Math 208-002
Exam #1
October 2, 2013
UNL Student ID Number:
Name:
Turn off your cell phones.
Page:
1
2
3
4
5
6
Total
Points:
8
20
20
20
22
10
100
Score:
Instructions:
1. No books or notes may be used on the exam.
2. Credit or partial credit will be given only when the appropriate explanation and/or work is
shown.
3. Read and follow directions carefully.
4. This exam has 10 questions, for a total of 100 points. There are 6 pages (in addition to this
cover sheet).
5. You will have 50 minutes to complete the exam.
6. ALL CELL PHONES MUST BE TURNED OFF DURING THE EXAM. Phones
that ring, vibrate, or otherwise cause a distraction will be confiscated and may result in the
student receiving no credit for the exam.
DO NOT OPEN YOUR EXAM UNTIL INSTRUCTED TO DO SO!!
1. Use the following contour diagram for the function z = f (x, y) to answer the following questions.
(4 points)
(a) Is fy (−2, 1) positive, negative, or zero? Briefly justify your answer.
(4 points)
(b) Is fx (1, 2) positive, negative, or zero? Briefly justify your answer.
(10 points) 2. For the function g(x, y) = y 3 +sin(2x), draw graphs of cross-sections of g with x fixed at x = 0,
x = π/4, and x = 3π/4.
(10 points) 3. Write ~v = 1~i + 2~j + 1~k as the sum of two vectors, one parallel, and one perpendicular, to the
vector
2
3
6
w
~ = ~i + 4~j − ~k.
7
7
7
Page 2
(10 points) 4. An airplane is flying at an airspeed of 600 km/hr in a cross-wind that is blowing from the
northeast at 50 km/hr. In what direction should the plane head to end up going due east?
Express your answer in degrees north of east.
(10 points) 5. The monthly mortgage payment, in dollars, P , for a house, is a function of 3 variables P =
f (A, r, N ), where A is the amount borrowed (in dollars), r is the interest rate (as a percent),
and N is the number of years before the mortgage is paid off. Give the units for each of the
three first-order partial derivatives, and briefly explain whether you expect the sign of each
partial to be positive or negative.
Page 3
(10 points) 6. Show that the following limit does not exist:
2x − y 2
.
(x,y)→(0,0) 2x + y 2
lim
(10 points) 7. Consider the three points A = (0, 1, −1), B = (1, 0, −1), and C = (2, 1, 0). Find the area of
the triangle ABC.
Page 4
8. The temperature of a gas at the point (x, y, z) is given by G(x, y, z) = x2 − 5xy + y 2 z.
(8 points)
(a) What is the rate of change in the temperature at the point (1, 2, 1) in the direction of
~v = 1~i + 2~j + 2~k?
(4 points)
(b) What is the maximum rate of increase in the temperature at the point (1, 2, 1)?
9. Consider the plane 4x − y − 2z = 6.
(5 points)
(a) Give an example of a two-variable function f (x, y) so that the graph of z = f (x, y) is the
given plane.
(5 points)
(b) Give an example of a three-variable function g(x, y, z) and a value of c so that the level
surface g(x, y, z) = c is the given plane.
Page 5
(10 points) 10. For the function f (x, y) = x2 y, find the quadratic Taylor polynomial Q(x, y) which approximates f near (1, 0).
Page 6