Provide an appropriate response.
-Find the average value of the function y = e-x over the interval [ 0, 3]. Express an exact answer in terms of e.

Question

Quizplus · Accepted Answer

To find the average value of the function, we need to integrate it over the given interval and divide by the length of the interval.

∫e^-x dx evaluated from 0 to 3 = [-e^-x] evaluated from 0 to 3 = -(e^-3 - e^0) = e^0 - e^-3 = 1 - 1/e^3

Length of the interval = 3 - 0 = 3

Therefore, the average value of the function y = e^-x over the interval [0, 3] is:

(1 - 1/e^3)/3 = (3 - e^3)/(3e^3)

Simplifying, we get:

(3 - e^3)/(3e^3) = (3/e^3) - (1/3)

So the exact answer in terms of e is (3/e^3) - (1/3), which corresponds to choice C. We need to find the average value of y = e^-x over the interval [0,3], which is given by:

(1/3-0) * ∫0^3 e^-x dx

= (1/3) * [-e^-x] from 0 to 3

= (1/3) * [-e^-3 + 1]

= (1-e^-3)/3

Since e^-3 cannot be simplified further, we leave it in the exact form, which is C: 11ebb6d7_48ab_bad4_b437_419c7f446637_TB8593_11.

Provide an Appropriate Response