Average function value, Mathematics

Average Function Value

The average value of a function f(x) over the interval [a,b] is specified by,

favg = (1/b-a) ab f(x) dx


We know that the average value of n numbers is only the total of all the numbers divided with n therefore let's start off with this. Let's take the interval [a,b] and divide this in n subintervals each of length,

x = (b -a)/n

Now by all of these intervals select the points x1*, x2*,...., xn* and consider that this doesn't really issue how we select each of these numbers as long as they arrive from the suitable interval.

 We can then calculate the average of the function values f(x1*), f(x2*),.....,f(xn*) by computing,

(f(x1*), f(x2*),.....,f(xn*))/n

Here, from our definition of ?x we can find the formula for n as given in below.

n = (b -a)/ ?x

and we can plug it  in (4) to have,

(f(x1*), f(x2*),.....,f(xn*))/((b -a)/ ?x)

= ([f(x1*), f(x2*),.....,f(xn*)]?x)/(b -a)

= (1/(b -a)) ([f(x1*), f(x2*),.....,f(xn*)]?x)

= (1/(b -a))  490_mean.png    f(xi*)?x

Let's here raise n. Doing that will mean that we are taking the average of increasingly function values in the interval and therefore the larger we select n the better it will approximate the average value of the function.

If we did so take the limit as n goes to infinity we must find the average function value. Or,

favg = limn→∞ (1/b-a)  490_mean.png       f(xi*) ?x = (1/(b -a))      490_mean.png                 ab f(xi*) dx

We can factor the 1/(b -a) out of the limit where we have done and here the limit of the sum must look familiar as which is the definition of the definite integral. Therefore, putting in definite integral we find the formula as we were after.

favg = (1/(b -a)) ab f(x) dx

Posted Date: 4/13/2013 4:07:20 AM | Location : United States

Related Discussions:- Average function value, Assignment Help, Ask Question on Average function value, Get Answer, Expert's Help, Average function value Discussions

Write discussion on Average function value
Your posts are moderated
Related Questions
use the simplex method to solve the following lp problem. max z = 107x1 + x2 + 2x3 subject to 14x1 + x2 - 6x3 + 3x4 = 7 16x1 + x2 - 6x3 3x1 - x2 - x3 x1,x2,x3,x4 > = 0

Carmen bought 3 pounds of bananas for $1.08. June paid for her purchase of bananas. If they paid the same price per pound, how many pounds did June buy?

please i need the solution for halm''s differential equation

4x+3y+7=0 and 3x+4y+8=0 find the regression coefficient between bxy and byx.

Describe Adding and Subtracting Fractions in details? To add or subtract fractions, here are some steps: 1. Find the lowest common denominator (LCD) or any common denominato

Factor following polynomials.                               x 2 + 2x -15 Solution x 2 +2x -15 Okay since the first term is x 2 we know that the factoring has to ta

A garden shop wishes to prepare a supply of special fertilizer at a minimal cost by mixing two fertilizers, A and B. The mixture is to contain at least 45 units of phosphate at lea

Let f : R 3 → R be de?ned by:                                        f(x, y, z) = xy 2 + x 3 z 4 + y 5 z 6 a) Compute ~ ∇f(x, y, z) , and evaluate ~ ∇f(2, 1, 1) . b) Brie?y

Solve : 4x2+2x+3=0 Ans) x^2 + (1/2)x = -(3/4) (x+1/4)^2 = 1/16 - 3/4 = -11/16 implies x = (-1+i(11)^(1/2))/4 and its conjugate.

Computer monitors are calculated by their diagonals. If a monitor is advertised to be 19 in, Determine the actual viewing area, considerthe screen is square? (Round to the nearest