Math 117
Practice Exam 1
_______________________________________________
Name
Instructor: Nancy Goodisman
1. Let f ( x)  x  1 if x  1
x 2 if x  1.
a.
Find the left-hand limit of f(x) as x approaches 1 from the left.
lim f ( x) = _________
x 1
b. Find the right-hand limit of f(x) as x approaches 1 from the right.
lim f ( x) = _________
x 1
c. Does the lim f ( x ) exist? Why or why not?
x1
2. Continuity and Differentiability
a. Draw the graph of a function that is continuous at x = 2, but is not differentiable at
x = 2.
b. Is it possible for a function to be
differentiable but not continuous
at a value of x?
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)  2 x  x 2 .
3. Show that the lim
4. Let f ( x) 
2x
. Find the slope of the tangent line when x = 3.
x 1
2
5. The demand function p  500  10q relates the price, p, and the quantity sold, q.
a.
Give the function that relates total revenue, R, to the number sold, q.
b. What is the total revenue when 30 units are sold?
c. Give an expression for the average revenue per unit sold.
d. What is the average revenue per unit when 30 units are sold?
e. Give an expression for the marginal revenue.
f. What is the marginal revenue when 30 units are sold?
g. Explain the difference between the average revenue per unit sold and the marginal
revenue.