- Quizzes are returned.
- Checking to see if you are hung up on $f$ and $g$, or whether you can write the formula with other names....
- Compositions -- identifying and constructing -- is the difficult part.
- "inside" and "outside" function identification is the tricky part -- the rest is just applying known rules.
- You have an exam coming up, Friday. It will cover through the chain rule (section 2.5).
- The Sehnert lecture is coming soon: I hope that you'll attend.
Today is the deadline to sign up for the dinner. If you want to come, please sign up on the sheet I'll hand around, and I'll RSVP for you. Feel free to bring a friend!
- We approached this topic via some examples:
- One last important application of implicit functions: orthogonal families. #49, p. 163.
- In calculus you encounter the famous ladder problem. Example 2 on
p. 177. 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.
- The problems considered here are the fun ones, the story problems:
- Everyone loves story problems, right?! Let's go over the strategy,
and then try some.
I jokingly use the acronym "UPCE" (oopsie!) for the general problem solving strategy:
- Understand
- Plan
- Carry out
- Evaluate
- Here is our author's more detailed approach to these related rates
problems:
- Read the problem carefully.
- Draw a diagram if possible.
- Introduce notation. Assign symbols to all quantities that
are functions of time.
- 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 (as in Example 3).
- 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.
"Warning: a common error is to substitute the given numerical information (for quantities that vary with time) too early." (p. 179). Substitute only after the differentiation is complete.
- Read the problem carefully.
- Examples:
- #4, p. 180
- #6
- #23 (similar triangles)
- #31 (ladders! Pythagoras....)
- #34
- #37 (law of cosines)
- #38 looks like fun....