Question

Find the average value of the function f(x) = 4x -8 over the interval 1,3?

a) −8
b) −4
c) −2
d) 4

Answer 1

The average value of the function f(x) = 4x - 8 over the interval [1, 3] is 0, which is calculated using the formula for average value of a function. The result is a positive value that does not match any of the negative options provided.

Step-by-step explanation:

The student is asking to find the average value of the function f(x) = 4x - 8 over the interval [1, 3]. To calculate this, we apply the formula for the average value of a function over an interval [a, b]:

Average value = (1 / (b - a)) * ∫ab f(x) dx

Let's perform the calculation step by step:

1. Find the integral of the function over the interval [1, 3].
   ∫13 (4x - 8) dx
2. Calculate the anti-derivative of f(x): (1 / (3 - 1)) * [(4/2)x2 - 8x] evaluated from 1 to 3.
3. Plug in the values of x = 3 and x = 1 into the anti-derivative and subtract.
   (1 / 2) * [(4(3)2/2 - 8(3)) - (4(1)2/2 - 8(1))]
4. After simplification, the result is 0, which means f(x) has an average value of 0 over the interval [1, 3].

Therefore, the answer is not listed among the options provided, since all the given options are negative, and the actual average value is 0.