Day 34, Mat129: Calculus I

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Announcements
- Your 4.2 homework is due.
- Your 4.1 homework is returned: graded 2, 13, 22
  - #2:
    1. Apologies if I "corrected" your values -- they were read off a graph, so there's some uncertainty. I was thinking that I took them off a table....
    2. Don't just say "over", "under" -- explain!
    3. Midpoint is not the average of LRR and RRR -- trapezoidal is.
  - #13: answer the entire question -- upper/lower....
  - #22: this is essentially a pattern identification problem. Look at the top of p. 288:
    
    $\Delta{x}=\frac{b-a}{n}$
    Right endpoints are given by
    
    $x_i=a+ i \Delta{x}$
- We have our third midterm exam coming up, on Monday.
  I'm going to drop the lowest of your three midterms. That's not an excuse to blow this one off, but rather an opportunity to improve on your test average.
  The test will cover up from 3.3 through 4.2, which we basically finished last time.
- We'll meet in here on Friday, for a review. If you run out of questions, we'll move on.
- Quiz at the end of the hour....
Section 4.3: the Fundamental theorem of calculus
- The fundamental theorem 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 antiderivative of $f$; that is, a function such that $F'=f$.
  2. If $f$ is continuous on $[a,b]$, then the function $F$ defined by \[ F(x)=\int_{a}^{x}f(t)dt \;\;\;\;\; a \le x \le b \] is continuous on $[a,b]$ and differentiable on $(a,b)$, and $F'(x)=f(x)$.
  It says that to integrate, all you have to do is think of an antiderivative F of the integrand f.
- One can use linearization to derive/motivate the FTC.
- Examples:
  - #3, p. 318
  - #22
  - #23
  - #8
  - #13 (a composition, requiring the chain rule)
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.
- Let me bring to your attention the table of integrals on page 322, and at the back of the book (Reference Page 6 and following). This is a table of functions and their antiderivatives, and was the old way of evaluating integrals (before software got so good).
- Examples:
  - #1, p. 326
  - #5 (linearity of the integral)
  - #45. Think about it graphically.
  - #50; furthermore, what does
    
    $100+\int_{0}^Tn'(t)dt$
    represent?

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