workshop 3 problem 2 - -1 Problem Averages Consider the...

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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. 1 (t 1 ) f f 10 (t 10 ) T t 0 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 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 k =1 f ( t k t = 1 T N k =1 f ( t k 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: < f ( t ) > = lim N →∞ 1 T N k =1 f ( t k t = 1 T T 0 f ( t ) dt In other words, the average of a function over an interval [0 , T ]
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