Day 15, Mat128: Calculus A

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Today:

Announcements
- Your quiz is returned.
  - Note: $x$ values are in the domain; $f(x)$ values are in the range. My favorite function is a soda machine. The difference between domain -- coins, money -- and the range -- soda -- is clear. It's not so clear in many of our examples, because the domain and range are both sets of real numbers.
  - "Differentiability implies continuity, but continuity does not imply differentiability." Continuous doesn't necessarily mean smooth, which is essential for the existence of a tangent line.
  - All these functions are continuous on their domains.
- You have a homework due.
We'll start with a few more problems from 2.1: last time we did #3; let's do 11 and 13:
- #3, pp. 110
- #11, pp. 111
- #13, pp. 111
Today we begin our discussion of Section 2.2: The derivative as a function

Let's use the definition of the derivative function,
\[ f^\prime(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]
to determine the derivative functions of these important functions:
1. Constant functions: $f(x)=c$
2. Linear functions: $f(x)=mx+b$
3. Quadratic functions: $f(x)=ax^2+bx+c$
4. An important application of the tangent line
The first two derivatives are easy: each function has a straight line graph, so the tangent line is the graph of the function itself. The slope of each of those graphs is easy to see (either 0 or m). The derivatives are hence on full display. Not so much for the quadratic, and it's the first function that has an explicit dependence on $x$.
How well can you recognize derivatives and the functions that give rise to them? Let's try #3, p. 122.
If a function's derivative is another function, does that function have a derivative?
The derivative of the derivative of a function is called the second derivative of the function. And how do we interpret these "higher derivatives" in the context of a motion?
Let's think of a quadratic motion, e.g. the motion of an eraser thrown across the room. Let s be the height of the eraser:
\[s(t)=at^2+bt+c\]
Each of the coefficients has an important, intuitive role to play:
- $c$ is the initial height, at time $t=0$;
- $b$ is the initial velocity, rate of change in height at time $t=0$;
- $2a$ is the acceleration (due to gravity) of the height (the derivative of the velocity is acceleration).
  - It is constant across this motion, if we neglect other forces (e.g. air resistance), or the fact that gravity is actually a function of the distance from the center of the Earth.
  - Forces result in accelerations, per Newton's equation $f=ma$.
Of course we could continue on and define higher derivatives (third, fourth, etc.). The notation for the $n^{th}$ derivative is frequently $f^{(n)}(x)$.
Here are other notations for the derivative function:
\[ f^\prime(x) = \frac{dy}{dx} = \frac{df}{dx} = \frac{d}{dx}\left(f(x)\right)=\left(f(x)\right)'=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]
There are several different ways of writing the derivative, and you need to get used to them. The form $\frac{d}{dx}\left(f(x)\right)$ reminds us that the operation of differentiation is itself a function: it takes a function in its domain and returns another function -- the derivative.
Examples:
- #2, p. 122.
- #25
- #48, p. 125.
If we have time, we'll start into Section 2.3 by using the definition of the derivative to prove several of the formulas in this section. We've already proven the first two rules (constant functions $f(x)=c$ and the linear function $f(x)=x$).
- Rules that help us avoid having to use the definition each time (with Proofs):
  - The sum rule
    works as we'd hope: "The derivative of a sum is the sum of the derivatives."
  - The constant multiple rule (makes sense)
  - The product rule (has a tricky bit)
  - The power rule (via dominoes -- i.e., mathematical induction)
  - NOTE: at this point we have all the rules necessary to differentiate all polynomials, without needing to resort to the definition!
    The derivative of the monomial $a_nx^n$ is $na_nx^{n-1}$.
    And a polynomial is just a sum of these. So
    $s'(t)=(at^2+bt+c)'=2at+b$
    and
    $s''(t)=(2at+b)'=2a$
- Other rules:
  - The quotient rule!

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