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MATH 251-Spring 2006 Midterm 2 Exam, Exams of Calculus

A midterm exam for math 251 at simon fraser university, held on march 7, 2006. The exam consists of 6 questions, worth a total of 50 points, covering topics such as limits, critical points, directional derivatives, differentials, and lagrange multipliers.

Typology: Exams

2012/2013

Uploaded on 02/18/2013

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MATH 251-3, Spring 2006 Simon Fraser University
Midterm 2 7 March 2006, 5:30โ€“6:20pm
Instructor: Ralf Wittenberg
Last Name:
First Name:
SFU ID:
Signature:
Instructions
1. Please do not open this booklet un-
til invited to do so.
2. Write your last name, first name(s) and
student number in the box above in block
letters, and sign your name in the space
provided.
3. This exam contains 6 questions on 6 pages
(after this title page). Once the exam be-
gins please check to make sure your exam
is complete.
4. The total time available is 50 minutes, and
there are 50 points, so allow about a minute
per point; for example, you should aim to
spend about 10 minutes on a 10-point ques-
tion. Attempt all problems!
5. This is a closed book exam. Only non-
programmable scientific calculators are al-
lowed.
6. Use the reverse side of the previous page if
you need more room for your answer, and
clearly indicate where the solution contin-
ues.
7. Show all your work, and explain your an-
swers clearly.
8. Good luck!
Question Maximum Score
1 8
2 8
3 10
4 8
5 10
6 6
Total 50
pf3
pf4
pf5

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MATH 251-3, Spring 2006 Simon Fraser University

Midterm 2 7 March 2006, 5:30โ€“6:20pm

Instructor: Ralf Wittenberg

Last Name:

First Name:

SFU ID:

Signature:

Instructions

  1. Please do not open this booklet un- til invited to do so.
  2. Write your last name, first name(s) and student number in the box above in block letters, and sign your name in the space provided.
  3. This exam contains 6 questions on 6 pages (after this title page). Once the exam be- gins please check to make sure your exam is complete.
  4. The total time available is 50 minutes, and there are 50 points, so allow about a minute per point; for example, you should aim to spend about 10 minutes on a 10-point ques- tion. Attempt all problems!
  5. This is a closed book exam. Only non- programmable scientific calculators are al- lowed.
  6. Use the reverse side of the previous page if you need more room for your answer, and clearly indicate where the solution contin- ues.
  7. Show all your work, and explain your an- swers clearly.
  8. Good luck!

Question Maximum Score

Total 50

  1. In each of the following cases find the limit, if it exists, or show that the limit does not exist:

(a) [4 points]

lim (x,y)โ†’(0,0)

3 xy x^2 + 2y^2

(b) [4 points]

lim (x,y)โ†’(0,0)

3 xy^2 x^2 + 2y^2

  1. Consider the function

F (x, y, z) = x^2 z โˆ’

y z^2

(z > 0).

(a) [5 points] Find the directional derivative of F at the point P (2, โˆ’ 3 , 1) in the direction of the vector v = i + 5j โˆ’ 2 k.

(b) [2 points] Find the maximum rate of change of F at the point P (2, โˆ’ 3 , 1). In the direction of which unit vector u is the directional derivative a maximum?

(c) [3 points] Find an equation to the tangent plane to the surface

F (x, y, z) = x^2 z โˆ’

y z^2

at the point P (2, โˆ’ 3 , 1).

  1. Let w =

x^2 y^3 z^4

(a) [4 points] Find the differential dw.

(b) [4 points] Suppose x increases by 1%, y increases by 2% and z increases by 3%. Using your answer from (a): By approximately what percentage will the value of w increase or decrease?

  1. [6 points] Use the method of Lagrange multipliers to find the maximum value of f (x, y) = xy on the ellipse 4x^2 + 9y^2 = 36.