QuestionFind the average rate of change of $y=2 \times 3^{x}$ from $x=0$ to $x=4$ . Options: A 40.5 B 162 C 158 D 40 E 4

Studdy Solution

STEP 1

Assumptions1. The function is $y= \times3^{x}$ . The interval is $[0,4]$
3. We need to find the average rate of change over this interval

STEP 2

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

STEP 3

We need to find the values of the function at the endpoints of the interval, $f(0)$ and $f()$ .

STEP 4

First, let's find $f(0)$ . Substitute $x=0$ into the function.
$f(0) =2 \times3^{0}$

STEP 5

Calculate the value of $f(0)$ .
$f(0) =2 \times3^{0} =2 \times1 =2$

STEP 6

Now, let's find $f(4)$ . Substitute $x=4$ into the function.
$f(4) =2 \times3^{4}$

STEP 7

Calculate the value of $f(4)$ .
$f(4) =2 \times3^{4} =2 \times81 =162$

STEP 8

Now that we have the values of $f(0)$ and $f(4)$ , we can substitute these into the formula for the average rate of change.
$Average\, rate\, of\, change = \frac{f(4) - f(0)}{4 -0}$

STEP 9

Plug in the values for $f(4)$ and $f()$ to calculate the average rate of change.
$Average\, rate\, of\, change = \frac{162 -2}{4 -}$

STEP 10

Calculate the average rate of change.
$Average\, rate\, of\, change = \frac{162 -2}{4 -0} = \frac{160}{4} =40$ The average rate of change of the function $y=2 \times3^{x}$ over the interval $[0,4]$ is40.

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