Day 43, Mat128: Calculus A

Last time:

Next time:

Today:

Announcements
- The "take-home" quiz is returned.
  - My key
- Here are the notes from last time: two story problems.
- No quiz Friday: we're going to review. Come with questions.
- I'll have solutions for all of our assignments available for you to use to check your work.
Last time:
- We reviewed the "dogs do calculus" problem (linear regression) from last time (we needed to spend a few more minutes evaluating the solution), and
- Then tried a few problems from the handouts for these two sections 3.3 and 3.4 (solution to AC3.4.9)
Today: Let's continue on with some problems whose solutions truly involve calculus:
- Today we hit Section 3.5: Related Rates.
  The basic idea is that we're going to find an equation relating the rates of change of two different quantities, one of which is known.
  That will allow us to determine the other rate.
- In calculus you are destined to encounter the famous ladder problem (in our textbook, you'll find it here, with a corresponding geogrebra example). We wish to establish the rate of one thing, relative to the rate of the other. These are so-called "related rates". So the ladder's top rung is falling vertically at one speed, as the feet are slipping out at another. Speeds are, in this case, rates of change of positions.
- Related rates problems frequently involve implicit differentiation. One assumes that a pair of variables, which are related in an equation are varying in response to some associated independent variable (frequently time).
- Here is a more detailed approach to 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. It's generally the last thing that you do....
- To give ourselves some idea of what's going on, let's check out the geogebra example mentioned in Activity 3.5.3 (I really appreciate that these animations are now available!)
- The preview for the section sets the stage, with a spherical balloon.
  Let's look over that one, for some of these details.
- Then we'll do two problems from that section, that illustrate how related rates problems work: we'll focus on the cone problems, and a swimming pool.
  - Exercise 1: Height of a conical pile of gravel, and #7: sand dump;
  - Exercise 5
Links:
- Do Dogs Know Calculus?
- Cherry blossoming data (from the EPA). Some "Key Points" cited in that webpage:
  - 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.
- More generally, Leaf and Bloom Dates (EPA)
- A Keeling Analysis in Mathematica

Website maintained by Andy Long. Comments appreciated.

Footnotes