Day 27, Mat128: Calculus A

Last Time

Next Time

Announcements:
- If you need to Zoom to class, here's the link. Old Zooms are here.
- Your exams are due Monday.
- We have just one more section of material: today's section 3.5
- Whiteboard of today's trough problem (two approaches to the same optimization problem).
Last time:
Today:
- The problems considered here are the fun ones, the story problems. Not everyone thinks these are fun, however:
- Let's start with an example optimization problem from section 3.4 (Activity 3.4.5), and attack it in two different ways.
  (There's always more than one way to attack a problem!)
  
  $A(h)=h(2+\sin({1-h^2}))$ $A(\theta)=\sin({\theta})(2+\cos({\theta}))$
- Then we'll move on to related rates problems (section 3.5).
  - These are in the same vein as the optimization problems: story problems, but with a twist -- 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).
  - In Stewart's calculus, he suggested this 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.
- 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.
    These sorts of animations are a really useful part of "textbooks" today.
- 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:
  1. UPCE: Understand, Plan, Carry out, Evaluate
  2. Draw pictures
  3. Give meaningful names (letters) for variables.
  4. Establish reasonable domains for variables
  5. Establish a reasonable coordinate system
  6. Doodle relationships!
  7. Establish a relationship between the variables whose rates will be related.
  8. Compute derivatives, usually implicitly, which are now also related.
  9. Solve for the rate of interest (frequently given at a particular point in time).
Links:
- A Chain Rule tutorial
- Our free, on-line textbook, called Active Calculus.
- asciisvg.html

Website maintained by Andy Long. Comments appreciated.


$A(h)=h(2+\sin({1-h^2}))$	$A(\theta)=\sin({\theta})(2+\cos({\theta}))$