Day 10, Mat128: Calculus A

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Today:

Announcements
- Friday help sessions:
  - Two weeks ago I had two students show up
  - One weeks ago I had one students show up
  - Q: How many will show up this week? (What does the data suggest?)
- The most important definition in calculus is the definition of the derivative (but it comes in various flavors):
  
  Here is the forward derivative of $f$ at $a$:
  
  $f'(a)=\displaystyle{\lim_{h\to 0}}{\frac{f(a+h)-f(a)}{h}}$
  Here is the forward derivative of $f$ at $x$:
  
  $f'(x)=\displaystyle{\lim_{h\to 0}}{\frac{f(x+h)-f(x)}{h}}$
  There are still those who have not memorized these definitions. You need to.
Today:
- We'll review some parts of the quiz, featuring some highlights from your papers.
  - I'd like Zach H. and Radip to give us their solutions to #2
  - Chris will lead us on #1; Chris was as parsimonious in can be when it comes to "Explain"ing: in general you should give full sentences, explaining exactly what you see and why what you see excludes a particular graph.
- We'll continue talking about the second derivative (section 1.6).
- If the derivative is good at showing where a function is increasing or decreasing (see the formal definition in section 1.6), the second derivative is important for showing the inflection of a function (its concavity).
- Notice also that the second derivative can be thought of as the derivative of the derivative: \[ f''(x)=\lim_{h\to 0}{\frac{f'(x+h)-f'(x)}{h}} \] Hence, all your table work (forward, backward, centered difference formulas) still applies.
- The two most important examples, to illustrate the role of the second derivative, are these poster children for concavity:
  - $f(x)=x^2$: $f'(x)=2x$, and $f''(x)=2$: $f''(x)>0 \longrightarrow$ concave up:
  - $f(x)=-x^2$: $f'(x)=-2x$, and $f''(x)=-2$: $f''(x)<0 \longrightarrow$ concave down:
  Activity 1.6.2 allows us to explore your preview exercise more deeply.
- Let's start in on your Section 1.6 worksheet. Remember that the function of problem 4 -- $f(x)=x^3$ -- is our poster child for inflection point (where the concavity changes smoothly from concave up to down, or vice versa):

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