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w3es_average

# w3es_average - 1 Problem Averages Consider the plot below...

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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 k =1 f ( t k ) c) Multiply and divide the expression above by the interval length Δ t . What is N Δ t equal to? Rewrite the expression using this fact. Solution. We use the fact that N Δ t = T to obtain < f ( t ) > = 1 N Δ t N k =1 f ( t k t = 1 T N k =1 f ( t k t d) Take the limit as N goes to infinity. Note that 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: < f ( t ) > = lim N →∞ 1 T N k =1 f ( t k t = 1 T T 0 f ( t ) dt

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w3es_average - 1 Problem Averages Consider the plot below...

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