Day 21, Mat128: Calculus A

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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 $a > 0$ : $\frac{d}{d x} [F (x)] = \frac{d}{d x} [a^{x}] = \ln (a) a^{x} = \ln (a) F (x)$
  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 $π$ ). Then $\frac{d}{d x} [F (x)] = F (x) (= F^{'} (x) = F^{″} (x) = F^{‴} (x) = . . . .)$
  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 $F (x) = \frac{f (x)}{g (x)}$ . $\frac{d}{d x} [F (x)] = \frac{d}{d x} [\frac{f (x)}{g (x)}] = \frac{\frac{d}{d x} f (x) g (x) - f (x) \frac{d}{d x} g (x)}{[g (x)]^{2}} = \frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{[g (x)]^{2}}$ 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.) $f (x) = 4 x^{3} - 7 x^{2} + 5 x - 1 = \frac{4 x^{3} - 7 x^{2} + 5 x - 1}{1}$
  See? So what's the derivative of $F (x) = \frac{x^{2} - 3 x - 4}{x - 2}$
  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).
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Updated on 02/26/2024 19:05:05

Footnotes

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