Day 12 in MAT229

Today:

• Announcements:
  • We have an exam tomorrow. It will cover everything from chapter 6.

  • I may not have it graded by Friday -- just a head's up.

  • I'm returning your homework 2 set. A couple of things:
    1. Problem 1:
       • Best to leave tangent lines in the form \[ l(x)=f'(x_0)(x-x_0)+f(x_0) \] where your job is to compute the derivative and evaluate the function and the derivative at $x_0$.

       • It's not enough to just declare that a critical number represents a minimum: you must show me, using
         • The first derivative test
         • The second derivative test
         • asymptotics! (Will)
         A graph does not suffice, because without fully understanding the derivative you couldn't graph the function -- so you're relying on your calculator. We want to rely on your brain.

    2. Problem 2:
       • Don't be a "dx dropper" on integrals. An integral must have a differential in the integrand, or it's abstract nonsense.

       • Check your answer to see if it makes sense. You have a picture of the area -- does the decimal value of your integral make sense?

       • Issues with the volume: you do this by "washers" (or by subtraction of a smaller volume from a bigger volume).

         Incorrect: \[ \pi\int_{1}^{4}\left(\frac{1}{\sqrt{x}}-\frac{1}{x}\right)^2 dx \] Correct: \[ \pi\int_{1}^{4}\left(\left(\frac{1}{\sqrt{x}}\right)^2-\left(\frac{1}{x}\right)^2\right) dx \]

         You can roughly approximate your volume -- does it make sense?

  • From the worksheet on Friday: (I'm going to hold off on these, since they're not on the exam -- more to come on Thursday.)
    • But I did notice lots of dx dropping.
    • Some confusion between anti-differentiation and differentiation.

• Today we will focus on the Review for exam 1 (pdf summary)
  • By the way, never forget the most important definition in calculus. It can always appear, on any exam.

  • Ground rules:
    1. No notes or books or cell phones are allowed. A scientific calculator is fine (that can do logs, exponentials, sines, sqrts, etc.).
    2. You have the full period to work on it. When you are finished, you can give me your exam and leave.

  • I'll start with a few questions that students have asked:
    • Caitlin asked about other arc functions. I have never used arcsec and arccsc; I have used arccot. But I generally think that you only define functions that are particularly useful.

      I told Caitlin this: "My objective is to teach you how to construct inverses -- and show some useful examples. But we could spend the rest of the semester doing this for more and more functions, and I'm not thinking that that would be a useful way to spend our time. So maybe we've constructed enough inverses, and we should move on to other things. But I hope that you understand how we constructed the inverses we did, and could make a good attempt to construct new ones, if need be."

    • Someone else asked what the test would look like, and I said it will look like your homework, the things we do in class, and the IMath. No big surprises.

  • I'll start with any questions you may have....