MATH 2263 Multivariable Calculus
Exam 1 – October 3, 2013
Name:
Time Limit: 50 Minutes
This exam contains 6 pages (including this cover page) and 5 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
top of every page, in case the pages become separated.
You may not use your books, notes, or a calculator on this exam.
You are required to show your work on each problem on this exam.
Problem
Points
1
20
2
20
3
20
4
20
5
20
Total:
100
Score
MATH 2263 Multivariable Calculus
Exam 1 - Page 2 of 6
October 3, 2013
1. P (1, 0, 1), Q(−2, 1, 3), R(4, 2, 5).
(a) (7 points) Find a nonzero vector orthogonal to the plane through the points P , Q and R.
(b) (7 points) Find the equation of the plane through the points P , Q and R.
(c) (6 points) Find the area of the triangle P QR.
MATH 2263 Multivariable Calculus
Exam 1 - Page 3 of 6
October 3, 2013
2. A surface is created by rotating the parabola z = y 2 − 1 about the z−axis.
(a) (13 points) Sketch this surface and find its equation.
(b) (7 points) Sketch the curve of the intersection of this surface with the xy-plane and find
its equation.
MATH 2263 Multivariable Calculus
Exam 1 - Page 4 of 6
3. (a) (5 points) Find the domain of the function f (x, y) =
October 3, 2013
xy + y 2
.
x2 + y 2
(b) (15 points) Find the set of points at which the following function


xy + y 2

 2
2
f (x, y) = x + y


5
is continuous and justify your answer.
if (x, y) 6= (0, 0)
if (x, y) = (0, 0)
MATH 2263 Multivariable Calculus
Exam 1 - Page 5 of 6
4. (a) (5 points) Find the partial derivatives of the function f (x, y) =
(b) (15 points) Estimate
√
October 3, 2013
p
x3 + y 2 .
2.023 + 0.972 . (Hint: use the linear approximation for f (x, y).)
MATH 2263 Multivariable Calculus
Exam 1 - Page 6 of 6
October 3, 2013
5. The temperature at any point (x, y, z) is given by T (x, y, z) = 50 − x2 − 2y 2 − 3z 2 . A bee flew
from (1, 0, −2) to (3, 1, 0) in 1 second along a straight line at a constant speed.
(a) (7 points) How fast is the temperature of the bee changing with respect to distance
at (3, 1, 0)?
(b) (7 points) How fast is the temperature of the bee changing with respect to time at
(3, 1, 0)?
(c) (6 points) To get warm as soon as possible, in which direction will the bee go from (3, 1, 0)?