Find the average value fave of the function f on the given interval.f(t) = 6te−t2,    [5, 7]fave =

The average value of a function f(x) on the interval [a, b] is given by the formula:

fave = 1/(b - a) * ∫ from a to b f(x) dx

Here, f(t) = 6te^(-t^2), and the interval is [5, 7].

So, we need to compute the integral of f(t) from 5 to 7 and divide by the length of the interval, which is 7 - 5 = 2.

The integral of f(t) = 6te^(-t^2) can be computed using the method of integration by substitution. Let u = -t^2. Then du = -2t dt, or dt = -du / (2t).

Substituting these into the integral, we get:

∫ from 5 to 7 f(t) dt = ∫ from 25 to 49 -3e^u du = [-3e^u] from 25 to 49 = -3e^49 + 3e^25.

So, the average value of f on the interval [5, 7] is:

fave = 1/2 * (-3e^49 + 3e^25) = -1.5e^49 + 1.5e^25.