Day 19, Mat128: Calculus A

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Announcements
- Your quiz Friday will cover section1.8: the tangent line approximation.
- You have a new assignment, corresponding to Section 2.3.
- These two sections will be the last two covered on our next exam, which is coming next Friday 3/1 (just before Spring break).
- We're not skipping 2.2 -- we'll come back to it.
Last time:
- We plotted the data for another potato problem, created the local linear approximation (i.e. estimated a derivative at the point closest ($t=60$) to the time of interest ($t=63$) minutes), and then used the line to estimate the temperature at that time.
- We can use concavity to determine if our approximation is a likely an over- or under-estimate.
Any questions on local linearization?
Today we begin a new chapter.
- We're going to plow through a bunch of "proofs" featured in sections section 2.1 (and one rule from section 2.3: the product rule); the objective is to create a bunch of useful new tools out of our old (but wonderful!) tool, the limit definition of the derivative: \[ F'(x)=\lim_{h \rightarrow 0}\frac{F(x+h)-F(x)}{h} \] Notice the switch to an uppercase $F$: we'll be talking about sums of other functions, etc., and we often want to use $f$ as the name of one of those.
  Here we go! For the following proofs, assume that functions $f$ and $g$ are differentiable at $x$.
  - Theorem: Let $F(x)=c$ where $c$ is a constant. Then $F'(x)=0$.
    This one is obvious, "by slopes".
  - Theorem: Let $F(x)=x$. Then $F'(x)=1$.
    This one is also obvious, "by slopes".
  - Theorem (sum rule -- "The derivative of a sum is the sum of the derivatives." It rolls off the tongue....):
    Let $F(x)=f(x)+g(x)$. \[ \frac{d}{dx}[F(x)] = \frac{d}{dx}[f(x)+g(x)] = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) = f'(x) + g'(x) \]
  - Theorem (product rule -- not simply the product of the derivatives! No tongue-rolling, please!):
    Let $F(x)=f(x)\cdot g(x)$. \[ \frac{d}{dx}[F(x)]=\frac{d}{dx}[f(x)\cdot g(x)]=f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x) =f(x)g'(x) + g(x)f'(x) \]
  - Corollary (constant multiple rule): multiplying a function by a constant scales it in the $y$ direction. This makes it steeper by a factor of $c$.
    Let $F(x)=cf(x)$. \[ \frac{d}{dx}[F(x)]=\frac{d}{dx}[cf(x)]=0*f(x)+c\frac{d}{dx}f(x)=cf'(x) \] This is a consequence of the product rule and the constant rule (which is why we call it a corollary).
  - Corollary (difference rule -- "The derivative of a difference is the difference of the derivatives." Tongue-roller....):
    Let $F(x)=f(x)-g(x)$. \[ \frac{d}{dx}[F(x)] = \frac{d}{dx}[f(x)+(-g(x))] = \frac{d}{dx}f(x) + \frac{d}{dx}(-g(x)) = \frac{d}{dx}f(x) + (-1)\frac{d}{dx}g(x) = f'(x) - g'(x) \]
    This is a consequence of the sum rule and constant multiple rules.
    Once we've proven a rule, we can put it to use!
  - Theorem (Power rule): This one was the star of your preview exercise. Hopefully you saw the pattern!
    Sadly this is about all that some people retain from calculus. If you're going to retain something, retain the limit definition of the derivative, because the power rule is proven based off the limit definition (and mathematical induction). Or remember the linear approximation, using tangent lines; but this, on it's own, is not nearly as swell!
    Let $F(x)=x^n$, where $n$ is an integer, $n \ge 1$. \[ \frac{d}{dx}[F(x)]=\frac{d}{dx}(x^n)=nx^{n-1} \]
    This proof will also make use of one of our new tools (the product rule). Once you build and prove a tool, it becomes a power tool:).
    The proof is actually "by dominoes" (or rather, the principle of mathematical induction....):
    1. If the first domino falls, and
    2. If the $(k)^{th}$ domino falling implies that the $(k+1)^{th}$ domino falls,
    then all of the dominoes are falling down.
    A more mathematical version of induction is as follows. Consider a proposition $P(n)$ (In this case we might think of the proposition being this:
    \[ P(n): \frac{d}{dx}\left(x^n\right) = nx^{n-1} \] Then
    1. if $P(1)$ is true, and
    2. if $P(k) \implies P(k+1)$ for $k \ge 1$,
    then $P(n)$ is true $\forall n \in \mathbb{N}$.
    The power rule works for any real exponent (except 0) -- it's just that we've only proven it for positive integers. You may use it for other powers (so write $\sqrt{x}=x^\frac{1}{2}$, for example, to make use of this nice new rule!).
    So here's the big news: we've got all the power we need to differentiate (easily) any polynomial -- and that's a huge and important group of functions!
    If we need the derivative of \[ f(x)=4x^3-7x^2+5x-1 \]
    it's easy now!
  - Theorem (exponential rule):
    Let $F(x)=a^x$, with $a>0$: \[ \frac{d}{dx}[F(x)] = \frac{d}{dx}[a^x] = \ln(a) a^x = \ln(a) F(x) \]
    This says that an exponential function has a slope function which is just a constant times the function itself (and the constant is the log, base $e$, of $a$).
    In particular, let $F(x)=e^x$, where $e \approx 2.71828$ (it's irrational, like $\pi$). Then \[ \frac{d}{dx}[F(x)] = F(x) \hspace{1in} (= F'(x) = F''(x) = F'''(x) = ....) \]
    This is certainly one of the most amazing rules. It says that $e^x$ is an exponential function which is its own derivative: whose values are its slopes, as well (and all its higher derivatives as well -- its concavity, its jerk, ...).
    Here's an animation which motivates the discussion...
  Next time we'll add rules for derivatives of the trigonometric functions, and rational (quotient) functions.
- For today's worksheet, we'll work with polynomials and powers (negative, and other than simple integers): Sections 2.1 and 2.3 worksheet.

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