m high?