AP Calculus AB
Spring Break Fun Sheet 
 x2  2 x  6 
1. lim  2

x 3
 x  4 x  21 
x4



2
 x  7 x  10 
2. lim 
x 2
Use the following function for questions 3 – 8.
 x2  3

f ( x)   5 x  9

11  x  1
x  2
2  x  3
x3
3. lim f ( x) 
4. lim f ( x ) 
5. lim f ( x) 
6. f (2) 
7. f (3) 
8. Is f(x) continuous? Justify.
x 2
x 2
9.

d
3x x  5 x 2
dx

x 3

d cos(3 x )
e 
dx
11.
d sec(5 x2 )
7
dx
d
4 x 2 3x  5
dx
12.
d
ln(4 x 2  1)
dx
13.
d
tan 3 (2 x 4 )
dx
14.
15.
d  4 x 2  3x  1 


dx 
e5 x

16.
d
arctan(4 x3 )
dx
17. lim
18.


20.  3xe5 x
21.

3x
4  7x
4
dx
2 x 3  5 x 2  2e 2 x dx

10.
19. sec x dx
22.

34 x  x 5  2
2x 2
3
x
dx
cot( x  h)  cot( x)

h 0
h
23.
 (x
2
2
2
dx
 sin x  2) dx
24.  2 cot (3 ) d

25.

(2 x  3)( x  1) dx
28.  2 x 3x 2  5 dx
27. 5 dx
26.
x 2  3x  5
 x  5 dx
29.  sec 2 (4 x) dx
30.

dx
31.
 cos (2x) sin(2x) dx
32.
33.
7
 (2 x  3) dx
34.

35.  2 xe3 x
3x
3
2x  1
2
2
3x e ( 2 x
2
1)
dx
 (sin
3
x cos x  2) dx
2
4
 5e
3 x 2 4
dx
36. The function f is differentiable for all real numbers. The point (2, 5) is on the graph of y = f(x), and the
slope at each point (x, y ) on the graph is given by
A) Find
dy 6 x 2  4
.

dx
y
d2y
and evaluate it at the point (2, 5).
dx 2
B) Find y = f(x) by solving the differential equation
dy 6 x 2  4
with the initial

dx
y
condition f (2) = 5.
37. A water tower contains 250 gallons of water at t = 0. During the time interval 0 < t < 14
hours, water is being pumped into the tank at a rate
 x2 
P(t )  cos    5
 31 
During the same time interval, water is being removed from the tank at the rate
2
G(t )  .27 x 2  4 x
A) How many gallons of water are removed from the tank during the time interval 0 < t < 14?
B) Is the level in the tank rising or falling at t = 7?
C) How many gallons of water are in the tank at time t = 14?
D) At what time t for 0 < t < 14 is the volume of water in the tank the least?
38. A particle moves along the x-axis with velocity at time t > 0 given by v(t )  2  ln(t  1) . Its
initial position is given by x(0) = 4.
A) Find the acceleration at t = 2.
B) Is the speed of the particle increasing at t = 2? Give a reason for your answer.
C) Find all the values of t at which the particle changes direction. Justify your answer.
D) What is the position of the particle at t = 10? Show any work that lead to your answer.
39. Let f be a function defined on the closed interval [0, 7]. The graph of f, consisting of four line
x
segments, is shown below. Let g be the function given by g ( x)   f (t ) dt .
1
A) Find g(-2), g(3) and g(4).
B) On what intervals is g decreasing? Justify your answer.
C) Find the average rate of change of g on the interval 1 < x < 4. Show work that leads to
your answer.
D) Find the x-coordinate of each point of inflection of the graph of g on the interval
-2 < x < 4. On what interval(s) is the graph of g concave down? Justify your answers.