Monday, November 7, 2011

Optimization Problems

Several Students have wondered how to get started on some of the later optimization problems.  Here are some pictures to help you see what's going on.  If you need more hints, email me or comment.

#34) A Poster is to have an area of 180 square inches.  You want to find the maximum area of the printed (red) area with the margins they require.  So you have an equation relating Width and Height with a fixed area.  How can you figure out an equation for the area of the red rectangle?

#38)  This is a classic ladder leaning on a  building problem.  James Stewart loves a leaning ladder.  We know that the fence is 8 feet high and is 4 feet from the building.  This is the physical situation:

In this problem I think it helps to look at a geometric picture.  If we put a line in parallel to the ground that touches the top of the fence, we get two similar triangles.  They have the same angle theta (red) and we know the bottom length of the small one from the distance from the building to the fence and we know the left leg of the larger one from the height of the fence.  The length of the ladder will be the sum of the two hypotenuses.  Can you figure out a way to optimize that length?

The Definite Integral

This is a cool GeoGebra demonstration designed by Jason McCullough at the University of California Riverside.  This shows in action what is happening on page 363 of your text, the ideas we're using in chapter 5 to define the Definite Integral.  
He includes a good description of how to use it.  Think about how this relates to the definition of Area on page 365 and the definition of the Definite Integral in page 372.  As you move the lower slider to the right you are getting better and better approximations of area as more rectangles are added.  

The Definite Integral, Area and Riemann Sums

As always, comment or email me if you have any questions.