MATH2321, Calculus III for Science and Engineering, Fall 2015
1
Exam 1
Name (Printed)
Date
Signature
Instructions
•
STOP.
Print your name, the date, and then sign the exam on the line
above.
•
This exam consists of 5 problems, each worth 20 points apiece, for a total of
100 points.
•
Work as many problems as you can within 65 minutes. You may work the
problems in any order, so use your time wisely. If you finish early, I encourage
you to check your work before you hand in the exam.
•
Work each problem in the space provided, or on the back of the preceding
page. Be sure to indicate if your work runs to the back of a page.
•
Circle or box your final answer to each problem, and show all work. I will
assign partial credit for incorrect answers based upon the work that you
submit.
•
You may use your calculator on this exam. The use of any other electronic
device is
NOT
permitted.
•
This exam is closed book. If you brought your textbook or any notes with
you today, keep them out of sight for the duration of the exam.
•
If the wording of any problem is unclear, raise your hand.
I will come to
your desk and attempt to clarify.
•
Do not turn the page until I give the signal to begin. Good luck!
MATH2321, Calculus III for Science and Engineering, Fall 2015
2
Problem 1.
[20 points]
Consider the surface in
R
3
which is described by the equation

z
2
+ 4 =
x
2
+ 2
y
2
.
(a.)
[4 points]
Graph the crosssection of the surface in the plane where
y
= 0. A
quick sketch is sufficient, but be sure to label the axes of your graph.
Answer
: The crosssection with
y
= 0 consists of those points in the
xz
plane
which satisfy
x
2
+
z
2
= 4
.
(1)
The graph is thus a circle centered at the origin with radius
r
= 2, as sketched in
Figure 1.
z
x
y=0
Figure 1: Crosssection for
y
= 0 in
xz
plane.
(b.)
[4 points]
Graph the crosssection of the surface in the plane where
x
= 0. A
quick sketch is sufficient, but be sure to label the axes of your graph.
Answer
: The crosssection with
x
= 0 consists of those points in the
yz
plane
which similarly satisfy
2
y
2
+
z
2
= 4
.
(2)
In this case, the graph is an ellipse at the origin with minor and major axes
along the
y
 and
z
axes, respectively. The
y
intercepts of the ellipse are
y
=
±
√
2, and the
z
intercepts are
z
=
±
2. See Figure 2 for a sketch.
(c.)
[8 points]
Graph the contours of the surface in the planes where
z
= 0
,
±
1
,
±
2. Label each contour with the corresponding value of
z
. Be sure to
label the axes of your graph.
MATH2321, Calculus III for Science and Engineering, Fall 2015
3
x=0
z
y
Figure 2: Crosssection for
x
= 0 in
yz
plane.
Answer
: In the horizontal planes where
z
= 0,
z
=
±
1, and
z
=
±
2, the
corresponding contour lines in the
xy
plane satisfy
z
= 0 :
x
2
+ 2
y
2
= 4
,
z
=
±
1 :
x
2
+ 2
y
2
= 3
,
z
=
±
2 :
x
2
+ 2
y
2
= 0
.
(3)
For
z
= 0 or
z
=
±
1, the contour line is an ellipse centered at the origin, with
major and minor axes in the
x
 and
y
directions, respectively. As the absolute
value of
z
increases from zero, the size of the corresponding ellipse decreases. For
z
=
±
2, the only solution is (
x, y
) = (0
,
0), so the contour is simply the point at
the origin. The contour diagram is graphed in Figure 3.
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