Math 201
Spring 2023
Exam 2
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Directions. Read each problem carefully and indicate answers as directed. Solutions must be supported by calculations or explanations: No points will be given for answers that are not accompanied
by, or are not consistent with, supporting work. Partial credit can be earned for steps that make reasonable progress towards a solution.
For conceptual questions, answer in complete sentences. For computational problems, box your final
answer. Calculators, notecards, cell phones, and all forms of collaboration are not allowed.
For two bonus points, write your current favorite book on the last page of your exam.
Problem #
Points
1
14
2
14
3
26
4
16
5
10
6
10
7
10
Total
100
Score
Problem 1
[14 points total]
[SHOW ALL WORK]
a. (4 pts) Solve the following equation for x:
3e5x
2
= 12.
b. (6 pts) The average leaf width w in millimeters (mm) in tropical Australia is a function of the
average rainfall x, also in mm, and is given by the equation
w(x) = 32.7 ln (x)
32.7 ln(244.5).
Give the rate of change of average leaf width w with respect to rainfall x when the average
annual rainfall is 2000 mm. Include units.
c. (4 pts) The graph of y = f (x) is given below.
What is f
1 (0)?
What is f
1 (1)?
Problem 2
[14 points total]
a. (9 pts) Consider the function f (x) = 2x2
to find f 0 (3).
[SHOW ALL WORK]
3x. Use the limit definition of the derivative
b. (5 pts) The graph of a function y = f (x) is given below. Give values for ✏, , and L so that the
picture supports the sentence: “If 0 < |x 3| < , then |f (x) L| < ✏.”
Problem 3
[26 points total]
[SHOW ALL WORK]
Answer the following derivative questions. You can use any derivative rules that we have seen in class
and do not need to simplify your answers.
a. (6 pts) Find r0 (t), where r(t) =
cos(t) + 3t2
.
log2 (t) 3
b. (7 pts) Let u(x) = f (g(x) + 1), where y = f (x) and y = g(x) are graphed below.
Compute the derivative u0 (1).
Problem 3 (cont.)
c. (6 pts) Evaluate
[26 points total]
[SHOW ALL WORK]
dh
, where h(x) = arctan 3x + x1/3 .
dx
d. (7 pts) On Day t, a clothing company sells q(t) shirts at a price p(t). If R(t) denotes the
company’s revenue on day t, then R(t) = p(t)q(t).
On Day 10, the company sells 500 shirts and the quantity sold is decreasing by about 4 shirts
a day. Meanwhile, the price per shirt is $50 and is increasing at about $3 per day. Use the
product rule to estimate the rate of change of revenue R(t) on Day 10. Include units.
Problem 4
[16 points total]
[SHOW ALL WORK]
The curve of points defined by the equation x2 + xy + y 2 = 1 is graphed below.
a. (4 pts) Use the equation to find all of the points on the curve with x-coordinate equal to 1 and
label them on the curve.
b. (8 pts) Find a formula for
dy
.
dx
b. (4 pts) Find the equation of the tangent line at one point with x-coordinate equal to 1 and
sketch the tangent line on the curve.
Problem 5
[10 points total]
[SHOW ALL WORK]
Let y = f (x) be a differentiable function on ( 4, 4). The graph of its derivative y = f 0 (x) is given
below. Use the graph to answer the following questions:
a. (4 pts) Give the intervals in ( 4, 4) where y = f (x) is increasing.
b. (2 pts) Give the x-value(s) in ( 4, 4) where y = f (x) has a horizontal tangent line.
c. (4 pts) By sketching a tangent line, estimate f 00 (2).
Problem 6
[10 points total]
[SHOW ALL WORK]
Let S(t) denote the water stored in Lake Sonoma, a reservoir in Northern California, t months after
January 2014. Here S(t) is measured in acre-feet (the amount of water needed to cover one acre of
area with one foot of water). Scientists estimated S(t) and S 0 (t) at several data points and recorded
the information in the following table:
t
S(t)
S 0 (t)
2
183,000
3,000
7
164,000
-8,000
a. (5 pts) Use linear approximation to estimate the change in S(t) from the beginning of March
2014 (t = 2) to the beginning of April 2014 (t = 3). Include units.
b. (5 pts) Use linearization to estimate the water level 9 months after January 2014. Include
units.
Problem 7
[10 points total]
[SHOW ALL WORK]
Choose two of the following statements and decide whether each is true or false. Write the full word
“True” or “False.” Then justify your answer. Put a big star next to the two statements that you want
me to grade.
(a) If f (x) is a continuous function on ( 1, 1), then f 0 (a) must exist at every point a in ( 1, 1).
(b) If a differentiable function f has the table of values below, then f 0 (1) ⇡ 3.
(c) If f and g are differentiable functions and f (2) 6= g(2), then it must be the case that f 0 (2) 6= g 0 (2).