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Practice Exam 2 - Calculus III - Fall 2008 | MATH 241, Exams of Advanced Calculus

University of Illinois - Urbana-ChampaignAdvanced Calculus
Prof. Jeremy A. Rouse

Material Type: Exam; Professor: Rouse; Class: Calculus III; Subject: Mathematics; University: University of Illinois - Urbana-Champaign; Term: Spring 2008;

Typology: Exams

Pre 2010

Uploaded on 12/09/2009

cohn1
cohn1 🇺🇸

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Math 241 - Exam 2 - March 4, 2008
Name:
Question Number Possible Points Score
1 20
2 30
3 30
4 30
5 30
Total 140
Instructions:
•Write your name on the exam now.
•You may begin when the bell rings.
•You may not use the book or notes.
•Show your reasoning unless otherwise specified.
•You do not need to simplify your answers.
1
pf3
pf4
pf5

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Download Practice Exam 2 - Calculus III - Fall 2008 | MATH 241 and more Exams Advanced Calculus in PDF only on Docsity!

Math 241 - Exam 2 - March 4, 2008

Name:

Question Number Possible Points Score

Total 140

Instructions:

  • Write your name on the exam now.
  • You may begin when the bell rings.
  • You may not use the book or notes.
  • Show your reasoning unless otherwise specified.
  • You do not need to simplify your answers.

1

  1. (a) (10 points). Express the equation for the cone z^2 = x^2 + y^2 in cylindrical coordinates.

(b) (10 points). Find the (scalar) curl of the vector field F~ (x, y) = (−(x − y)/(x^2 + y^2 ), (x −

y)/(x^2 + y^2 )).

  1. Let f (x, y) = x^4 − x^2 y^2 + y^4 − x^2 + y^2.

(a) (15 points). Find the critical points of f.

(b) (15 points). Use the second derivative test to determine if these critical points are local

maxima, local minima, or saddle points. What is the minimum value of the function?

  1. (a) (15 points). Drawn below are the plot of ∇f and the level curve g(x, y) = k. Mark the

points on the curve where maxima and minima of f occur, subject to g(x, y) = k.

y

1

x

0 0 0.

(b) (15 points). Explain using the physical interpretation of divergence why if F~ (x, y, z) =

a~i + b~j + c~k (here, a, b, and c are constants), then ∇ · F~ = 0.