Question

What is the average rate of change of the function over the interval x = 0 to x = 6?

f(x)=2x−1/3x+5 <--- This is a fraction
Enter your answer, as a fraction, in the box.

Answer 1

Answer:

15 2/3, or 47/3

Step-by-step explanation:

I'm going to assume, correctly or not, that you actually meant f(x) = 2x^2 - (1/3)x + 5.  Double check on this right now, please.

If I'm right, then evaluate f(x) at x = 0 and x = 8:

f(0) = 5

and

f(8) = 2(8)^2 - (1/3)(8) + 5 = 128 - 8/3 + 5 = 133  - (2 2/3), or:   130 1/3

Then the average rate of change of f(x) = 2x^2 - (1/3)x + 5 over the interval [0,8] is:

               130 1/3 - 5         125 1/3

a. r. c. = ------------------- = ---------------- = 15 2/3, or 47/3

                    8 - 0                  8

Answer 2

The average rate of change of the function $f(x)=2x-\frac{1}{3} x+5$ over the interval $x = 0$ to $x = 6$ will be $\frac{5}{12}$ .

What is average rate of change of the function ?

Average rate of change of the function is a rate of change of the function, on an average, over the given interval. i.e. values of $x$ and $y$ .

Using slope formula we find Average rate of change of the function.

i.e. $f(x)=\frac{f(x_{2} )-f(x_{1} )}{x_{2}-x_{1}}$

We have,

$f(x)=2x-\frac{1}{3} x+5$

$x_{1} = 0$ to $x_{2}= 6$

So,

First find $f(x_{1})$,

$f(x_{1})=2x-\frac{1}{3} x+5$

$f(0)=2*0-\frac{1}{3} *0+5$

$f(x_{1})=5$

Now, find $f(x_{2})$,

$f(x_{1})=2x-\frac{1}{3} x+5$

$f(2)=2x*2-\frac{1}{3} *2+5$

$f(2)=4-\frac{2}{3}+5=9-\frac{3}{2}$

$f(x_{2})=\frac{15}{2}$

Now, putting values of $f(x_{1})$ and $f(x_{2})$ in given formula of  $f(x)$,

i.e.

$f(x)=\frac{f(x_{2} )-f(x_{1} )}{x_{2}-x_{1}}$

$f(x)=\frac{\frac{15}{2}-5 }{6-0}$

$f(x)=\frac{\frac{15-10}{2} }{6}$

$f(x)=\frac{5} {12}$

So, the average rate of change of the function is $\frac{5}{12}$, which is find out using the slope formula.

Hence, we can say that the average rate of change of the function $f(x)=2x-\frac{1}{3} x+5$ over the interval $x = 0$ to $x = 6$ will be $\frac{5}{12}$ .

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