Day 6, Mat128: Calculus A

• I've graded your second quiz. A few comments:
  • The class is about evenly divided between those who clearly know what they're doing, and those who clearly do not. We want everyone to move from the latter category to the former!

    There were struggles with both

    1. limits, and
    2. straight lines.

  • I chose examples you might have seen: one from our worksheet, and one from the section 1.2 exercises (exercise 1). The message I hoped to convey is this: that

    if you do your reading and homework it will help you to do well in this class!:)

    The solutions to all the exercises in the text are available. In addition, they're interactive.

    You might consider trying some of the exercises at the end of a section, even if they're not assigned. It will help you to self-evaluate your understanding of the material.

  • Let's take a look at our key.

    I actually noticed a small error in one of the problems (that I missed); can you find it?

• From now on, Fridays, 2:00-3:15, I will be available to do exercises. We'll start in the atrium outside of my office (find me in either MEP443, or MEP441); if we get too busy, we'll move to a classroom (this one may be available).

1. One-sided limits
2. The Derivative at a Point

In particular, last time I asked you to try two of the exercises from Section 1.3: Derivatives at a point:

• #7 (more freedom!)
• #8 (we might say that we're at \(t=30.08\) years past the start of 1993)

• We extend the limit definition of the derivative beyond single points, leading us to this important definition, that of the derivative function:

  Definition 1.4.2. Let $f$ be a function and $x$ a value in the function's domain. We define the derivative of f, a new function called $f'$ by the formula \[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \] provided this limit exists.

• Let's take a look at a couple of useful animations, referenced in section 1.4:
  • Graphing the derivative function : what do you notice about the shapes of the two curves?

  • The Derivative as a Function: notice that this one gives rise to a discontinuity, in the "slope function" -- that is, the derivative function!

• Now let's have a look at the preview.

• I really like this exercise, referenced in the text. Fun and challenging!

• Next time we'll be working on some worksheets: I'm going to edit this a bit for clarity, but if you want to get started, you can give this a try:
  • (Word)
  • (pdf)