Day 14, Mat128: Calculus A

I had suggested completing your chapter 2 worksheets, which we will go over together to make sure that you get your questions answered.

Also feel free to raise any other questions today.

• 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 we obtained this theorem by using tools we'd already built; in particular, the product rule, and a trick to get the derivative of the multiplicative inverse of a function.

  A theorem that you prove on the way to proving some theorem is called a "lemma".

  • Lemma (derivative of the multiplicative inverse):
    Let $F(x)=\frac{1}{g(x)}$. Then \[ \frac{d}{dx}[F(x)] = \frac{-g'(x)}{g(x)^2} \]

• Theorem (trigonometric derivatives):
  Let $F(x)=\sin(x)$. \[ \frac{d}{dx}[F(x)]=\cos(x) \]

  We needed a couple of limits to establish this one, as well as one trig identity: \[ \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a) \]

• Corollary:
  Let $F(x)=\cos(x)$. \[ \frac{d}{dx}[F(x)]=-\sin(x) \]

  This is a consequence of the derivative of sine, by symmetry and periodicity.

  Further consequences: \[ \frac{d^2}{dx^2}[\sin(x)]=-\sin(x) \] \[ \frac{d^3}{dx^3}[\sin(x)]=-\cos(x) \] \[ \frac{d^4}{dx^4}[\sin(x)]=\sin(x) \]

  The sine (and cosine) functions are their own fourth derivatives.

• There is your worksheet about polynomials and powers) (key).
• There is the one labelled "Chapter 2 worksheet" (key).
• Let's review what we've done so far:
  • Linear things!
  • Average velocity (and "rate of change", more generally) -- slopes of secant lines
  • Instantaneous velocity -- slopes of tangent lines
  • Geometric, numeric and algebraic calculation and meaning of rates of change (slopes of lines: change in y over change in x)
  • Limits -- left, right, and at a point
  • Continuity (when the limit at a point agrees with the function value there)
  • The limit definition of the derivative at a point
  • Indeterminate forms
  • When derivative values don't exist (graphically, algebraically) -- you can't draw a tangent line.
  • The derivative's geometric sense (slope of a tangent line)
    1. what the sign means
    2. what zero values mean (or may mean!)

  • The derivative function (as a limit definition)
  • Centered, forward, backward definitions of the derivative (especially useful for estimating derivatives of data tables)
  • The second derivative and acceleration/concavity
  • The tangent line approximation (the linearization)
  • Building "power tools" using the limit definition of the derivative

    In particular, if I don't ask you to use the limit definition, you may use any of these rules to compute derivatives from now on:

    • the sum rule
    • product rule (and constant multiple rule)
    • power rule (using "induction", or dominoes, to prove it)
    • quotient rule (from the multiplicative inverse rule)
    • exponential rule
    • sine and cosine rules