- For today, you were to read Section 1.2 in
our text, and carry out Preview Activity 1.2.1.
- For next time: Let's keep rolling: read Section
1.3: The derivative of a function at a point, and carry out the
preview exercise.
Also carry out the following two exercises: Section 1.1, #8; Section 1.2, #7.
These will test your comprehension of important elements of these sections.
- At the end of the hour we'll have our first quiz, over the
material from last week.
- Roll
Last time we were looking at Section 1.1: How do we measure velocity?
- The key concept is average velocity; the interesting notion is instantaneous velocity!
The key to that "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 a change in $y$ over a change in $x$:
\[ AV = \frac{\textrm{change in y}}{\textrm{change in x}} \]
Now what happens as that "change in $x$" approaches 0? (that's the "limit" part!)
If all is well in the world, then the ratio becomes indeterminate because the "change in $y$" also approaches 0: $\frac{0}{0}$ is what makes this "indeterminate".
- Are there questions about your
worksheet for the section?
One thing we haven't done yet is see how Desmos can help us investigate secant lines
-
Let's start with that preview activity.
-
Key ideas are limits, from the left and from the
right, and then a definition of instantaneous
velocity. There's an important definition of limit in
this section, as well (Definition
1.2.2). Let's take a look at that first:
\[
\begin{equation*}
\lim_{x \to a} f(x) = L
\end{equation*}
\]
And because we want to talk about what's going on from the left of \(a\) and what's going on to the right of \(a\), we considered the so-called "one-sided limits": from the left:
\[ \begin{equation*} \lim_{x \to a^-} f(x) = L \end{equation*} \]
and from the right:
\[ \begin{equation*} \lim_{x \to a^+} f(x) = L \end{equation*} \]
So in this graph,
we can say that \[ \begin{equation*} \lim_{x \to 1^-} g(x) = 3 \end{equation*} \]
and from the right:
\[ \begin{equation*} \lim_{x \to 1^+} g(x) = 2 \end{equation*} \]
- Then we'll revisit the average velocity/instantaneous
velocity from last time:
\[ \begin{equation*} IV_{t=a} = \lim_{b \to a} AV_{[a,b]} = \lim_{b \to a} \frac{s(b)-s(a)}{b-a}\text{.} \end{equation*} \]
or, equivalently,
\[ \begin{equation*} IV_{t=a} = \lim_{h \to 0} AV_{[a,a+h]} = \lim_{h \to 0} \frac{s(a+h)-s(a)}{h}\text{.} \end{equation*} \]
- Let's recall how this definition works in the case of that
function we looked at last time, and the instantaneous velocity
at the point $a=2$:
\[
s(t)=64-16(t-1)^2
\]
Then we'll have a look at Exercise 1.2.4
- One of the interesting ideas is that of a discontinuity
in a function: not all functions are smooth, or even connected!
What that means is that the definition above may not make sense
in some cases. What can go wrong? Three things:
We interpret the derivative as the slope of the tangent line (the limit of secant lines!) at a point. You can frequently look at a graph and see where things go awry. For example, if there's no tangent line at a point, then there's no instantaneous velocity!
- Because there are some musicians in here, I
thought that we'd listen to the sound
of a function.... In particular, I want you to listen for
the "sounds of discontinuity".
- 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.