Day 23, Mat128: Calculus A

• If you need to Zoom to class, here's the link. Old Zooms are here.

• Next week, Wednesday, we'll have our third exam.

  By Monday I'll provide you a sample exam, to give you a sense of what it will look like. We'll also start with any questions you may have.

  The exam will cover the material from the Chain Rule on, but, as you no doubt are aware, earlier material will appear because calculus builds on itself pretty much.

  And you can always expect a problem involving the most important definition in calculus!

We talked about how to identify extrema, both

• local
• global

1. The first derivative test: Let $c$ be a critical number of continuous function $f$.
   • If $f^\prime$ changes sign from positive to negative at $c$, then $f$ has a local maximum at $c$.
   • If $f^\prime$ changes sign from negative to positive at $c$, then $f$ has a local minimum at $c$.
   • If $f^\prime$ does not change sign at $c$, then $f$ has neither a max nor a min at $c$.

2. Second derivative test: Suppose $f^{\prime\prime}$ is continuous near $c$.
   • If $f^{\prime}(c)=0$ and $f^{\prime\prime}(c)>0$, then $f$ has a local minimum at $c$;
   • If $f^{\prime}(c)=0$ and $f^{\prime\prime}(c)<0$, then $f$ has a local maximum at $c$.

3. Check the endpoints, as well!

I want to begin today by continuing with the example we were working on last time, which came from your worksheets for 3.1 (get that out if you've got it handy -- I have some extra copies otherwise).

Let's finish our table, and then take a look at the functions, to see how we did.

• graph of $h$
• graph of $h'$

• We'll start right in with the Preview. There are a number of important lessons that it contains:
  • All rational functions look like polynomials eventually, far enough away from zero.
  • We don't really need calculus to answer these questions -- we can use our understanding of functional forms (in particular, quadratics).
  • Symmetry!

• There's one theorem in here that we need to think about, called the Extreme Value Theorem. The basic idea is that, if the function is continuous on a closed and bounded domain, then it must have a global maximum and a global minimum. Is that obvious to you?

• I notice especially in this section that the authors uses different vocabulary from time to time. "Global" is the same as an "Absolute" extremum; "local" is the same as a "relative" extremum.

  That's not all bad, because you need to be aware that people use different words for the same concept -- but it might be confusing at first!

  This is similar to the different notations for the derivative.

• Then it's on to some activities whose solutions truly involve calculus:
  • Example 3.3.4, and
  • Activity 3.3.4

• Finally, I'll turn you loose on your worksheet for 3.3.

• A Chain Rule tutorial
• Our free, on-line textbook, called Active Calculus.