MATH 116 Midterm: Midterm 1 Winter 2015
Math 116 — First Midterm
February 9, 2015
Name:
Instructor: Section:
1. Do not open this exam until you are told to do so.
2. This exam has 9 pages including this cover. There are 8 problems. Note that the problems
are not of equal difficulty, so you may want to skip over and return to a problem on which
you are stuck.
3. Do not separate the pages of this exam. If they do become separated, write your name on
every page and point this out to your instructor when you hand in the exam.
4. Please read the instructions for each individual problem carefully. One of the skills being
tested on this exam is your ability to interpret mathematical questions, so instructors will
not answer questions about exam problems during the exam.
5. Show an appropriate amount of work (including appropriate explanation) for each problem,
so that graders can see not only your answer but how you obtained it. Include units in your
answer where that is appropriate.
6. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric
keypad). However, you must show work for any calculation which we have learned how to
do in this course. You are also allowed two sides of a 3′′ ×5′′ note card.
7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch
of the graph, and to write out the entries of the table that you use.
8. Turn off all cell phones and pagers, and remove all headphones.
9. You must use the methods learned in this course to solve all problems.
Problem Points Score
1 14
2 13
3 13
4 14
5 11
6 10
7 16
8 9
Total 100
Math 116 / Exam 1 (February 9, 2015) page 2
1. [14 points] While you are trying to fill your old bucket with water, it begins to leak. Suppose
the continuous function f(t) is the rate of change of the volume of water in the bucket, in
gallons per minute, tminutes after it begins to leak. A graph of f(t) for 0 ≤t≤3 is shown
below. The function f(t) is linear for 1 ≤t≤2. The region R1has area 5.8, and the region
R2has area 3. There are 7 gallons of water in the bucket at t= 1.
1 2 3 t
R1
R2
f(t)
(1,4)
(3,−4)
a. [5 points] Write an expression involving integrals for A(t), the volume of water in the
bucket, in gallons, tminutes after the bucket began to leak where 0 ≤t≤3. Your
expression may contain the function f.
b. [2 points] How much water was in the bucket when it began to leak? How much water
was in the bucket 3 minutes after it began to leak? Fill in the blanks below.
There were gallons of water in the bucket when it began to leak.
There were gallons of water in the bucket 3 minutes after it began to leak.
c. [3 points] Write an expression involving an integral for the average rate of change of the
amount of water in the bucket during the first three minutes after it began to leak, and
find the value of your expression, including units.
d. [4 points] For t≥3, suppose f(t) is linear with slope 1, but is only defined until the time
when the bucket is empty. For what value of tis the bucket empty? (Remember that f
is continuous as specified above).
Math 116 / Exam 1 (February 9, 2015) page 3
2. [13 points] Fred is designing a plastic bowl for his dog, Fido. Fred makes the bowl in the
shape of a solid formed by rotating a region in the xy-plane around the y-axis. The region,
shaded in the figure below, is bounded by the x-axis, the y-axis, the line y= 1 for 0 ≤x≤4,
and the curve y=−(x−5)4+ 2 for 4 ≤x≤21/4+ 5. Assume the units of xand yare inches.
x
y
a. [7 points] Write an expression involving one or more integrals which gives the volume of
plastic needed to make Fido’s bowl. What are the units of your expression?
b. [6 points] Fred wants to wrap a ribbon around the bowl before he gives it to Fido as a
gift. The figure below depicts the cross section of the bowl obtained by cutting it in half
across its diameter. The thick solid curve is the ribbon running around this cross section,
and the dotted curve is the outline of the cross section which is not in contact with the
ribbon. Write an expression involving one or more integrals which gives the length of the
thick solid curve in the figure (the length of ribbon Fred needs to wrap the bowl).
x
y
Document Summary
Section: do not open this exam until you are told to do so, this exam has 9 pages including this cover. Note that the problems are not of equal di culty, so you may want to skip over and return to a problem on which you are stuck: do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam: please read the instructions for each individual problem carefully. Include units in your answer where that is appropriate: you may use any calculator except a ti-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card.
