Solved on Mar 19, 2024

Calculate the average rate of change of f(x)=5x33xf(x) = -5x^3 - 3x on the interval [2,4][2, 4].

STEP 1

Assumptions
1. The function given is f(x)=5x33xf(x)=-5x^{3}-3x.
2. The interval for calculating the average rate of change is [2,4][2,4].

STEP 2

The average rate of change of a function over an interval [a,b][a,b] is calculated using the formula:
Average rate of change=f(b)f(a)ba\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}

STEP 3

First, we need to find the value of the function at the beginning of the interval, which is f(2)f(2).
f(2)=5(2)33(2)f(2) = -5(2)^{3}-3(2)

STEP 4

Calculate f(2)f(2) by evaluating the expression.
f(2)=5(8)3(2)f(2) = -5(8)-3(2)
f(2)=406f(2) = -40-6
f(2)=46f(2) = -46

STEP 5

Next, we need to find the value of the function at the end of the interval, which is f(4)f(4).
f(4)=5(4)33(4)f(4) = -5(4)^{3}-3(4)

STEP 6

Calculate f(4)f(4) by evaluating the expression.
f(4)=5(64)3(4)f(4) = -5(64)-3(4)
f(4)=32012f(4) = -320-12
f(4)=332f(4) = -332

STEP 7

Now that we have f(2)f(2) and f(4)f(4), we can calculate the average rate of change on the interval [2,4][2,4].
Average rate of change=f(4)f(2)42\text{Average rate of change} = \frac{f(4) - f(2)}{4 - 2}

STEP 8

Plug in the values for f(4)f(4) and f(2)f(2) to calculate the average rate of change.
Average rate of change=332(46)42\text{Average rate of change} = \frac{-332 - (-46)}{4 - 2}

STEP 9

Simplify the expression by subtracting and dividing.
Average rate of change=332+462\text{Average rate of change} = \frac{-332 + 46}{2}
Average rate of change=2862\text{Average rate of change} = \frac{-286}{2}
Average rate of change=143\text{Average rate of change} = -143
The average rate of change of the function f(x)=5x33xf(x)=-5x^{3}-3x on the interval [2,4][2,4] is 143-143.

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