- Ironically, I went from class Monday to my office, and was reading this article on sealevel rise along the southern Gulf Coast of the US when I came across this image:
Can you guess why I'm not so enamored of the cutpoint model in this case?
- Your quiz will cover section 3.1: Using derivatives to identify extreme values.
- A reminder that there are a couple of new IMath assignments, to get you thinking about finding extrema.
- We reviewed the quiz, and talked about section 3.1.
- We looked at an example, of why finding extrema is important: the modeling of time of cherry blossoming in Washington, D.C.
- We started a worksheet about extrema.
- Let's look over the worksheet, second page:
worksheet for 3.1.
- Then we'll move on to related rates problems (section
3.5).
- These 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.
- In calculus you are destined to encounter the famous
ladder problem (in our textbook, you'll find it here).
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:
- Read the problem carefully.
- Draw a diagram if possible.
- Introduce (good) notation. Use sensible variable names. Assign symbols to all quantities that are functions of time (usually time will be our independent variable).
- Express the given information and the required rate in terms of derivatives.
- 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.
- Use the Chain Rule to differentiate both sides of the equation with respect to t.
- Substitute the given information into the resulting equation and solve for the unknown rate.
- 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:
- The preview for the section
- 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!)
- Then it's your turn: 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 these lines:
- UPCE: Understand, Plan, Carry out, Evaluate
- Draw pictures
- Give meaningful names (letters) for variables.
- Establish reasonable domains for variables
- Establish a reasonable coordinate system
- Doodle relationships!
- Establish a relationship between the variables whose rates will be related.
- Compute derivatives, usually implicitly, which are now also related.
- Solve for the rate of interest (frequently given at a particular point in time).
- These 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.
- Time for a quiz....
- 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)