Day 10, Mat128: Calculus A

• Our test Friday will cover everything through section 1.3.

  It will be kind of like two quizzes: half on 1.3, and the rest will cover 1.1, 1.2, and straight line equations.

  Again, you can use your phone as a calculator, but ONLY as a calculator.

• I've given you a new assignment: to become a black belt on one of the activites suggested by our author.
• Here's a key for your 1.3 homework.

• We went over the quiz.
• We had a look at one of the problems on the a worksheet for Section 1.3: we used the limit definition to compute the derivative of a quadratic \[ f(x)=x^2-x \] at a point, \(x=2\).
• Then we started on section 1.4 by observing that we could forget about which particular point we were computing at, and just do it for an arbitrary point \(x=a\). Once we realized that, that we could compute \(f'(a)\) for \(f(x)\), getting \[ f'(a)=2a-1 \] we actually have a function. So, for all real numbers \(x\), \[ f'(x)=2x-1 \] And we noted that the derivative of a quadratic (polynomial of degree 2) is a linear polynomial (of degree 1).

  This is a general rule: the derivative of a polynomial of degree \(n\) is a polynomial of degree \(n-1\) for all \(n\in\mathbb{N}\).

• We checked out a few of the animations that our author references in this section (and you should try them yourself, and understand them!).

We transition from the derivative value at a point, to the derivative function over an interval, or domain.

• 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.

• Now let's have a glance at the preview, which extends problem 4 from the previous worksheet to see the upshot.

• Our author references several useful animations in section 1.4, and we had a look at a couple of them last time. There's one we didn't check out, and I want to have a look at that one:
  • Graphing the derivative function: what do you notice about the shapes of the two curves?

    Graphing the derivative function

    On the left is shown the graph of $f(x)$ while the right shows the graph of the derivative $f'(x)$.
    

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

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

• Let's get into our worksheet for Section 1.4. I've got a doctored up version which we'll start with.

• Desmos page for secant lines.
• My freshman physics professor at Miami University, Prof. Don C. Kelly wrote The Little Book of Calculus. Self-Help Guide to Calculus. It's a neat "little" (50 some pages) resource produced by him and his kids/grandkids. Check out page 12, limits, in particular.