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Top 10 Reasons Why I Am Teaching Related Rates Tomorrow

- When a ladder slides down a wall, it is more important to know the speed of both the top and the bottom of the ladder than it is to stop it from falling.

- To practice the chain rule and implicit differentiation in a situation where the independent variable is not named
*x* and the dependent variable is not named *y*.

- To consider the case two dependent variables that are both functions of the same independent variable (an idea first introduced in the context of parametric equations) but that are also constrained by their relationship to each other.

- Because all serious standard calculus courses (except those that are secretly real analysis in disguise) must cover related rates. It is a rite of passage.

- The geometry necessary to set up a related rates problem is closely related to the geomtry necessary to do those Calc II problems with the slicing up of solids. Both the "find the volume of the pyramid" type and the "find the work" type tend to use a similar set-up.

- To shatter my students' illusions (brought about by Chapter 3) that the heart of every calculus problem is the "take the derivative" step.

- Leaky conical tanks are ubiquitous.

- To help me sort my students. The students who find a problem about leaky conical reservoir full of water to be entirely different from a problem about a leaky conical paper cup full of orange juice are doomed. Those who consider "the cone problems" and "the ladder problems" and the "distance between the ships problems" as totally separate classes of problems but that can solve each of them when they match the prototypes presented in class are the C students. The B students will recognize that these are all problems about triangles and will figure out how to apply similar triangles and/or the Pythagorean Theorem to these and other triangle-problems. Only the A students will be able to use the technique in a general setting.

- An evil, sadistic streak by which I enjoy torturing my students with difficult problems. I watch them suffer while refusing to "tell them how to do the problems" (by which they mean "provide a deterministic algorithm that works in EVERY case").

- The Math Department said, "If you are teaching Calc I, then you will cover related rates on Monday, October 31 and on Wednesady, November 2."