Day 24, Mat128: Calculus A

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!

In particular I want to emphasize one of those most important things we saw last time: that you can do some calculus without taking a derivative! Because we understand the structures of -- or classes of -- some functions (e.g. quadratics), we can solve problems based on that prior knowledge.

Knowledge should amount to "saved work", don't you think?

• A problem arose in which you needed to find the zeros of the second derivative function \[ y''(x)=e^{\sin(x)}(\cos(x)^2-\sin(x)) \]
• How can we find the zeros?

• Here's what a very powerful software program gave for the graph of one of your functions: \[ h(x)=\frac{\sqrt[3]{x-1}}{x+2} \]
  

• How can we know the truth? Calculus! Algebra! Know your functions....

• Section 2.8
• Curve Plotting
• Section 3.1
• Other keys were provided on the day21 page.

1. Section 2.5: The chain rule
   • $\left(f(g(x))\right)'=f'(g(x))g'(x)$
   • The hard part is often figuring out the composition.
   • I like to think "f of stuff prime" is "f prime of stuff, times the derivative of the stuff".

2. Section 2.6: Derivatives of Inverse Functions
   • $\ln(x)$ (perhaps most importantly)
   • Inverse trig functions
   • But any function, which is invertible. We know how to do it, and we derived the formula from implicit differentiation and the chain rule: \[ \left(f^{-1}(x)\right)' = \frac{1}{f'(f^{-1}(x)} \]
   • As we saw in the case of the inverse trig functions (the "arc" functions), we sometimes needed a trick (e.g. Pythagorean theorem) to express the result in simpler terms.

3. Section 2.7: Derivatives of functions given implicitly
   • Especially useful for curves traced out by several functions -- e.g. circles -- curves which fail the VLT.

4. Section 2.8: Using Derivatives to Evaluate Limits
   • L'Hôpital's rule

5. "Curve Sketching" (building a table, incorporating information on $f$, $f'$, $f''$, in order to graph $f$).
   • Domain of $f$ (and $f'$ and $f''$, if different)
   • Symmetry, periodicity, classes of functions
   • zeros of $f$, $f'$, $f''$
   • signs of $f$, $f'$, $f''$
   • Asymptotes -- vertical and horizontal
   • special function values

6. Section 3.1: Using derivatives to identify extreme values
   1. first derivative test
   2. second derivative test

1. One example straight from the sections listed above (to encourage you to revisit -- if you have already visited! -- the text); and
2. One problem straight from a worksheet (to encourage you to revisit those).