Day 08, Mat227

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- Your second assignment is due Today.
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!)
Now on to Section 4.4: the indefinite integral:
- Not too much to shout about here: the indefinite integral represents the family of anti-derivatives of a function:
  
  ${\int{f(x)dx{=}{F(x)}{+}{C}}$
  where ${F({x})}$ is any particular anti-derivative.
So, as mentioned in class, differentiation and integration are not quite inverse processes: the derivative of a function is unique, and an infinite number of functions have the same derivative; vice versa, a function has an infinite number of anti-derivatives.
Let me bring to your attention the table of integrals at the back of the book. This is basically a table of functions and their antiderivatives.
Examples:
- #1, p. 326
- #5
- #45
- #50; furthermore, what does
  
  ${100+\int_{0}^T{n'}({t}){dt}}$
  represent?

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