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Unformatted text preview: 12. Related Rates P. K. Lamm 10/07/11 (22:33) p. 1 / 7 Lecture Notes: 12. Related Rates These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. Example 1.1: A particle moves around the unit circle as a function of time. At time t = 3, x = 3 / 5 m, y =- 4 / 5 m, and x = 2 m/s. What is y (3)? 1. Draw a picture and label the variables. 2. What is wanted and what is given? Answer: Wanted: y (3) Given: x (3) = 3 / 5, y (3) =- 4 / 5, x (3) = 2. Since were given x (3) but we want y (3), it makes sense to ask the following: 3. Are x ( t ) and y ( t ) related? If so, how? Answer: If x ( t ) and y ( t ) are unrelated, then we should be able to pick them as we wish. In fact, however, the point ( x ( t ) ,y ( t )) must always lie on the unit circle. This means that the two functions are related and that we must always have x 2 ( t ) + y 2 ( t ) = 1 , for all t. (1) Note that the relationship between x ( t ) and y ( t ) holds for all time , not just for the time t = 3. This will become important in the next step: 4. Differentiate the relationship equation (1) with respect to the independent variable ( t ). The reason for this step is we have already related the functions x ( t ) and y ( t ), but what we really need to do is relate the rates x ( t ) and y ( t ). d dt x 2 ( t ) + y 2 ( t ) = 1 1 12. Related Rates P. K. Lamm 10/07/11 (22:33) p. 2 / 7 or, since x 2 ( t ) = ( x ( t )) 2 , a power of a function of t (with a similar statement for y 2 ( t )) we must use the Chain Rule to differentiate: 2 x ( t ) x ( t ) + 2 y ( t ) y ( t ) = 0 This last equation holds for all t , but we actually only need information about y ( t ) at t = 3: 5. Evaluate the rate equation at the time of interest ( t = 3 ): 2 x (3) x (3) + 2 y (3) y (3) = 0 6. Solve for the quantity ( y (3) ) of interest: 2 y (3) y (3) =- 2 x (3) x (3) y (3) =- x (3) x (3) y (3) , =- (3 / 5) 2...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.
- Fall '10