Day 07, Mat227

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Announcements:
- Your second assignment has been pushed back to Friday. Questions?
- For those who are new to Mathematica, my colleague Don Krug recommends the following:
  - A Student's Introduction to Mathematica, by Cliff Hastings: It shows you what Mathematica can do to help you in your classwork. this is a quicker, less in-depth introduction than the videos below.
  - The Hands-On guide to Mathematica is a good way to learn the basics (and more.) You needn't watch all these videos, but one or two should be enough to get the basics. Be sure to have Mathematica running and work along with Cliff as he goes through the some of the ways you can use Mathematica.
Section 4.3: the Fundamental Theorem of Calculus
We are on the cusp of alleviating the tedium of constructing rectangles. Or, as our author says: "Newton and Leibniz...saw that the Fundamental Theorem enabled them to compute areas and integrals...without having to compute them as limits of sums...." (p. 310).
- So here it is: the fundamental theorem that ties together integral and differential calculus.
  It comes in two parts:
  1. If ${f}$ is continuous on ${[a{,}b]}$ , then
    ${\int_{a}^{b}f(x)dx{=}F(b)-F(a)}$
    where ${F}$ is any anti-derivative of ${f}$ ; that is, a function such that ${F'(x){=}f(x)}$ .
    In words: definite integration is easy if you know an anti-derivative of the integrand:
    just evaluate an anti-derivative at the two endpoints, and subtract.
  2. If ${f}$ is continuous on ${[a{,}b]}$ , then the function ${g}$ defined by
    
    ${g(x)=\int_{a}^{x}f(t)dt \;\;\;\;\; a \le x \le b}$
    is continuous on ${[a{,}b]}$ and differentiable on ${(a{,}b)}$ , and ${g'(x){=}f(x)}$ .
    In words: It's easy to construct an anti-derivative of a function:
    just define it as the integral of ${f}$ from ${a}$ to ${x}$ .
    This happens to be the particular anti-derivative ${g}$ which is 0 at ${x=a}$ : ${g(a)=0.}$
  The Fundamental Theorem of Calculus says that in order to integrate, all you have to do is think of an anti-derivative F of the integrand f.
- Let's derive/motivate the FTC using the linearization.
- Examples:
  - #22 -- doing integrals is easy now!
  - #23
  - #3, p. 318 (don't forget about thinking of integrals as areas).
  - #8 (straightforward)
  - #11 (a little trick here....)
  - #13 (a composition, requiring the chain rule)
    
    ${h(x){=}\int_{2}^{1/x}{\sin^4({t})}{dt}}$
    Ask yourself how you can use the closely-related integral
    
    ${g(x){=}\int_{2}^{x}{\sin^4({t})}{dt}}$
    to solve your problem. How are ${g}$ and ${h}$ related?
  - #58: requires Mathematica (or your favorite technology -- but something!)

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