Day 27, Mat128: Calculus A

I noticed that a fair number of you are missing at least one IMath assignment. Although I will be dropping your two lowest, you might want to try out any that you didn't get to along the way.

For that reason, if you wish you may request that I open up some of the old assignments for you to take, if you didn't before. I'll start them at 90% credit -- but they are still worth doing, as practice for the final.

Email me with your requests for any that you want to try (tell me which assignments you want re-opened).

NB: Please don't pay any attention to your percentage score on IMath. Each assignment is equally weighted.

• We'll start right in with the Preview. There are a number of important lessons that it contains:
  • All rational functions look like polynomials eventually, far enough away from zero. (This rational function looks pretty boring....)
  • We don't really need calculus to answer these questions -- we can use our understanding of functional forms (in particular, quadratics).
  • We can use symmetry, even when the function is neither even nor odd. This function is symmetric about the line \(x=-1\).

  \[ f(x) = 2 + \frac{3}{1+(x+1)^2}\text{.} \] \[ f'(x) = \frac{-6 (x+1)}{\left((x+1)^2+1\right)^2} \]

• There's one theorem in here that we need to understand: the Extreme Value Theorem. The basic idea is that, if the function is continuous on a closed and bounded domain, then it must have a global maximum and a global minimum. Is that obvious to you?

• I notice especially in this section that the authors uses different vocabulary from time to time. "Global" is the same as an "Absolute" extremum; "local" is the same as a "relative" extremum.

  That's not all bad, because you need to be aware that people use different words for the same concept -- but it might be confusing at first!

  This is similar to the different notations for the derivative.

• Then it's on to some problems whose solutions truly involve calculus:
  • Example 3.3.4 (we'll follow along with this one), and
  • then try our hands at Activity 3.3.4

• Finally, I'll turn you loose on your worksheet for 3.3.

• Based on the entire 102 years of data, the average peak bloom date for Washington's cherry blossoms is April 4.
• Peak bloom date for the cherry trees is occurring earlier than it did in the past. Since 1921, peak bloom dates have shifted earlier by approximately seven days. The peak bloom date has occurred before April 4 in 16 of the past 20 years.
• While the length of the National Cherry Blossom Festival has continued to expand, the Yoshino cherry trees have bloomed near the beginning of the festival in recent years. During some years, the festival missed early peak bloom dates entirely.