  Calculus Sample Exam

Calculus Sample Exam
Chapter 1: Limits, Continuity, 1 sided Limits, and Limits to 
Directions: Use the graph to determine the limit, if it exists.

1. lim x 2  3
x 2

3  x, x  2
2. lim 
x 2 0,
x2

3. lim
x 4
Directions: Determine the limit algebraically, if it exists
4. lim
x 5
 5x 
5. limtan  
x 
 3 
x  139
x 2
x  10
x 10 x 2  100
6. lim
7. lim
6 1 cos x 
x2
x 0
8. Let f(x) = 4x – 5 and g(x) = x2. Find lim f  g  x    lim  f g x 
x 2
x 2
1
x 4
Directions: Find the limit.
9. lim
 x  x 
2

 3  x  x   4  x 2  3x  4

x
x 0
10. lim
f  x  x   f  x 
x
x 0
, where f(x) = 5x – 5
Directions: Use the Graph to determine limits and discuss continuity
11. a) lim
x 4
b) lim
x 4
c) lim
x 4
12. a) lim
x 4
b) lim
x 4
c) lim
x 4
Directions: Find the limit if it exists
13. lim
x 49
 x 3  1, x  0
14. lim f(x)  
x 0
 x  1, x  0
x 7
x  49
Directions: Determine the values where the function is not continuous. Evaluate whether
the points of discontinuity are removable or non-removable.
15. f(x) = -12x2 + 8x + 9
16. f  x  
x
x  5x
2
17. f  x  
x 1
x 1
Directions: Find the limit
 x 6 
18. lim 

x 8   x  8 
 x 2  x  1
19. lim 

3
x 1
 x 1 
1

20. lim  x 2  
x 0 
x
Chapter 2: Derivatives & Related Rates
Directions: Match the graph to its derivative.
21. _____
A.
22. _____
B.
23. _____
C.
24. _____
D.
25. _____
E.
Directions: Find the derivative.
26. f(x) = -7x2 + 2cos x
27. f  x  
x4  8
28. f  x  
x3
29. f  x  
30. f  x   x 5 cos x
31. f  x  
32. f  x  
sinx
8x  cos x
2
x3
2
7
x
 3cos x
9 x
x2  3

33. f  x   x 6  4
34. f  x   x 4 7  9x

4
Directions: Solve
35. A projectile is shot upwards from the surface of the earth with an initial velocity
of 126 meters per second. The position function is s(t) = -4.9t2 + vot + so, where vo
refers to the initial velocity, and so refers to the starting height. What is the
velocity after 6 seconds?
36. A ball is thrown straight down from the top of a 300 foot building with an initial
velocity of -22 feet per second. The position function is s(t) = -16t2 + vot + so,
where vo refers to the initial velocity, and so refers to the starting height. What is
the velocity of the ball after 1 second?
Directions: Find
dy
by implicit differentiation
dx
37. x8 + 9x + 7xy – y8 = 4
38. x = tan(x + y)
Directions: Solve
39. The radius of a sphere is increasing at a rate of 9 inches per minute. Find the rate
of change of the volume when r = 15 inches.
40. A point moves along the curve y = 2x2 + 1 in such a way that the “y” value is
decreasing at the rate of 2 units per second. At what rate is “x” changing when
3
x= ?
2
Chapter 3: Applications of Derivatives
Directions: Find the critical numbers for the function.
42. f  x    x  2   x  1
41. f  x   x 2x  1
4
3
Directions: State why Rolle’s Theorem does NOT apply for the given interval
43. f  x  
1
 x  3
2
, [2, 4]
44. f  x   x 2  3x , [0, 2]
Directions: Determine whether the Mean Value Theorem can be applied. If it can, find
f b   f  a
the values of “c” in which f '  c  
over the given interval.
ba
45. f  x 
x

2

 2  2x  1
2x  1
, [-1, 3]
46. f  x   8 
7
, [1,7]
x
Directions: Find the horizontal asymptote.
47. f  x  
5x
2
x 3
48. f  x  
2x 2  6x  1
1 x 2
49. f  x  
2
x

x 3 x 2
Directions: Analyze the function. Identify intercepts, vertical and/or horizontal asymptotes,
use the 1st and 2nd derivative tests to and identify and test critical values for
possible relative maximum and minimum points. Then sketch a graph.
50. f  x  
1
 x  2
2
Directions: Solve
51. The management of a large store has 1600 feet of fencing to enclose a
rectangular storage yard using the building as one side of the yard. If the
fencing is used for the remaining 3 sides, find the area of the largest possible
yard.
52. A page is to contain 45 square inches of print. The margins at the top and
bottom of the page are each 11/2 inches wide. The margins on each side are 1
inch. What should be the dimensions of print so that a minimum amount of
paper is used?
Directions: Use Newton’s Method to approximate the zero in the given interval to 0.001
53. x3 + 4x + 2, [-1, 0]
Directions: Find dy and y for each.
54. y = x3 – 2x when x = 2 and x = 0.1
55. f  x  
1
when x = 2 and x = -0.01
x
Directions: Find dy
56. y 
x2
2x 2  1
57. y  1 4x 2