This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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:...
View
Full
Document
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
 BERMAN

Click to edit the document details