Overview
We conclude our work in Chapter 3 (Using the Derivative) by seeing how the derivative may be employed to relate the rates of two different quantities that are related and each changing as time varies. The main idea here is: if two quantities are related to one another, and each is changing as time changes, then the rates at which each quantity is changing must be related. Hence, we consider a class of problems known as related rates problems.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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Understand the basic idea of a related rates problem: if the values of two (or more) changing quantities are related, then their respective rates of change must also be related.
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Be able to use basic geometry results such as the Pythagorean Theorem, formulas for the area of familiar figures, and trigonometry to establish relationships among quantities of interest.
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Understand the difference between differentiating an equation such as \(A = x^2\) with respect to \(t\) and differentiating with respect to \(x\).
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(Review) Recall the method of Implicit Differentiation (Section 2.7).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Use Implicit Differentiation/the Chain Rule to take the derivative (usually with respect to time) of the equation relating the variables to each other.
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Set up and solve a wide range of related rates problems, including those found in the activities in section 3.5 in Active Calculus.
To prepare for class
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Read the beginning of section 3.5 in Active Calculus, up until the Preview Activity.
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Do the Preview Activity for 3.5 (on WeBWorK if required by your teacher).
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Read the next part of section 3.5 in Active Calculus, up until the first activity.
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Watch the following videos (in the shown order), making yourself a list of at least 3 things you learned from the videos or your reading - and include this as part of your feedback:
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Do the following problem (on paper) and include its answer in your feedback:
Suppose we have a situation with a cylinder of radius \(r\) and height \(h\), so that its volume is \(V = \pi r^2 h\). Suppose further that \(V\), \(r\), and \(h\) are all changing as time changes, and thus each is implicitly a function of \(t\). Find an equation that relates \(\frac{dV}{dt}\), \(\frac{dr}{dt}\), and \(\frac{dh}{dt}\). Write one sentence to say what derivative rule(s) you used to establish the equation that relates the rates.
After class
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Watch the following video for a Related Rates problem involving some trigonometry:
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Do some experimentation with the following interactive applets (by Marc Renault) showing important examples of Related Rates problems (some of which might match WeBWorK problems you have been assigned!):
