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Math206A Multivariable Calculus Exam - Prof. P. Wong, December 12, 2007, Exams of Mathematics

Birla Institute of Technology and ScienceMathematics

The final exam for the math206a multivariable calculus course taught by prof. P. Wong, held on december 12, 2007. The exam covers various topics in multivariable calculus, including critical points, change of variables, directional derivatives, green's theorem, line integrals, work, divergence, gauss' theorem, and stokes' theorem.

Typology: Exams

2012/2013

Uploaded on 03/07/2013

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MATH206A MULTIVARIABLE CALCULUS - PROF. P.
WONG
FINAL EXAM - DECEMBER 12, 2007
NAME:
Instruction: Read each question carefully. Explain ALL your work
and give reasons to support your answers.
Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
1. 15
2. 20
3. 15
4. 20
5. 15
6. 15
7. 20
Total 120
1
pf3
pf4
pf5
pf8

Partial preview of the text

Download Math206A Multivariable Calculus Exam - Prof. P. Wong, December 12, 2007 and more Exams Mathematics in PDF only on Docsity!

MATH206A MULTIVARIABLE CALCULUS - PROF. P.

WONG

FINAL EXAM - DECEMBER 12, 2007

NAME:

Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers. Advice: DON’T spend too much time on a single problem.

Problems Maximum Score Your Score

  1. 15
  2. 20
  3. 15
  4. 20
  5. 15
  6. 15
  7. 20 Total 120

1

2 FINAL EXAM - DECEMBER 12, 2007

  1. Let f : R^2 → R and g : R → R^3 be given by

f (x, y) = 3x^3 + y^2 − 9 x + 4y and g(t) = (t

2 6 , e

t/ (^3) , 1 − t).

(8 pts) (i) Classify all critical points of f (local max/min, saddle points, etc.).

(7 pts.) (ii) Find the Jacobian matrix J(g ◦ f )(0, 1) of g ◦ f at (0, 1).

4 FINAL EXAM - DECEMBER 12, 2007

  1. Consider the following function F : R^3 → R given by F (x, y, z) = xy + z^2.

(5 pts) (i) Find the directional derivative DuF (1, 1 , 1) of F at the point (1, 1 , 1) in the direction of u = 2i − j + 2k.

(5 pts) (ii) Find a direction (give a unit vector) in which F decreases most rapidly at the point (1, 1 , 1).

(5 pts) (iii) Find an equation of the plane tangent to the level surface F (x, y, z) = 2 at the point (1, 1 , 1).

MATH206A MULTIVARIABLE CALCULUS - PROF. P. WONG 5

  1. (8 pts) (i) Let C be the path formed by the triangle with ver- tices (0, 0), (1, 0)), and (0, 1), oriented counterclockwise. Use Green’s theorem to evaluate the line integral ∮ C

2 xy dx + (x + 1)^2 dy.

(12 pts) (ii) Let F (x, y) = (−e−x^ ln y, e− yx ). Determine whether the vector field F is path independent. If so, find a function f so that ∇f = F.

MATH206A MULTIVARIABLE CALCULUS - PROF. P. WONG 7

  1. Let F (x, y, z) = (x^2 y, 2 xz, yz^3 ). (5 pts) (i) Find the divergence div F of F.

(10 pts) (ii) Use Gauss’ (or Divergence) theorem to evaluate the flux of the vector field F (^) ∫∫

∂S

© F · n dσ

where ∂S is the surface of the rectangular box S determined by

0 ≤ x ≤ 1 , 0 ≤ y ≤ 2 , 0 ≤ z ≤ 3.

8 FINAL EXAM - DECEMBER 12, 2007

  1. Let F (x, y, z) = (z, x, y). Suppose M is the portion of the parabo- loid z = x^2 + y^2 + 2 that lies inside the solid cylinder x^2 + y^2 ≤ 1.

M

(5 pts.)(i) Find curl F.

(5 pts.)(ii) Write a parametrization for the surface M. Be sure to indicate the domains for the parameters.

(10 pts.)(iii) Use Stokes’ theorem to evaluate the path integral ∮ ∂M

F · dx [Hint: Use parts (i) and (ii).].