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The instructions and problems for a calculus exam, covering topics such as partial derivatives, gradients, rates of change, and level sets. Students are required to find partial derivatives, gradients, and linear approximations, explain why a graph does not have a tangent plane at a point, and solve optimization problems using lagrange multipliers.
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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
−
−
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).
f (x, y) = x^3 − 3 x − y^2.
(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/