Question

Find the average value of the negative-valued function y=f(x), given that the area of the region bounded by the curve f(x) and x-axis from x=−9 to x=16 is 2.

Answer 1

The average value of the negative-valued function
`\(y=f(x)\)`over the interval
`\([-9, 16]\),` given that the area of the region bounded by the curve
`\(f(x)\) and the x-axis is 2, is \(-(1)/(4)\).`

Step-by-step explanation:

The average value of a function
`\(f(x)\) over an interval \([a, b]\)` is given by the formula
`\((1)/(b-a) \int_(a)^(b) f(x) \,dx\)`. In this case, the interval is
`\([-9, 16]\)`, and we are given that the area under the curve from
`\(x=-9\) to \(x=16\) is 2.`

The average value is negative because the function is negative-valued. The negative sign indicates that the function spends more time below the x-axis than above it over the given interval.

The integral of the function over the interval is
`\(\int_(-9)^(16) f(x) \,dx = 2\).` To find the average value, we divide this integral by the width of the interval, which is
`\(16-(-9) = 25\).`

`\((1)/(16-(-9)) \int_(-9)^(16) f(x) \,dx = (1)/(25) * 2 = -(1)/(12)\).`

Therefore, the average value of the negative-valued function
`\(y=f(x)\)`over the interval
`\([-9, 16]\) is \(-(1)/(25) * 2 = -(1)/(12)\).`