4.1 Related Rates Problems

4.1 Related Rates Problems - These notes closely follow the...

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These notes closely follow the presentation of the material given in James Stewart’s textbook Calculus, Concepts and Contexts (2nd edition). These notes are intended primarily for in-class presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working through the exercises therein. Related Rates Problems In solving a related rates problem, one attempts to find the rate of change of some quantity based on the rate of change of some related quantity. The basic strategy for solving related rates problems is outlined on page 267 of our textbook. Example Suppose that one leg of a right triangle remains of fixed length while the other leg grows at a constant rate k 0 and that all the while the triangle remains a right triangle. Find a formula that gives the rate at which the hypotenuse is growing. Solution Let a be the length of the fixed leg of the triangle. Let x be the length of the other leg and let y be the length of the hypotenuse. Note that a is a constant quantity but x and y are functions of time ( t ). We are given that dx / dt k and we want to determine dy / dt . Since the triangle remains a right triangle at all times, then by the Pythagorean Theorem, a 2 x 2 y 2 for all t 0. (Recall that x and y are functions of t . ) Differentiating both sides of this equation with respect to t , we obtain d dt a 2 x 2 d dt y 2 or 2 x dx dt 2 y dy dt . Since dx
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This note was uploaded on 11/13/2010 for the course MA 141 taught by Professor Wears during the Spring '07 term at N.C. State.

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4.1 Related Rates Problems - These notes closely follow the...

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