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MTH 301
Hour Exam 2
Name: Date:
11 Problems. 100 Points. Follow directions carefully, and show your work. Please do not leave any question blank, and turn off cell phones and other noisemakers to avoid disturbing your classmates.
I have verified that this exam contains 11 problems and 7 printed pages. Initial.
Print the name of the people sitting either side of you :-
Short Answer - minimum explanation and calculations necessary (5 points each).
For the first four questions, let f (x, y) = x^2 y
- Calculate the partial derivatives fx and fy at the point (1, 2).
- Find the gradient vector ∇f at (1, 2).
- Find the rate of change of f (x, y) at the point (1, 2) in the direction of ~u = ~i − ~j.
- Find the linear approximation to f (x, y) at the point (1, 2).
- Draw a tree diagram for f (x, y) where x and y are both functions of a single variable t and use it to write down a formula for df dt.
- Describe in words or sketch the graph with the following contour dia- gram assuming the contours are equally spaced and getting larger as you move away from the origin.
−
−
y
4
4
2
(^02)
−
−
−5 x
5
5
3
1
−1^3
−
−4 −3 −1 0 1
Long Answer - show work and provide explanations, an answer without supporting work is not worth much (20 points each).
- Find and classify the critical points of the function
f (x, y) = x^3 − 3 x − y^2.
- Let f (x, y, z) = x^2 + y^2 + z^2. Answer the following questions about f (x, y, z).
(a) Describe the level set f (x, y, z) = 1.
(b) Find the gradient vector ∇f.
(c) Find the gradient vector at the point (1/
(d) Use your answers to find an equation for the tangent plane to the sphere of radius 1 centered at the origin at the point (1/
