Survey of Calculus Exam #1
Thursday, September 16, 2004
Name ______________________________
You must show all your work on this paper. Solutions without correct supporting work will not be accepted.
1. Use the graph given to evaluate the limits: (3 points each)
a.
lim f ( x)  ________
x 2
b.
lim f ( x)  ________
x 2
2. Evaluate the limits without using a graphing calculator or making tables. (8 points each)


a. lim t  5t 1 / 2 = __________
x25
x3
= __________
x  3 x  8 x  15
b. lim
2
3. Given f ( x)  3x 2  2 x  8 , find f (x) using the definition of the derivative. (10 points)
4. Given f ( x)  2 x 2  x  2 , find: (5 points each)
a. the average rate of change between x  2 and x  5 .
b. the instantaneous rate of change at x  2 .
5. Find the derivative. Do not simplify your answers. Circle your answers.
a.
5
f ( x)  3x  4  5 x 3
x
c.
f ( x)  x 2 3 x 5  2
x 4  3x 2  1
b. g ( x) 
x2  5
5
6. Draw the graph of a function that satisfies the conditions given (if possible):
a. The function is continuous at x  2 , but not differentiable at x  2 .
b. The function is defined at x  1, but lim f ( x ) does not exist.
x1
7. Given y 
27
3
x
, find
d2y
dx 2
(7 points each)
x 8
.
(7 points)
(5 points each)
100
cents for 50  x  100 ,
100  x
a. Find the instantaneous rate of change of the cost with respect to purity.
(5 points)
8. If the cost of purifying a gallon of water to a purity of x percent is C ( x) 
b. Evaluate this rate of change for a purity of 95% and interpret your answer. (5 points)
10. A company’s profit function is P( x)  6 x  200 dollars, where x is the number of units.
a. Find the average profit function. (3 points)
b. Find the marginal average profit function. (4 points)
c. Evaluate the marginal average profit function x = 10 and interpret your answer. (3 points)