Question

A) Find the average value of the function over the given interval. (Round your answer to four decimal places.)
f(x) = 16 - x^2 , [-4, 4]
Answer: ???

B) Find all values of x in the interval for which the function equals its average value. (Enter your answers as a comma-separated list. Round your answers to four decimal places.)
Answer: x = ???

Answer 1

Answer:

(a) The average rate of change is 0

(b) The values of x are: $x =4\ or\ x = -4$

Step-by-step explanation:

Given

$f(x) = 16 - x^2; [-4,4]$

Solving (a): The average rate of change (m)

This is calculated using:

$m = \frac{f(b) - f(a)}{b - a}$

Where

$[a,b] = [-4,4]$

So:

$m = \frac{f(4) - f(-4)}{4 - -4}$

$m = \frac{f(4) - f(-4)}{8}$

Solve f(4) and f(-4)

$f(4) = 16 - (4^2) = 16 - 16 = 0$

$f(-4) = 16 - (-4^2) = 16 - 16 = 0$

So:

$m = \frac{f(4) - f(-4)}{8}$

$m = \frac{0 - 0}{8}$

$m = \frac{0}{8}$

$m = 0$

Solving (b): All values of x where $f(x) = m$

In (a), we have:

$m = 0$

So:

$f(x) = 0$

So, we have:

$f(x) = 16 - x^2$

$16 - x^2=0$

Express 16 as $4^2$

$4^2 - x^2=0$

Apply difference of two squares

$(4 - x)(4 + x) = 0$

Split

$4 - x =0; 4 + x = 0$

Solve for x

$x =4\ or\ x = -4$

Answer 2

We will find that the average of the function over the given interval is 0, and the two values of x such that the function is equal to the average are x = -4 and x = 4.

How to find the average of a function over an interval?

To find the average of a function f(x) over an interval [a, b] we need to compute:

$\frac{f(b) - f(a)}{b -a}$

In this case we have:

f(x) = 16 - x^2

And the interval is:

[-4, 4]

We will get:

$\frac{(16 - (4)^2) - (16 - (-4))^2}{4 - (-4)} = \frac{0}{8} = 0$

So the function has an average of 0 over that interval.

B) Here we just need to solve:

f(x) = 16 - x^2 = 0

We will get:

16 - x^2 = 0

16 = x^2

±√16 = x

±4 = x

So the solutions are x = 4 and x = -4.

If you want to learn more about averages of functions, you can read:

https://brainly.com/question/16033358