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MTH 301 Hour Exam 2: Calculus Problems - Prof. Aaron D. Wootton, Exams of Calculus

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.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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koofers-user-14s 🇺🇸

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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) = x2y
1. Calculate the partial derivatives fxand fyat the point (1,2).
2. Find the gradient vector fat (1,2).
1
pf3
pf4
pf5

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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

  1. Calculate the partial derivatives fx and fy at the point (1, 2).
  2. Find the gradient vector ∇f at (1, 2).
  1. Find the rate of change of f (x, y) at the point (1, 2) in the direction of ~u = ~i − ~j.
  2. Find the linear approximation to f (x, y) at the point (1, 2).
  1. 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.
  2. 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).

  1. Find and classify the critical points of the function

f (x, y) = x^3 − 3 x − y^2.

  1. 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/