QuestionCalculate the average rate of change of $f(x) = x^2 - 2x + 8$ from $x_1 = 1$ to $x_2 = 4$ .

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=x^{}- x+8$ . The initial $x$ value is $x_{1}=1$
3. The final $x$ value is $x_{}=4$

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 find the values of $f(x_{1})$ and $f(x_{2})$ . Let's start with $f(x_{1})$ .
$f(x_{1}) = f(1) = (1)^{2} -2(1) +8$

STEP 4

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

STEP 5

Now, let's find the value of $f(x_{2})$ .
$f(x_{2}) = f(4) = (4)^{2} -2(4) +8$

STEP 6

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

STEP 7

Now that we have the values of $f(x_{1})$ and $f(x_{2})$ , we can plug them 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.
$\text{Average rate of change} = \frac{}{3} =3$ The average rate of change of the function from $x_{1}=1$ to $x_{2}=4$ is3.

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