Solved on Mar 19, 2024

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

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Calculate the average rate of change of f(x)=−5x3−3xf(x) = -5x^3 - 3xf(x)=−5x3−3x on the interval [2,4][2, 4][2,4].

STEP 1

Assumptions1. The function given is f(x)=−5x3−3xf(x)=-5x^{3}-3xf(x)=−5x3−3x.2. The interval for calculating the average rate of change is [2,4][2,4][2,4].

STEP 2

The average rate of change of a function over an interval [a,b][a,b][a,b] is calculated using the formula:Average rate of change=f(b)−f(a)b−a\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}Average rate of change=b−af(b)−f(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).f(2)=−5(2)3−3(2)f(2) = -5(2)^{3}-3(2)f(2)=−5(2)3−3(2)

STEP 4

Calculate f(2)f(2)f(2) by evaluating the expression.f(2)=−5(8)−3(2)f(2) = -5(8)-3(2)f(2)=−5(8)−3(2)f(2)=−40−6f(2) = -40-6f(2)=−40−6f(2)=−46f(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).f(4)=−5(4)3−3(4)f(4) = -5(4)^{3}-3(4)f(4)=−5(4)3−3(4)

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

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

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

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

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

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

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

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

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