QuestionFind the average rate of change of $f(x)$ from $x_{1}=2$ to $x_{2}=4$ for $f(x)=\sqrt{3x+1}$ , rounded to the nearest hundredth.

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\sqrt{3 x+1}$ . We are finding the average rate of change from $x_{1}=$ to $x_{}=4$

STEP 2

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

STEP 3

First, we need to find the value of the function $f(x)$ at $x_{1}=2$ . Substitute $2$ into the function $f(x)$ .
$f(x_{1}) = f(2) = \sqrt{3 \cdot2 +1}$

STEP 4

Calculate the value of $f(2)$ .
$f(2) = \sqrt{3 \cdot2 +1} = \sqrt{7}$

STEP 5

Next, we need to find the value of the function $f(x)$ at $x_{2}=4$ . Substitute $4$ into the function $f(x)$ .
$f(x_{2}) = f(4) = \sqrt{3 \cdot4 +1}$

STEP 6

Calculate the value of $f(4)$ .
$f(4) = \sqrt{3 \cdot4 +1} = \sqrt{13}$

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.
$\frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}} = \frac{\sqrt{13} - \sqrt{7}}{4 -2}$

STEP 8

Calculate the average rate of change.
$\frac{\sqrt{13} - \sqrt{7}}{4 -2} = \frac{\sqrt{13} - \sqrt{7}}{2}$

STEP 9

The exact value of the average rate of change is $\frac{\sqrt{13} - \sqrt{7}}{2}$ . However, the problem asks for the answer rounded to the nearest hundredth. Use a calculator to find this value.
$\frac{\sqrt{13} - \sqrt{7}}{2} \approx.12$ The average rate of change of $f(x)$ from $x_{}=2$ to $x_{2}=4$ is approximately $.12$ .

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