MATH 165
Exam 2
Fall 2018
Student name:
Main instructor:
Recitation instructor:
Section:
This exam is closed book and closed notes. No electronic devices, including calculators, are allowed.
Drawing pictures to help understand and solve the questions is encouraged. Answer each question
completely using exact values; you do not need to simplify your answers unless otherwise indicated.
Show your work neatly, including good notation showing the steps in your work, as well as writing
legibly; answers without work and/or justifications will not receive credit. Circle your final
answer for each problem. Each problem is worth 10 points, for a total of 60 points.
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DO NOT BEGIN THIS EXAM UNTIL INSTRUCTED TO START
Do not write
in these boxes
on the exam
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1. Find the equation of the tangent line to the curve x2 + exy − y2 = −3 at the point (0, 2).
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1
2. Evaluate lim (1 + arctan(3x)) x .
x→0
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3. A recipe for a cake includes yeast which causes the cake to expand. The cake always
remains in the shape of a cylinder where the height is one-third the length of the radius.
While the cake is expanding you also make some frosting, continually adjusting the amount
based on the surface area of the cake to be covered, so it is important to determine how the
surface area is changing to make sure an appropriate amount of frosting is made.
Determine how fast the surface area of the cake is changing (with units) when the cake is
3 inches high and the volume of the cake is expanding at a rate of 34 in3 /min.
(If the bottom of the cake is not frosted, then SA = πr2 + 2πrh and V = πr2 h.)
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4. Let f(x) =
4x + 2
for −5 6 x 6 4.
x2 + 2
(a) Determine the intervals for which the function is increasing and which it is decreasing.
(b) For each critical point and endpoint of f(x) classify it as one of the following: absolute max;
absolute min; local max; local min; neither max nor min. Justify your answers.
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xln x
5. Find f (1) where f(x) = 2
.
x (x + 1)(x + 2)
0
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6. Last week you sold some pipe to your best customer, Mario, who used the pipe to connect
Peach’s Castle (P) to Yoshi’s Island (Y). Peach’s Castle is located on the edge of a large lake,
while the island is located by going 5 kilometers along the shoreline, and then 6 kilometers
into the lake. There were two types of pipes used, one that can run along the shore, and a
sturdier (more expensive) version that can be used in the water.
This week Mario’s brother Luigi is coming in to get some pipe for a different project. As you
are preparing the bill you realize you misplaced the cost of the pipe used in the water. But you
do recall the following two things, (i) the cost of the pipe that runs on the shore is 100 gold
coins per kilometer and (ii) that the minimum cost for Mario’s project happened when the
pipe went along the shoreline exactly 52 kilometers and the remainder went in the water in a
straight line to the island.
(a) If W is the number of gold coins per kilometer for pipe that can be used in the water,
determine W. (Hint: set up an optimization problem for the total cost of the piping in terms of
x, the amount of pipe on the shoreline, and also involves W; see the picture below.)
Y
6 km
x km
P
5 km
(b) Using the information from (a) determine how many gold coins Mario spent on pipes to
complete his project.
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