Math 117
Practice Exam 1a
_______________________________________________
Name
Instructor: Nancy Goodisman
1. Let f ( x) 
a.
x 2  25
x5
Find the left-hand limit of f(x) as x approaches 5 from the left.
lim f ( x ) = _________
x 5 
b. Find the right-hand limit of f(x) as x approaches 5 from the right.
lim f ( x ) = _________
x 5 
c. Does the lim f ( x ) exist? Why or why not? If the lim f ( x ) exists, what is it?
x 5
x 5
d. Does the function, f, (given above) have any discontinuities? If so, what are they?
Justify your answer.
f ( x  h)  f ( x )
gives the same result for f ' ( x) as using the
h 0
h
differentiation rules for the derivative of the function f ( x)  3x 2  x .
2. Show that the lim
3. Let f ( x) 
x2
.
3x  10
a. Find f (x) .
b. Find the equation of the tangent line when x = 5.
4. Let f ( x)  ( x  2)( x  3) .
a. For what values of x is f(x) positive? Graph the solution on the x axis below.
_____________________________________________ x
b. For what values of x is f(x) negative? Graph the solution on the x axis below.
_____________________________________________ x
5. Use the chain rule to find the rate of change in y with respect to x if
y  100 x  x 3 .
6. The cost function, C(q)  (q  50) q  1000 gives the total cost in dollars of
producing q units of a product.
a. What is the total cost of producing 100 units using the formula given above?
b. What is the average cost per unit when 100 units are produced if total cost is
given by the formula above?
c. Find the derivative of the total cost function that is given.
d. In this case, what is the marginal cost when production level is at 100 units?
e. What information is given by the marginal cost at 100 units that would be of
interest to the manager of the factory or the financial officer of the company?
f. (Extra credit) Find and explain the marginal average cost at 100 units.