Name:
MATH 208
Exam # 2
This test is worth 100 points. Show all of your work to receive full and partial credit, making
sure that all work and answers are legible. Books, notes, formula sheets, etc. are not allowed.
(6 points)
1. Find a vector normal to the surface −x3 + 2y 2 − z = 4 at the point (1, 0, −1).
(8 points)
2. Below is a contour diagram for f (x, y).
(a) Which of ∇f (0, 3) and ∇f (1, 2) is bigger in magnitude?
(b) Draw ∇f (0, 3) and ∇f (1, 2) on the contour diagram.
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MATH 208
3. Consider the function f (x, y) =
√
2x − y.
(a) (10 points) Find all four second order partial derivatives of f (x, y) at the point (3, 5).
(b) (7 points) Find the closest quadratic approximation of f (x, y) near the point (3, 5). (You do NOT
need to expand and collect like terms.)
(12 points)
4. Given z = 2xey , x = u2 + v 2 , y = u2 − v 2 , use the chain rule to find ∂z/∂v.
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MATH 208
5. Let f (x, y) = 4 + x3 + y 3 − 3xy, with contour diagram shown below.
(a) (5 points) Use the diagram to predict the location of the critical points and whether each is a local
maximum, local minimum, or saddle point.
(b) (12 points) Use the formula given for f (x, y) and the Second Derivative Test to confirm your
predictions.
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MATH 208
6. (a) (10 points) Use Lagrange multipliers to find the points (x0 , y0 ) at which f (x, y) = 3x − 4y has a
maximum or minimum subject to the constraint x2 + y 2 = 1.
(b) (3 points) The maximum of f (x, y) subject to this constraint is:
(c) (3 points) The minimum of f (x, y) subject to this constraint is:
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MATH 208
Z
1
Z
7. Consider the iterated integral
0
1
sin(x2 )dxdy.
y
(a) (8 points) Sketch the region of integration.
(b) (8 points) Reverse the order of integration.
(c) (8 points) Evaluate the integral you found in part b). Write your answer as a decimal rounded to
two decimal places. (You may use the approximation cos(1) ≈ 0.54.)
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