Day 21, Mat128: Calculus A

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Next time:

Today:

Announcements
- We're still working on sections 2.1 and 2.3: these will be the last two covered on our next exam, which is coming up Friday 3/1 (Spring break eve). Coverage: 1.4-2.1, 2.3.
- We'll start 2.2 this week (probably not today). Please read that section, and try the preview exercise.
Last time:
- We continued plowing through a number of "proofs" of some derivative rules: on Wednesday we proved the first four rules using the limit definition of the derivative:
  - rules for constant functions $f(x)=c$ and the function $f(x)=x$;
  - sum rule;
  - product rule;
  - constant multiple and difference rules were "free", consequences of the previous rules; and then we proved the
  - power rule using mathematical induction and the product rule.
  Then on Friday we
  - did an example (derivative of a polynomial), and then
  - proved the exponential function rule.
  We still need to prove the quotient rule before we move along.
But first let's talk about that local linearization quiz....
- My key
- Your key
- Median: 6 (but that skewed a little because there were five zeros, for those absent).
- The local linearization is just that linear function that has the tangent line as its graph: \[ y-f(a) = f'(a)(x-a) \] then \[ y = L(x) = f(a) + f'(a)(x-a) \]
- Please forget about slope-intercept form! Use point-slope form: it shows everything going on locally at $x=a$.
Today we continue with a quick review of the exponential rule, and then we move along to the quotient rule.
- 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...
- Theorem (quotient rule -- not simply the quotient of the derivatives!):
  Let $F(x)=\frac{f(x)}{g(x)}$. \[ \frac{d}{dx}[F(x)]=\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{\frac{d}{dx}f(x)g(x) - f(x)\frac{d}{dx}g(x)} {[g(x)]^2} = \frac{f'(x)g(x) - f(x)g'(x)} {[g(x)]^2} \] I got a rhyme from a colleague, to help us remember this formula: "Lo dee hi minus hi dee lo, over the denominator square they go!"
  There are two ways to go about proving it:
  - The limit definition (of course!)
  - But this can be proved as a corollary of the product rule, by this one simple little trick....
- So here's the big news: we've got all the power we need to differentiate (easily) any rational function -- a ratio of polynomials -- and that's an even huger and importanter group of functions!:)
  (By the way, the polynomials are merely a subset of the rational functions: they're the rational functions with constant polynomials for denominators.) \[ f(x)=4x^3-7x^2+5x-1=\frac{4x^3-7x^2+5x-1}{1} \]
  See? So what's the derivative of \[ F(x)=\frac{x^2-3x-4}{x-2} \]
  Derivatives of rational functions are easy now! (No more limit definition derivations for you -- at least most of the time). But we still have a few more rules to prove....
- Next up: rules for derivatives of the trigonometric functions -- but those won't be on your exam.
- If we've got some time, we'll check out the worksheet, working with polynomials and powers (negative, and other than simple integers): Sections 2.1 and 2.3 worksheet.
  One thing to be aware of is that "surds" (those horrible square and cube root signs) really represent non-integer powers (and so the power rule can be used).
Links:

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Footnotes

Derivative of $f$: \[ 12 x^2-14 x+5 \]
Derivative of $F$: \[ \frac{x^2-4 x+10}{(x-2)^2} \]