Day 2, Mat128: Calculus A

Last Time

Next Time

Today:

Announcements
- You have a reading -- and work -- assignment for next time (Wednesday: remember no class on Monday):
  please read Section 1.2 in our text, and carry out Preview Activity 1.2.1 (bring your work to class).
- You will have a quiz next time (Wednesday), at the end of the hour, over what we covered last time and what we'll cover today.
- Roll
Questions?
Today:
- Getting on IMath: you'll need the website, the course ID, and the password. They are:
  - http://imath.nku.edu
  - ID: 526
  - Password: 3.1415926
  You'll have an assignment on IMath about once per week. This first one is just to get you used to the animal. There is no penalty for multiple attempts, so you should be able to get 100%.
  I'll be dropping a few of your lowest scores, just like I'll drop a couple of your lowest quiz scores.
  Keep an eye on the deadlines for IMath homework, but I'm pretty easy about extending them when necessary. The important thing is to try them, and practice.
- Let's start by reviewing what we discussed last time: "Linear Things"
- Then we'll hit Section 1.1: How do we measure velocity?
  - First impressions: what do you think of the text, and of this preview activity?
    Did you notice that you can "activate" the exercises, and so check your answers?
  - Of course we should look over the Preview activity.
    I'm going to have you partner up to compare your previews. Then we'll go over the "solution".
  - The key concept presented in section 1.1 is average velocity; but the really interesting notion is instantaneous velocity! That's where calculus comes in.
    The key to instantaneous velocity is something called an "indeterminate form", and limits. We'll get a look at that in the activity we do next, but it's hinted at by thinking about average velocity: as we noted last time, it's related to slopes, and a change in \(y\) over a change in \(x\) (which we can think of as "rise over run"):
    \[ AV = \frac{\textrm{change in y}}{\textrm{change in x}} = \frac{\Delta y}{\Delta x} \]
    Now what happens as that "change in \(x\)" approaches 0? (that's the "limit" part: \(\Delta x \longrightarrow 0\))
    If all is well in the world, then the ratio becomes indeterminate because the "change in \(y\)" also approaches 0! And everyone knows what \(\frac{0}{0}\) is, right?
    (Just kidding -- that's the "indeterminate" part!)
  - Let's have a go at one of the activities from the section (1.1.2), during which we'll try to make sense of these notions.
    We'll combine it with Example 1.1.2, however, to make the calculations easier. We're computing AV over and over again, so let's use the expression for \(s\) (and Algebra!:) to simplify things (algebra is actually supposed to help you make your life easier):
    \[ AV = \frac{\textrm{change in y}}{\textrm{change in x}} = \frac{s(0.8+h)-s(0.8)}{(0.8+h)-0.8} = \frac{\hspace{3in}}{h} \] Now what?
  - Then it's on to today's group activity, which involves solidifying your understanding of these issues (including units). Your worksheet for today.
    If you don't finish this in class, then finishing it becomes part of your homework!
Links:
- Desmos page for today's worksheet

Website maintained by Andy Long. Comments appreciated.