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Unformatted text preview: 1 Problem. Averages Consider the plot below in which we have drawn a continuous function of time and evaluated it at the center of 10 intervals. a) Write an expression for the average of these 10 points. Solution. < f ( t ) > = f ( t 1 ) + f ( t 2 ) + + f ( t 10 ) 10 b) Generalize this expression to N points and write it in sigma notation. Solution. < f ( t ) > = f ( t 1 ) + + f ( t N ) N = 1 N N X k =1 f ( t k ) c) Multiply and divide the expression above by the interval length t . What is N t equal to? Rewrite the expression using this fact. Solution. We use the fact that N t = T to obtain < f ( t ) > = 1 N t N X k =1 f ( t k ) t = 1 T N X k =1 f ( t k ) t d) Take the limit as N goes to infinity. Note that t goes to dt , and the sum becomes an integral. This limit is defined as the average value of a function on the interval [0 ,T ] . Solution. Note that the expression in part (c) is a Riemann sum. Thus, when we take the limit as N...
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This note was uploaded on 10/01/2008 for the course MATH 1910 taught by Professor Berman during the Fall '07 term at Cornell University (Engineering School).
- Fall '07