Day 34, Mat128: Calculus A

• Math Major/Minor lunch today -- run up and grab something if you're hungry!

  When: Friday 11/06/15, 11:30-1:30pm
  Where: MP 4th Floor Center Atrium
  

One of the main purposes of calculus is to help us to locate extrema -- maxes and mins -- places where a function takes its largest or smallest values.

• Our author begins by distinguishing between global extrema and local extrema:
  • global (absolute) maximum at $x=c$: \[ f(c) \ge f(x) \hspace{1in} \forall x \in D \]
  • local maximum at $x=c$: \[ f(c) \ge f(x) \hspace{1in} \forall x \in (c-\epsilon,c+\epsilon) \] where $\epsilon>0$.

    Notice that this says that a high or low point on the boundary won't be considered a local extremum by our author -- although it can be global!

    Minima are similarly defined, of course.

• Last time we explored two theorems through some of our own examples:
  • Extreme Value Theorem: If $f$ is continuous on a closed interval $[a,b]$, then $f$ attains a global maximum and global minimum on $[a,b]$.

    Notice that the interval is finite.

  • Fermat's Theorem: If $f$ has a local max or min at $c$ and $f^\prime(c)$ exists, then $f^\prime(c)=0$.

• closed interval method: to find the extrema of a function on a closed interval [a,b], do the following:
  1. Find the critical numbers for f (x values 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 example problems:
  • #8, p. 204
  • #22
  • #27
  • #31 (and let's decide whether they're local maxes or mins)
  • #45
  • #63