Day 29, Mat128: Calculus A

Last Time

Next Time

Announcements:
- If you need to Zoom to class, here's the link. Old Zooms are here.
- Your exams are returned today. Again, I averages the two (as percentages), and then added five percentage points.
  - Test 3 key
  - Mean and median: about 84
  - A lot of you are still missing the point on #4: L'Hopital's rule only works on indeterminate forms. If the limit is determinate, you just calculate the limit; it's only if the limit is indeterminate that you use L'Hopital's.
  - On #3, I asked you "explicitly" to use "implicit" differentiation: you do not solve for y as a "function of x" (because it's not!). It's a boatload easier if you can do these implicitly.
  - I wish that I could convince everyone to use point-slope form for your lines (especially tangent lines): generally, \[ y = y_0 + m(x-x_0) \] or, in the specific case of the graph of a function $f$ with derivative $f'$, \[ y = f(x_0) + f'(x_0)(x-x_0) \]
    A tangent line is a short story about a point and a slope.
    I've got a few of you on-board, thankfully; why not all?
- Your final:
  1. Will be Wednesday, 5/11, 2:30 pm.
  2. It will be a two-hour, cumulative exam.
  3. It will be half new and half old material.
  4. You will be allowed a page of notes, front and back.
  5. You might check the daily agendas for the reviews of our previous exam material. Certainly look over the old exams, and keys.
  6. New material (covered since the last exam): Optimization and Related Rates
Last time:
- We worked a few more examples of related rates problems, and an optimization problem.
- Then I illustrated Newton's method, which illustrates how we use tangent lines to find roots of functions (e.g. zeros of derivative functions); I used a Mathematica demonstration of Newton's method, illustrating how one might find the root of a derivative function.
Today:
- We review; and we'll start with questions that you have.
- Let me remind you of the most important definition in calculus, and that it is built on an indeterminate form.
  
  $f'(a)=\lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}}$
  
  This animal measures the instantaneous rate of change of the function $f$; and it turns out that it's really useful for finding extrema of functions (places where the rate of change is zero!), in particular.
- And while the limit definition is the thing on which all of calculus is based, the good news is that there are lots of rules that make our lives easier when it comes to calculating derivatives, of all of our favorite functions.
- What happens if you use L'Hopital's rule on the most important definition in calculus?
  
  $f'(a)=\lim_{x\to a}{\frac{f(x)-f(a)}{x-a}}$
- I've prepared a "sample final"; it will give you some idea of the kinds of problems you might encounter.

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