Day 20, Mat128: Calculus A

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Next time:

Today:

Announcements
- Today's quiz will cover section1.8: the tangent line approximation.
- You have an 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 next week. It's about trig functions, and that's important!
Last time:
- We plowed through a number of "proofs" of some derivative rules: 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.
Any questions on local linearization?
Today we continue with a few more rules.
- Let's review the power rule, in particular: the use of induction can certainly seem a little mysterious on first glance. It begins with induction:
  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 -- what's wrong with 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!).
  By the way, once we've proven the chain rule, we'll be able to prove the power rule for any real exponent, using exponential functions and their inverses (logs).
- 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...
- 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!"
  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 week we'll add 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.
- Then it's time for a quiz!
Links:
- 1.8 homework key

Website maintained by Andy Long. Comments appreciated.

Footnotes

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