Question

Find the average rate of change of f(x)=3x2−8 on the interval[1,a]. Your answer will be an expression involving a.

a. 6a−8.
b. 6a2−8.
c. 3a2−8a−11.
d. 3a2−8.

Answer 1

The average rate of change of the function f(x) = 3x² − 8 on the interval [1, a] is calculated using the difference quotient, resulting in an expression that simplifies to 3a + 3, which does not match the provided options. There may be an error in the options or in the initial function.

Step-by-step explanation:

To find the average rate of change of the function f(x) = 3x2 − 8 on the interval [1, a], we use the formula for the average rate of change of a function over an interval [x1, x2]:

Average rate of change = ° − f(x1) / x2 − x1

For the given function:

• f(1) = 3(1)2 − 8 = 3 − 8 = -5
• f(a) = 3(a)2 − 8

Now calculate the average rate of change:

Average rate of change = [f(a) − f(1)] / [a − 1]

Average rate of change = [3(a)2 − 8 − (-5)] / [a − 1]

Average rate of change = [3(a)2 − 8 + 5] / [a − 1]

Average rate of change = [3(a)2 − 3] / [a − 1]

Average rate of change = [3a2 − 3] / [a − 1]

Now factor out a 3 from the numerator:

Average rate of change = 3[a2 − 1] / [a − 1]

Note that a2 − 1 can be factored into (a + 1)(a − 1), so:

Average rate of change = 3(a + 1)(a − 1) / [a − 1]

Cancel out the (a − 1) terms:

Average rate of change = 3(a + 1)

Average rate of change = 3a + 3

However, this is not one of the provided options. It appears there may have been an error in the original calculation or in the provided options. Based on the initial function and the steps shown above, the correct average rate of change should be 3a + 3. Please double-check the options and the function.