One of the main things students struggle with on exams is identifying the method to use on each problem. When they are learned in class, they are segregated by type, and their distinguishing features are not always highlighted. Here’s a brief cheat sheet on related rates and optimization.
| optimization | related rates | |
|---|---|---|
| sample keywords | maximize, minimize, most, smallest | how fast, rate of change/increase/decrease |
| sample problem | What dimensions maximize the area enclosed in a rectangular fence built with 100 feet of fencing? | A shape is deforming such that it is always a rectangle of perimeter 100 ft, but the width is increasing by 5 ft/min. How fast is the area changing when the width is 20 feet? |
| goal | maximize area | find rate of change of area |
| answer form | dimensions in feet | rate in ft2/min |
| equation |  |
likewise,  |
| equation, unabridged |  (area viewed as function of width) |
![A(t) = 50w(t) - [w(t)]^2](https://s0.wp.com/latex.php?latex=A%28t%29+%3D+50w%28t%29+-+%5Bw%28t%29%5D%5E2&bg=ffffff&fg=000&s=0&c=20201002 "A(t) = 50w(t) - [w(t)]^2") (area and width viewed as related functions of time) |
| final answer | a 25′ square fence maximizes the area | the area is changing by 50 ft2/min |
R. Weber,
I am a retired EE ,77 yrs old, that has become interested in matrices, and
through these studies have encountered Computer Graphics using matrices.
QUESTIONS:
1. Can you suggest an inexpensive textbook that introduces Computer Graphics <used, less than $20), with a level that discusses homogeneous coordinates ?
2. Which free Internet net site might I use to learn Computer Graphics?
I especially want to learn about the mathematical justification for using homogeneous coordinates
Thank you,
Thanks for your comment and apologies for the lateness of my reply – I’ve been neglecting this blog. I’m afraid I don’t have any recommendations for you on either count, although my understanding of homogeneous coordinates is that the justification is more engineering than mathematics: they make the implementation computationally simpler because every movement is multiplication (and not only that, multiplication by a matrix with a large number of zeroes). Best of luck!