QuestionCalculate the average rate of change of $f(x)=x^{2}-2x+8$ from $x=1$ to $x=4$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=x^{}-x+8$ . The x-values are $x_{1}=1$ and $x_{}=4$

STEP 2

The average rate of change of a function from $x_{1}$ to $x_{2}$ is given by the formula $\text{Average rate of change} = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}$

STEP 3

First, we need to calculate the value of the function at $x_{1}$ and $x_{2}$ .
For $x_{1}$ $f(x_{1}) = f(1) = (1)^{2} -2(1) +8$

STEP 4

Calculate the value of $f(x_{1})$ .
$f(x_{1}) = (1)^{2} -2(1) +8 =1 -2 +8 =7$

STEP 5

Now, calculate the value of the function at $x_{2}$ .
$f(x_{2}) = f(4) = (4)^{2} -2(4) +8$

STEP 6

Calculate the value of $f(x_{2})$ .
$f(x_{2}) = (4)^{2} -2(4) +8 =16 -8 +8 =16$

STEP 7

Now that we have the values of $f(x_{1})$ and $f(x_{2})$ , we can substitute these into the formula for the average rate of change.
$\text{Average rate of change} = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} = \frac{16 -7}{4 -1}$

STEP 8

Calculate the average rate of change.
$\text{Average rate of change} = \frac{16 -7}{4 -1} = \frac{}{3} =3$ The average rate of change of the function from $x_{1}$ to $x_{2}$ is3.

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