Day 10 in Mat122

Today:

• Return Homework 5.2: #11, 77

• Directed reading handout: Section 6.1 (html) (pdf)

• Question on 5.3: "#47 and/or #51"

• Section 5.4: The Fundamental Theorem of Calculus, Part II
  • Here's the news: Area function A(x) is

  • Now we arrive at the FTC, Part II:

  • Examples:
    • #46 p. 284

  • A word (or two!) about compositions, and the chain rule....

  • Examples:
    • #9, p. 282
    • #29, p. 283

• Section 5.6: The substitution rule
  • The substitution rule is the chain rule backwards: I'd prefer that we think as follows: and, even better,

    Forgetting for a moment that we might know how to solve this!;), we can always do the change of variables

    and hope that the integral on the right is easier to solve (certainly less cluttered).

    Writing it in this last way may be mysterious, because of the change of variable to u (and the change in the limits); but it's the disappearance of g'(x) that's really curious. It falls right out of the change of variables, however:

    Solving for du, we get du=g'(x)dx, and that piece disappears.

  • Examples:
    • #2, p. 295
    • #7, p. 295
    • #13, p. 296
    • #31, p. 296
    • #41, p. 296
    • #57, p. 296
    • #63, p. 297
    • #64, p. 297