Day 31, Mat128: Calculus A

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Today:

Announcements
- Your related rates homework is returned: graded 19, 28, 40 (actually I took your "best out of two" for a grade).
  Half of you are making it easy on me, by not submitting anything. Failing to practice related rates problems is not good strategy, however.
  There are good opportunities for help, especially including the calc lab. Please consider getting help, and submitting your homework -- rather than skipping the hard problems, and hoping for the best....
Today we begin talking about section 3.1: Maximum and Minimum Values.
- Example application of extrema:
  - The Chickens of Manitoba
  - How was the flock egg production over time?
  - How long should we let the flock live to maximize the average daily return?
- Maxima and minima are both types of extrema. Our author begins by distinguishing between global (or absolute) extrema and local extrema:
  - Local is the biggest thing in the neighborhood;
  - Global is the biggest thing anywhere!
- Take a moment and sketch a continuous function on the interval [-2,2]. Make it look kind of interesting -- maybe even ugly.
  1. Does your function have a global maximum and a global minimum?
  2. Does your function have local extrema which are not global extrema?
  3. If your function does have extrema, how would you characterize the derivative of the function at those extrema?
- By the way, our author requires that the function be defined in an open interval about a local extremum. So in this figure from p. 199, the right endpoint is not considered a local maximum:
- Now I want you to sketch a function defined on the interval [-2,2] that does not have any global extrema. What is true about your function?
- Now I want you to sketch a smooth function on [-2,2] that has a value of x=a where $f'(a)=0$ but no extremum there. What is true about your function?
By now we should have a good idea of what these two theorems say:
- Extreme Value Theorem
- Fermat's Theorem
We have hopefully deduced these. We might also have deduced the closed interval method, to find the extrema of a function on a closed interval [a,b]:
1. Find the critical numbers for f (that is, those values of x at which the derivative is zero or doesn't exist).
2. Find the values of f at the critical numbers of f in (a,b).
3. Find the values of f at the endpoints of the interval (at a and b).
4. The largest and smallest of the values from steps 1 and 2 are the absolute or global extrema.
Now let's look at some standard problems:
- #8, p. 204
- #22
- #31, p. 205 (and let's decide whether they're local maxes or mins)
- #45 (closed interval method)
- #63

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