MA125 Calculus I
Exam 3
Name: _________________________________________
There are 7 questions. For full credit, show all steps of your work. A calculator may not be
used.
1. Find the critical points of the following functions.
a.
𝑔(𝑥 ) = 2𝑥 3 − 54𝑥
b.
𝑓(𝑥 ) = 𝑥 2 −
16
𝑥
2. Consider the function
Given that
𝑓(𝑥 ) =
30𝑥
𝑓′(𝑥 ) = (𝑥 2
−4)2
3𝑥 2 −27
.
𝑥 2 −4
and
𝑓′′(𝑥 ) =
−30(3𝑥 2 +4)
,
(𝑥 2 −4)3
a) Find the critical points of 𝑓(𝑥).
Critical points:
b) Find the intervals where 𝑓(𝑥) is increasing and decreasing.
Increasing on these intervals:
Decreasing on these intervals:
c) Find the intervals where 𝑓(𝑥)is concave up and concave down.
Concave up on these intervals:
Concave down on these intervals:
(Sketch the function 𝑓(𝑥) on the next page.)
d) Sketch the function




𝑓 (𝑥 ) =
3𝑥 2 −27
. For full credit, show:
𝑥 2 −4
all x and y-intercepts,
all extrema, (label as ‘Max’ or ‘Min’)
all inflection points (label as ‘I’) and
any vertical and horizontal asymptotes.
3. Suppose that 120 feet of fencing are used to enclose a corral which is made up of two separate rectangular spaces of
equal size as shown in the figure below. Find the dimensions of the corral that can be made with 120 feet of fencing
that would have maximum area.
4. Evaluate the following limits. Check if L’Hôpital’s Rule is applicable and use the rule if it applies. (If it
doesn’t apply, evaluate the limit using another method.)
a.
lim𝑥→(−3)
b.
lim𝑥→0
c.
lim𝑥→0
9𝑥+36
𝑥 2 −9
𝑥
sin 𝑥
𝑒𝑥
𝑥
7
5. Write down the general anti-derivative of 𝑥 2 + 𝑒 𝜋𝑥 − 𝑥.
6. Find the function satisfying 𝑦 ′ = sin(5𝑥) that goes through the point (0,3).
7. Comparing conclusions of the first and second derivative tests. Consider the following information
given for a function 𝑓(𝑥):

The function 𝑓(𝑥) has two critical points, at 𝑥 = 0 and at 𝑥 = 1.

This is the sign chart for 𝑓′(𝑥):

This is the sign chart for 𝑓′′(𝑥):
a. What does the first derivative test tells us about each of the two critical points?
b. What does the second derivative test tells us about each of the two critical points?