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Calculus III Worksheet for Temperature Distribution and Critical Points, Assignments of Mathematics

Rose-Hulman Institute of Technology (RHIT)Mathematics

This worksheet, from calculus iii course taught by professor broughton in winter 2003-04, provides problems related to temperature distribution, critical points, and graphs of functions with specific domains. It also includes a table for hessian matrix values and their corresponding comments.

Typology: Assignments

Pre 2010

Uploaded on 08/18/2009

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MA 113 - Calculus III - Worksheet #9
Professor Broughton - Winter 2003-04
Name: Box #:
In this work sheet perform the elementary calculations by hand and then check
with Maple
1. Suppose the temperature distribution on a plate determined by −2≤x≤2,
−2≤x≤2 is given by
T= 20 −3x2+ 3xy −4y2+x−y.
•Sketch a graph of the function paying attention to the domain.
•Find all the critical points in the interior of the plate and find the values
of Tthere.
pf3
pf4
pf5

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MA 113 - Calculus III - Worksheet

Professor Broughton - Winter 2003-

Name: Box #: In this work sheet perform the elementary calculations by hand and then check with Maple

  1. Suppose the temperature distribution on a plate determined by − 2 ≤ x ≤ 2, − 2 ≤ x ≤ 2 is given by

T = 20 − 3 x^2 + 3xy − 4 y^2 + x − y.

  • Sketch a graph of the function paying attention to the domain.
  • Find all the critical points in the interior of the plate and find the values of T there.
  • Find the maximum and minimum values on the boundary..
  • What are the absolute maximum and minimum and what are the val- ues?
  1. This question is the same as 1 above except that the domain is a circular plate of radius 2 centered at the origin
  • Sketch a graph of the function paying attention to the domain.
  • Find the maximum and minimum values on the boundary.
  1. For each of the functions f (x, y) below sketch the graph near a critical point, and compute the determinant the Hessian matrix at the critical point.

H =

fxx fxy fyx fyy

  • f (x, y) = xy exp(− 2 x^2 − 2 y^2 )
  • f (x, y) = x^2 − xy + y^2 + x + y
  • x^3 − 3 xy^2
  1. Fill in the table.

value of H max or min comment H > 0 H < 0 H = 0