Day 9, Mat128: Calculus A

• Our test Friday will cover everything through section 1.3.

• We had a quiz over section 1.2: I had posted solutions to the homework problems Thursday evening, but it didn't seem that those percolated down through the quizzes....
  • The second problem was right off the homework, and Alykaa asked a question about it right before the quiz. "Any more questions?", I asked: but, hearing none, moved on. But clearly there were some questions, because that problem didn't go very well.

    You were to create the graph of a single function (some of you violated the VLT -- vertical line test -- by drawing a set of graphs of functions -- perhaps each taking care of one of the conditions; but we want one function which meets all the conditions).

  • The first problem was not right off the homework, but it was based off a couple of problems from the homework: the idea being that an expression, like \[ f(x)=\frac{x^2-4}{x-2} \] can be re-expressed upon factoring as \[ f(x)=\frac{(x-2)(x+2)}{x-2} \] but only reduced to \[ f(x)=x+2 \] so long as we retain the domain restriction (we can't put \(x=2\) into the original formula for \(f\)). So we can write \[ f(x)= x+2 \] on \(\Re - \{2\}\), and there's a hole in the graph at \(x=2\) -- but otherwise it's a straight line, a linear function, the nicest sweetest graph in the whole world. You only need two points to draw it perfectly (while recalling that you need to put a hole in the graph at \(x=2\)).

    Another way to express \(f\) is as a piece-wise defined function:

    \[ f(x) = \left\{\begin{array}{cc}{x+2}&{x \in \Re - \{2\}}\cr{undefined}&{x=2}\end{array}\right. \]

  • My biggest concern was on problem 1, as there was some wild "algebra" going on. The biggest problem I see at this point is that some are cancelling terms in a quotient, rather than factors: \[ \frac{ab}{a} = b \] (subject to the domain issues above), but \[ \frac{a+b}{a} \ne b \] in general.

    One student was puzzled by my asking for a graph "in the vicinity" of -2. The student wanted me to nail that down, but I didn't want to! In reality, I was hoping that folks would draw the graph of a linear function with a hole in it at \(x=2\)! That's the only really interesting point on the graph!:) But I took any plot, so long as it was correct "in the vicinity" of -2.

    And since it's a straight-line graph, you just needed to calculate it at two points, and connect them....

    A lot of problems in life are like that: we call them "under-determined" in mathematics. It's for you to figure out how to determine them! And it's related to the problem of the "indeterminate form" which is the derivative....

  • Here's a key for the quiz.

  • The median score was an 8.

• We reviewed the notion of derivative at a point, which is just an example of a limit.

  The most important definition in calculus is the definition of the derivative (here is the derivative of $f$ at $a$): \[ f'(a)=\lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}} \]

• Then we had a look at the a worksheet for Section 1.3, noting the focus on
  • tangent lines (and their equations)
  • slopes (and their estimates)
  • algebraic resolution of the indeterminacy of the limit definition of the derivative.
  Let's revisit that worksheet and do problem #4 in detail. That will lead us into our topic for today: the derivative function.

We want to make the 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 look at the preview, which extends problem 4 from the previous worksheet to lead us into the derivative function.

• Our author references several useful animations in section 1.4:
  • 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!

• With whatever time we have left, and next time we'll be working on a worksheet. Give this a try: handout for Section 1.4.

• 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.