Day 26, Mat128: Calculus A

I'll be dropping your three lowest quizzes -- why not end on a high note?

• First of all, this quiz covered section 3.1, which has three sections:
  1. Critical numbers and the first derivative test
  2. The second derivative test
  3. Summary
  I thought that questions 1 and 2 would be pretty safe questions for this section!

• One problem I noticed quite a bit on this quiz was the use of third person pronouns without references: "It is zero" -- but I have no idea what "it" is: is it the value of \(x\), the value of \(f(x)\), the value of \(f'(x)\), the value of \(f''(x)\), etc.

  Be careful to state explicitly what quantity you're referring to.

• The fourth problem is one where a change in the form of the expression of the function makes your work a little easier: \[ f(x)=\frac{(x-1)^2}{x} = \frac{x^2-2x+1}{x} = x-2+\frac{1}{x} \] Then \[ f'(x)= 1-\frac{1}{x^2} \] so the critical numbers are \(x = \pm 1 \), and \(x=0\) (although since this one isn't in the domain, it's not a surprise).

  Here's the graph of that function (blue), its derivative (black), and its "slant asymptote" \(x-2\) (red, dashed):

  The real heartbreaker on this one was that a fair number of you miscalculated the derivative as the quotient of the derivatives...:(

• Key

Let's spend a little time on that. I want to review the two problems we did last time first, however, just to finish and summarize.

So a gentle recap of section 3.5.

• These "related rates" problems are in the same vein as optimization problems (finding extrema): as one might imagine, we're relating the rates of change (derivatives) of several different variables to each other.

• One approach to working on related rates problems:
  1. Read the problem carefully.
  2. Draw a diagram if possible.
  3. Introduce (good) notation. Use sensible variable names. Assign symbols to all quantities that are functions of time (usually time will be our independent variable).
  4. Express the given information and the required rate in terms of derivatives.
  5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution.
  6. Use the Chain Rule to differentiate both sides of the equation with respect to t.
  7. Substitute the given information into the resulting equation and solve for the unknown rate.
  8. Don't forget your units.

  Warning: a common error is to substitute the given numerical information (for quantities that vary with time) too early. Substitute only after the differentiation is complete.

• Let's look at two examples from the text:
  1. The preview for the section
  2. Activity 3.5.3 -- and in particular, we should check out the geogebra example mentioned in that activity. (I really appreciate that these animations are now available!)

• Now let's take a look at your worksheet for 3.5, which is a set of three problems (one of them an activity from your textbook).

  You should solve each, thinking along the lines of my favorite formulation: UPCE: Understand, Plan, Carry out, Evaluate

  1. Draw pictures
  2. Give meaningful names (letters) for variables.
  3. Establish reasonable domains for variables
  4. Establish a reasonable coordinate system
  5. Doodle relationships!
  6. Establish a relationship between the variables whose rates will be related.
  7. Compute derivatives, usually implicitly, which are now also related.
  8. Solve for the rate of interest (frequently given at a particular point in time).

• 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 \(x=-1\).

  \[ g'(x) = \frac{6 (x+1)}{\left((x+1)^2+1\right)^2} \]

• 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.