Math 117
Applications of Derivatives Exam
11/12/20
Name:
Directions:
Show all your work.
Indicate your final answer by boxing, circling, or otherwise labeling it.
You may use a graphing calculator or a calculator on your computer.
You may not use a calculator or similar application to perform algebra, differentiate
a function, or explicitly calculate a limit.
You may not communicate with others about this exam.
CHEATING WILL NOT BE TOLERATED.
1
2
Problem Points Earned Total Possible Points
1
15
2
5
3
10
4
10
5
10
6
15
7
5
8
15
9
15
Total
100
3
(1) (15 points) Find the absolute maximum and minimum values for the
function
f (x) = x3 + 3x2 + 1
on the interval 1 ≤ x ≤ 4.
4
(2) (5 points)
The Mean Value Theorem states that if f is a function that meets two
conditions on an interval [a, b], then there is some number c within that
interval such that
f (b) − f (a)
f 0 (c) =
b−a
What are those two conditions?
5
(3) (10 points) For the function
f (x) = 2x3 − 3x2 − 36x
(a) Identify all open intervals on which f is increasing.
(b) Identify all open intervals on which f is decreasing.
(c) Identify all open intervals on which f is concave up.
(d) Identify all open intervals on which f is concave down.
6
(4) (10 points) The function f is continuous and f (x) is defined for all real
numbers x. Its derivative f 0 has
f 0 (2) = f 0 (5) = 0
f 0 (−3) undefined
f 0 (x) > 0 when x < −3, 5 < x
f 0 (x) < 0 when − 3 < x < 2, 2 < x < 5
Find the locations of all local maxima and minima for the function f .
7
(5) (10 points) The function f is continuous and f (x) is defined for all real
numbers x. Its derivative f 0 has
f 0 (−8) = f 0 (−4) = f 0 (5) = 0
and its second derivative f 00 has
f 00 (x) > 0 when x < −6, 4 < x
f 00 (x) < 0 when − 6 < x < −1, −1 < x < 4
Find the locations of all local maxima and minima for the function f .
8
(6) (15 points) Evaluate the following limits. Make sure to show your work.
ln(x)
+ 2x + 5
x
(b) lim
x→0 sin(x)
(a) lim
x→∞
x2
9
(7) (5 points) Evaluate the limit below. Make sure to show your work.
lim x ln(x)
x→0+
10
(8) (15 points) Sketch the graph of a function that has all of the following
properties.
• f (x) has a vertical asymptote at x = 0.
limx→0 f (x) = −∞
• f 0 (x) is undefined for x = 0.
f 0 (−2) = 0.
For x < −2, 0 < x, f 0 (x) > 0.
For −2 < x < 0, f 0 (x) < 0.
• f 00 (x) is undefined for x = 0.
f 00 (−4) = 0.
For x < −4, f 00 (x) > 0.
For −4 < x < 0, 0 < x, f 00 (x) < 0.
• lim f (x) = 2 and lim f (x) = −2
x→−∞
x→∞
11
(9) (15 points) A farmer wants to build an enclosure for her chickens. She
has 4,000 feet of fencing to make the enclosure with. The north side of the
enclosure will be a preexisting stone wall, so no fencing for it is needed.
Find the dimensions of the enclosure that maximize its size.