MA125 Calculus I
Exam 3
Name: _________________________________________
There are 6 questions. For full credit, show all steps of your work. A calculator may not be
used.
1. Consider the function 𝑓(𝑥) = 𝑥 3 − 12𝑥.
a. Find the critical points of 𝑓(𝑥).
Critical points:
b. Find the absolute maximum and minimum of 𝑓(𝑥) on the interval [−1, 10].
Maximum:
Minimum:
2. If f is differentiable at 𝑥 = 𝑎, the linearization of 𝑓(𝑥) at 𝑥 = 𝑎 is given by
𝐿(𝑥) = 𝑓(𝑎) + 𝑓 ′ (𝑎)(𝑥 − 𝑎). Use linear approximation to estimate sin(0.04).
3. Consider the function
Given that
′
𝑓 (𝑥) =
𝑓(𝑥 ) = 𝑥
2⁄
3 (𝑥 2
8(𝑥 2 −1)
1
3𝑥 ⁄3
and
𝑓
− 4).
′′ (
𝑥) =
2
8(5𝑥 +1)
4
9𝑥 ⁄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.)
2⁄
d) Sketch the function 𝑓(𝑥) = 𝑥 3 (𝑥2 − 4). For full credit, show and label the x and y-intercepts,
all extrema, inflection points and any vertical and horizontal asymptotes.
4. The best fencing plan A rectangular plot of farmland will be bounded on one side by a river and on the
other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest
area you can enclose, and what are its dimensions?
5. Initial Value Problem Find the function satisfying 𝑦 ′ = 5𝑥 − sin 𝑥 that goes through the point (0,10).
6. 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→4
3𝑥−12
b. lim𝑥→𝜋
cos 𝑥
2
c. lim𝑡→0
𝑥 2 −16
2𝑥−𝜋
2𝑡 2
𝑒 𝑡−1
Bonus Question:
Write down an equation for a function that has zeros at 𝑥 = ±3, a vertical asymptote at 𝑥 = 7 and a horizontal
asymptote at 𝑦 = −2.