Some students may even say that from their experience or physical intuition this actually makes sense, which raises the question of whether this is truly a physical phenomenon or a purely mathematical one that has been revealed from calculus. The kinetic energy of an object is $K = rac$ we see that as $x$ increases (while being less than $L$) and $dx/dt$ is fixed the point where the ladder touches the wall is dropping faster and faster. For example, you can make up all sorts of compelling related rates problems by starting with any physics or chemistry equation and imagining a situation where you might want to take its derivative: The reality, of course, is that students are familiar with many of the basic equations and concepts of science and economics, and there's no rule against using these in problems. 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.

$related rate calculus problems$

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. I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates.

Related rate calculus problems