Day 26 in Mat121

Today:

• Quiz today, over implicit differentiation

• Directed reading handout, Section 4.2 (html) (pdf)

• Section 3.9: Related Rates
  • Story Problems! Hooray!

  • "Ladders and trains" strategy....
    • Draw a picture
    • Assign variables and restate the problem
    • Find an equation that relates the variables and differentiate
    • Use the given data to find the unknown derivative

  • Examples:
    • #3, p. 166
    • #7, p. 167
    • #37, p. 168

  • Questions over the Directed reading handout, Section 3.9 (html) (pdf)?

• Section 4.1: linear approximations and applications
  • linearizations rely on the fact that as we zoom in on a function, the approximation of the tangent line gets better and better....
  • We're approximating -- we're going to make errors. That's okay!

    • I've replaced the square-root function with a succession of linear approximations, switching from one to another whenever they intersect (to create a continuous function).

    • Errors are related to the second derivative: if the function is straight as it passes a point, then the tangent line will be a perfect approximation; if it's curving, then the tangent line approximation will tend to get worse as the function curves away. The speed with which it curves away is related to the size of the second derivative: where K=max|f^''(x)| on the interval on which the approximation is used, and h is the distance from x=a at the point of tangency.

  • Examples:
    • #4, p. 179
    • #8, p. 179
    • #18, p. 180

  • Questions over the Directed reading handout, Section 4.1 (html) (pdf)?

• Great calculus demos (and more on linearization).