Question

find the average rate of change for $f(x) = 3x^2 - 1$ on the interval 1, 4.
interpret your answer within the context of the problem
note:
$y = f(x) = 3x^2 - 1$
solution:
step1: find y when x = 1:
step 2: find y when x = 4:
average rate of change =
interpret your answer:

Explanation:

Step1: Calculate f(1)

Substitute $x=1$ into $f(x)=3x^2-1$:
$f(1)=3(1)^2 - 1 = 3 - 1 = 2$

Step2: Calculate f(4)

Substitute $x=4$ into $f(x)=3x^2-1$:
$f(4)=3(4)^2 - 1 = 3\times16 - 1 = 48 - 1 = 47$

Step3: Apply average rate formula

Use $\frac{f(b)-f(a)}{b-a}$ for $[a,b]=[1,4]$:
$\frac{f(4)-f(1)}{4-1}=\frac{47-2}{3}=\frac{45}{3}=15$

On the interval $[1,4]$, the function $f(x)=3x^2-1$ increases by an average of 15 units for every 1-unit increase in the value of $x$.

Answer:

The average rate of change is 15.