Optimization
https://maa.org/taxonomy/term/40392/all
enAn Advanced Calculus Approach to Finding the Fermat Point
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/an-advanced-calculus-approach-to-finding-the-fermat-point
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>The author uses the concept of maximum and minimum values to find Fermat points for a triangle. (The Fermat point of a triangle is the point such that the distances from the vertices have a minimum sum.)</em></div></div></div>Hidden Boundaries in Constrained Max-Min Problems
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/hidden-boundaries-in-constrained-max-min-problems
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><i>Hidden boundaries in finding a global minimum</i></p>
</div></div></div>An Extremely Cautionary Tale
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/an-extremely-cautionary-tale
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Example of an extremal problem gone awry</em></p>
</div></div></div>Lagrange Multipliers Can Fail to Determine Extrema
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/lagrange-multipliers-can-fail-to-determine-extrema
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Some simple examples in which Lagrange multipliers fail to locate extrema</em></p>
</div></div></div>A Bug Problem
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-bug-problem
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>A bug is on the inside of a container that has the shape of a paraboloid \(y=x^2\) revolved about the \(y\)-axis. If a liquid is poured into the container at a constant rate, how fast does the bug have to crawl to stay dry?</em></div></div></div>The HM-GM-AM-QM Inequalities
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-hm-gm-am-qm-inequalities
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>Common inequalities are derived using LaGrange multipliers.</em></div></div></div>A Quick Proof that the Least Squares Formulas Give a Local Minimum
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-quick-proof-that-the-least-squares-formulas-give-a-local-minimum
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Use the second derivative test to establish that the regression parameters give a minimum.</em></p>
</div></div></div>The Flip-Side of a Lagrange Multiplier Problem
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-flip-side-of-a-lagrange-multiplier-problem
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><em>An attempt is made to define, for optimization problems, a dual or "flip-side" problem also solvable.</em></div></div></div>A Visual Proof of Eddy and Fritsch's Minimal Area Property
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-visual-proof-of-eddy-and-fritschs-minimal-area-property
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Describes and generalizes a technique for minimizing certain areas and volumes.</em></p>
</div></div></div>A Surface with One Local Minimum
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-surface-with-one-local-minimum
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>A smooth surface in \(\Re^2\) or \(\Re^3\) has one critical point that is a local, but not global minimum. Must that surface have another critical point? While the answer in the 2-D version is "yes", in 3-D the answer is "no", as illustrated by examples in this paper. The authors further analyze for which surfaces the answer is "yes" in 3-D.</em></p>
</div></div></div>"The Only Critical Point in Town" Test
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-only-critical-point-in-town-test
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>If a function of one variable has a unique critical point, then it is not only a local max/min, but global. Does the same hold for functions of two variables? The authors provide a counterexample.</em></p>
</div></div></div>From Fences to Hyperboxes and Back Again
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/from-fences-to-hyperboxes-and-back-again
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Some popular optimal value and constraint problems, like fence-area problem and box-volume problem, are generalized to \(n\)-dimensions.</em></p>
</div></div></div>Rutgers Maple Help on Intermediate to Advanced Maple
https://maa.org/programs/faculty-and-departments/course-communities/rutgers-maple-help-on-intermediate-to-advanced-maple
Sage Worksheets for Multivariable Calculus
https://maa.org/programs/faculty-and-departments/course-communities/sage-worksheets-for-multivariable-calculus
CalcPlot3D, an Exploration Environment for Multivariable Calculus
https://maa.org/programs/faculty-and-departments/course-communities/calcplot3d-an-exploration-environment-for-multivariable-calculus