Math 2374 Final Exam PRINT NAME____________________________

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Time Limit: 3 Hours WORKSHOP INSTRUCTOR________________

This exam contains 7 problems on 7 pages, not including this cover page. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated.

You may not use your books and notes on this exam, but you may use your graphing calculator. However, all answers must be justified by valid mathematical reasoning. This includes the evaluation of definite and indefinite integrals.

Show your work, in a reasonably neat and coherent way, in the space provided on the following pages. If you need more space, use the back of the preceding page.

Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written.

A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit.

Full credit will be given only for work that is presented neatly and logically; work scattered all over the page without a clear ordering will receive very little credit.

Do not give numerical approximations to quantities such as or p or .

Do simplify expressions such as e⁰ = 1, cos (p /2) = 0, etc.

The transformation from spherical coordinates to rectangular coordinates is given by:

x = r sinf cosq , y = r sinf sinq , z = r cosf

The Jacobian determinant is

Do not write in the table below.

1. Let
   
   1. Using Green’s Theorem, evaluate the line integral of F counter-clockwise around the circle of radius 3 centered at the point (4,4).
   2. Using any valid method, evaluate the line integral of F around the circle of radius 3 centered at the origin.
2. Let f(x,y) = x²y3
   1. Find the derivative matrix of f at (3,2).
      
   2. Find the direction in which f increases most rapidly at the point (3,2).
      
   3. Find the equation of the tangent plane to the graph of this function at the point (3,2,72). Write your answer in the form z = ax +by +c.
3. Let f(x,y) be a differentiable function satisfying these conditions:
   • (2,3) is a critical point of f, f(2,3) = 5, and the Hessian matrix for f at (2,3) is
   1. What is the second degree Taylor polynomial of f at (2,3)?
      
   2. What does the second derivative test say about this critical point?
      
   3. Give an example of a function f satisfying these conditions such that f has a local minimum at (2,3). Be sure to explain why your example does what you claim it does.
   4. Give an example of a function satisfying these conditions such that f has a saddle at (2,3). Be sure to explain why your example does what you claim it does
   .
4. Let F(x,y,z) = so that curl F =

Let S be the hemisphere given by , z > 0. Using spherical coordinates, set up, but do not evaluate, an integral which represents the flux of curl F across S, using the outward pointing normal. Your answer should be expressed as the integral of some function, with appropriate endpoints of integration. Your answer should not contain any dot products, cross products, etc. These should all be worked out and simplified. Then use Stokes’s Theorem to evaluate the integral.