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.
4
3
2
1
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-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
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(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.
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(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.
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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.
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(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).
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