A few of these problems are compelling - for example, computing the speed of an airplane based on ground observations of its altitude and apparent angular velocity - but most of them do feel a bit contrived. Thus, you can find related rates problems involving various area and volume formulas, related rates problems involving the Pythagorean Theorem or similar triangles, related rates problems involving triangle trigonometry, and so forth. Every related rates problem inherently involves differentiating a known equation, and the only equations that the calculus book assumes are the equations of geometry. The main reason that related rates problems feel so contrived is that calculus books do not want to assume that the students are familiar with any of the equations of science or economics. Interpreting the time derivative of a quantity as a rate of change. The skills that students are practicing in related rates problems are:ĭifferentiating a known equation implicitly with respect to time. Can anyone share any related rates questions which don't seem quite as contrived, and which might naturally seem interesting and motivated to a typical class of college freshmen? Basically, all the related rates questions seemed to be cooked up in response to the fact that calculus students now knew a method to solve them. which are realistic models of questions of natural interest in business, biology, etc. This is in stark contrast to many other topics addressed in first-year calculus - optimization, basic differential equations, etc. The baseball question (or something very similar) is actually an exercise in Stewart, and I struggled in vain to imagine a situation in which the manager of a baseball team would need to know the answer. My students do, but only because they know these questions will appear on their exams. How fast are they approaching each other after one second? How fast is the distance between the tip of his shadow and the top of the post changing when he is 40' away?Ī baseball player runs from first base to second at 20 ft/sec, and simultaneously another baseball player runs from third base to home at the same speed. How fast are they moving apart when the woman has been walking for ten seconds?Ī 6' man walks away from a 20' lamppost at a speed of 5 ft/sec. Ten seconds later, a woman starts walking south at 4 ft/sec from a point 20 ft due east of Point A. I taught my students to answer questions such as the following (taken, more or less, from the textbook):Ī man starts walking due north at 5 ft/sec from a Point A. If the problems involves more than two rates, the process is the same but you will need to be given more information to plug in.I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. Likely the equation will be a simple linear function of the rate so just divide by everything else. Once the only unknown that remains is the rate you want to find then you can solve for that rate. Often you will reuse the equation that was differentiated to solve for the final variable. Compute the radius from this information. If the problem is meant to be harder then you might be given the diamter or the volume of the sphere. If the problem is meant to be easy then the radius will be given directly. The question should mention something about when to compute the rate of change. Write this rate in a form similar to $\frac$ is $r$, and you should be able to figure this out from the problem. Read the problem and find a rate that is given.
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