Day 4, Mat128: Calculus A

Last time:

Next time:

Today:

Announcements
- Roll
- I hope that you had a good MLK, Jr. Day.
- You have an assignment for next time: please prepare!
- A reminder that you have a quiz Friday, covering straight lines equations, and section 1.1: rate of change.
Questions?
Review from last time:
- We talked about average and instantaneous rates of change. The average velocity is the slope of a secant line through the position curve, at two distinct points: \[ AV_{[a,b]} = \frac{s(b)-s(a)}{b-a}\text{.} \]
- If we let the two points creep closer and closer to a single point of interest, with shorter and shorter time intervals, then the average velocities may approach some "limiting" value (which is quite likely the instantaneous velocity) as the time interval heads for zero: \[ AV_{[a,a+h]} = \frac{s(a+h)-s(a)}{h}\text{.} \] (that is, as \(h \to 0\)).
  Last time I attempted this with a specific value of \(a\), and a specific function \(s\), which was the subject of the notes from last time that I put on-line: using algebra to remove indeterminacy in AV as we seek the instantaneous velocity.
  We'll review that in a minute.
- We see that the slope equation leads to an indeterminate form as the two points coalesce (as \(h \to 0\)), but that algebra can help turn the indeterminate form into a determinate one, allowing us to determine the instantaneous velocity if we're lucky.
- The slope \(m\) of a tangent line to a curve, which osculates ("kisses") the curve will generally be given by what we will call the derivative of \(f(x)=y\) at \(x_0\): \[ m = f'(x_0) \] This we can think of as the instantaneous rate of change of the function \(f\) at that point. If you've read ahead into Section 1.2,
  then you know that we will write this formally as \[ \lim_{h \to 0}AV_{[a,a+h]} = m\text{.} \]
Section 1.1: How do we measure velocity?
- How is the homework going for Section 1.1 (or how has it gone)? Do you have any questions about that?
- Let's review the problem we left off with last time: I sent out a summary, where I did it in a little more detail.
- Then it's on to today's group activity, which involves solidifying your understanding of these issues (including units). Your worksheet for today.
  If we don't finish this in class, then finishing it becomes part of your homework, and we'll review next time, before our quiz.
Links:
- Notes from last time: using algebra to remove indeterminacy in AV as we seek the instantaneous velocity.
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