Day 39, Mat128: Calculus A

Sorry about that, but we're finally seeing some good reasons for learning calculus! We're seeing the power....

• Isabella Carr, et al. (two posters!): Locomotor activity in female and male rats after acute oxycodone; and Sex Does Not Influence Oxycodone-Conditioned Place Preference in Rats
• Idalia Martin (poster): Predictive Analysis of Computer Hardware Prices through Machine Learning

• Pretty much a disaster. (Did you notice that these were all limits off of your worksheet?)
• Median: 5
• Key
• Please do use algebra, and arithmetic: e.g. \[ \frac{x^2}{x}=x \] and \[ \frac{2x}{x}=2 \] There's a reason they call it "simplification" -- it simplifies your life! Don't you want to simplify your life? The power is in your hands, and in your algebra....

• There seems to be some confusion about what it means to be determinate, and what it means to be indeterminate.

  determinate:

  e.g. \(\frac{0}{3}\), \(\frac{3}{0}\), \(\frac{\infty}{6}\), \(\frac{6}{\infty}\), \(\frac{4}{3}\), etc.

  indeterminate:

  e.g. \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(\infty*0\), \(1^\infty\), etc. (those last two need to be converted to quotients before you can apply L'Hôpital's rule).

• Speaking of applying L'Hôpital's rule, you must only apply it if the limit is indeterminate.

  If you apply it to determinate things, you'll almost always get garbage.

• Some of you appear to have forgotten about continuity, and how to compute limits of continuous functions. It's easy, so long as the limit is defined: simply replace the variable with it's limiting value, and you'll have your answer: \[ \lim_{x \to a}f(x)=f(a) \]

We also focused on reviewing the table we use to summarize the study of a function using its derivative.

So let's review that table again! And see how well it captures the essence of the function \(h(x)\).

• 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 Section 3.3 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?)

  This is a so-called "existence theorem": as our author says, "The theorem does not tell us where these extreme values occur, but rather only that they must exist."

  Then it's our job to find them; to root them out....

• 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.

  By the way, on #2, \(f(0)\approx 2.40\), and \(f(5)\approx 2.31\).

  (Why did we make it so close, without giving you the function??:)

  Let's look at #4.

• 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.