NAME
MATH 1400 - Elements of Calculus
Spring 2021
Exam 1
Show your work clearly and completely.
1.(4 pts each) Based on the graph of f(x) given, estimate the following limits or say if they
don’t exist.
f(x)
2 1 1 2 30
1
2
3
(a) lim
x!1
f(x) =
(b) lim
x!1 f(x) =
(c) lim
x!1 f(x) =

2.(4 pts each) Find each limit.
(a) lim
x!2(x
3 5x2 + 7) =
(b) lim
x!3
x+ 1
x2 + 2
=
(c) lim
x!10
p
x 2 =
3.(7 pts) Find the limit
lim
x!1
x2 + 4x+ 3
x2 x 2
4.(10 pts) For f(x) below, use 1 or 1 when appropriate to describe the behavior at each
zero of the denominator (i.e. If a limit exists at that point, compute it. If it doesn’t exist,
describe the left and right limits) and identify all vertical asymptotes.
f(x) =
x 2
x2 + x 6
5.(4 pts each) Find all horizontal asymptotes, if any, of each function.
(a) f(x) =
2x4 x3 + 1
3x4 + x3 + 5x2 + 2
(b) f(x) =
x3 1
2x2 + 4
(c) f(x) =
4x+ 1
x2 + 7
6.(4 pts each) Looking again at the graph of f(x) from Problem 1
f(x)
2 1 1 2 30
1
2
3
(a) List all values of x between -2 and 3 where f(x) is NOT continuous.
(b) List all values of x between -2 and 3 where f(x) IS continuous, using interval notation.
7.(7 pts) The revenue (in dollars) from the sale of x infant car seats is given by
R(x) = 90x 0.025x2 0  x  1, 500.
Find the average change in revenue if production is changed from 600 car seats to 900 car
seats.
8.(4 pts each) Using the function
f(x) =
x2 + x 2
x2 4x+ 3 =
(x 1)(x+ 2)
(x 1)(x 3)
complete the following steps to solve the inequality
x2 + x 2
x2 4x+ 3 > 0.
(a) Find all partition numbers of f .
(b) Plot the partition numbers on a number line dividing the number line into intervals.
5 4 3 2 1 0 1 2 3 4 5
(c) List a test number for each interval identified in (b).
(d) Complete the sign chart in part (b) by indicating whether the function is positive or
negative on each open interval.
(e) Give the solution to the inequality above using either inequality notation or interval
notation.
9.(12 pts) We’ve seen rules to compute derivatives more easily, but for this problem use the
four-step process (from Section 9.4) to find the derivative of f(x) = x2 + 2x 5. 