# The expression f ave [ a , b ] = ( c − a b − a ) f ave [ a , c ] + ( b − c b − c ) f ave [ c , b ] . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 6.5, Problem 22E
To determine

## To show: The expression fave[a,b]=(c−ab−a)fave[a,c]+(b−cb−c)fave[c,b].

Expert Solution

### Explanation of Solution

Formula:

The expression for the average value of f on the interval fave[a,b] is,

fave[a,b]=1baabf(x)dx (1)

Given information:

The average value of the function, a<c<b is the average value of f on interval [a,b] and fave[a,b]=1baabf(x)dx.

Calculation:

Since a<c<b, modify integral limit a to b by the addition of integral limits a to c and c to b.

Modify equation (1).

fave[a,b]=1baacf(x)dx+1bacbf(x)dx

Multiply and divide by terms (ca) and (bc).

fave[a,b]=caba[1caacf(x)dx]+bcba[1bccbf(x)dx] (2)

Express the function fave[a,c] by the use of equation (1).

fave[a,c]=1caacf(x)dx (3)

Here, fave[a,c] is average value of f on the interval [a,c].

Express the function fave[c,b] by using Equation (1).

fave[c,b]=1bccbf(x)dx (4)

Here, fave[c,b] is average value of f on the interval [c,b].

Substitute equations (3) and (4) in (2).

fave[a,b]=cabafave[a,c]+bcbafave[a,c]

Thus, the expression fave[a,b]=cabafave[a,c]+bcbafave[a,c] is derived.

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