calculus@internet
Area 6.5.1
Maxima and Minima from Contour Maps



Local Maxima and Minima of F(x,y)

The input values (a,b) where a function F(x,y) has a local minimum output value or a local maximum output value is called a critical point.

Critical points live in the domain of the function F(x,y). So do the contours of F(x,y). Therefore, it seems reasonable to look for maxima and minima in the contour diagrams.

The general process in finding local maxima and minima from contours is to graph the contour diagram of a function F(x,y), and then "look for the maxima and minima" by looking for concentric circular-type regions culminating in a point - a critical point. Here we see the comparison between the 2D contour graph and the 3D surface graph, and its maxima and minima for this region.

The advantage of the contour graph is that you can spot more easily the input coordinate values than you can with a 3D graph. Here is an exercise for you to work.

Exercise #1: Finding Critical Values from Contour Diagrams


Exercise #2: Finding Critical Values from Contour Diagrams


Exercise #3: Finding Critical Values from Contour Diagrams

Turn-in Alert!

You will turn-in this notebook to your instructor's mailbox.

What is the 2D Gradient Doing at a Critical Point?

I am not going to tell you quite yet. Let's let you figure this one out. Our goal will be to complete the following theorem statements:
#1. If (a,b) is a critical point for the function F(x,y), then the gradient of F evaluated at (a,b) is ...
#2. If (a,b) is a critical point for the function F(x,y), then the partial derivatives of F evaluated at (a,b) are ...
#3. If (a,b) is a critical point for the function F(x,y), then the tangent plane at (a,b) is ...

Turn-in Alert!

You will turn-in this notebook to your instructor's mailbox.

Vector Field Generator

Here is a Theorist Notebook that has a 2D Vector Field Generator in it. You may use this notebook to generate either an arbitrary 2D Vector Field, or the Gradient Field of a 3D Surface (i.e., THE DERIVATIVE of the 3D Surface).

Lagrange Multiplier Notebook

This notebook will allow you to graph a function F(x,y) and the constraining cylindricized curve automatically.


Road Map

  • Area 6: Vector Calculus Home Page
  • Area 6.1: Basic Graphing Commands
  • Area 6.2: Investigation of Surfaces
  • Area 6.3: Vectors
  • Area 6.4: Gradients and Partial Derivatives
  • Area 6.5: Optimization
  • Area 6.5.1: Maxima and Minima from Contour Maps (You Are Here)
  • Area 6.5.2: Maxima and Minima from 3D Graphs
  • Area 6.5.3: Lagrange Multipliers
  • Area 6.5.4: Generalized Constraint Equations
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    © 1995, Robert R. Curtis and Phil R. Smith. Printed version: © 1995, The Finley Press, Stockton, CA, USA
    curtis@ms.sjdccd.cc.ca.us ; prsmith@ucdavis.edu