Question

Given a function f (x) = 3x2 + 4, what is the average rate of change of f on the interval [2, 2 + h]?3h + 123h2 + 12h3h2 + 12h + 1616

Answer 1

We are given the following function:

$f(x)=3x^2+4$

We are asked to determine the average rate of change on the interval [2, 2 + h]. To do that we will use the following formula:

$r=\frac{f(b)-f(a)}{b-a}$

in the interval:

$\lbrack a,b\rbrack$

Therefore, we need to evaluate the function at the points x = 2 and x = 2 + h. Evaluating in x = 2 we get:

$f(2)=3(2)^2+4=16$

Now we evaluate at x = 2 + h:

$f(2+h)=3(2+h)^2+4$

Now we solve the square:

$f(2+h)=3(4+4h+h^2)+4$

Now we apply the distributive property:

$f(2+h)=12+12h+3h^2+4=12h+3h^2+16$

Now we use the average rate of change formula:

$r=\frac{f(2+h)-f(2)}{(2+h)-(2)}$

Substituting the values:

$r=\frac{12h+3h^2+16-16}{h}$

Simplifying:

$r=\frac{12h+3h^2}{h}$

Now we take common factor on the numerator:

$r=\frac{h(12+3h)}{h}$

We can cancel out the "h":

$r=12+3h$

Therefore, the average rate of change is 12 + 3h.