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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. 1 (t 1 ) f f 10 (t 10 ) T t f(t) 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. Solution. < f ( t ) > = f ( t 1 ) + ··· + f ( t N ) N = 1 N N X k =1 f ( t k ) c) Multiple and divide by the interval length Δ t and notice what N Δ t is equal to. Write the expression using this fact. Solution. We use the fact that N Δ t = T to obtain < f ( t ) > = 1 N Δ t N X k =1 f ( t k )Δ t = 1 T N X k =1 f ( t k )Δ t d) Take the limit as N goes to infinity, Δ 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 → ∞ we obtain a definite integral:...
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This homework help was uploaded on 02/15/2008 for the course MATH 1910 taught by Professor Berman during the Fall '07 term at Cornell.
- Fall '07