Stewart6e3_9

# Stewart6e3_9 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 3.9 Related Rates of Change In applications, one frequently deals with problems in which more than one quantity is changing with respect to change in yet another. Even if some of the rates of change are unknown, the relationships between the rates can often be exploited in order to determine the unknown values. (The Chain Rule plays a crucial rˆ ole here.) The best way to learn about the method of “related rates” is to do a bunch of examples! Example. Suppose a sphere’s volume is increasing at a rate of 100 cm 3 per second. How fast is the circumference of the sphere changing when the diameter of the sphere has diameter D = 50 cm? Solving this problem requires first understanding what is being asked (that is, reading the problem), and then drawing a picture or using a visual aid ( introducing notation where necessary) in order to help us express the relevant rates of change in in terms of derivatives. From here, it’s a matter of writing an equation relating the quantities involved, differentiating , and then finally plugging in the information given. (The italicized words here summarize the steps your text recommends in solving these problems.

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