Question

The function f(x)=9-x² is graphed on the interval [-3,3] as shown belo a) Determine the average value of f(x) on the given interval

Answer 1

The average value
`(\(\bar{f}\))` of the function
`\(f(x) = 9 - x^2\)` on the interval
`\([-3,3]\) is \(\bar{f} = (56)/(9)\)` or approximately (6.22).

Step-by-step explanation:

To find the average value of
`\(f(x)\)` on the interval
`\([-3,3]\),` we use the formula for average value:
`\(\bar{f} = (1)/(b-a) \int_(a)^(b) f(x) \,dx\),` where (a) and (b) are the endpoints of the interval. In this case, (a = -3) and (b = 3). The function
`\(f(x) = 9 - x^2\)` , so the integral becomes
`\(\bar{f} = (1)/(6) \int_(-3)^(3) (9 - x^2) \,dx\).`

Evaluating this integral, we get
`\(\bar{f} = (1)/(6) \left[9x - (x^3)/(3)\right]_(-3)^(3) = (56)/(9)\).` Therefore, the average value of
`\(f(x)\)` on the interval
`\([-3,3]\) is \((56)/(9)\)` or approximately (6.22).

In conclusion, the average value of
`\(f(x) = 9 - x^2\)` on the given interval represents the height of the rectangle whose base is the interval ([-3,3]) on the x-axis and whose top edge touches the curve of the function. Calculating this value involves finding the definite integral of the function over the given interval and dividing by the width of the interval.