Question

What is the average rate of change of the function $f\left( x \right)=2{{x}^{2}}-3x-1$ on the interval $\left[ 2,2.1 \right]$?

Answer 1

For solving this type of questions you should know about the average rate of change on this interval. And if we can calculate the average rate of change on the intervals by the formula $\dfrac{f\left( b \right)-f\left( a \right)}{b-a}$, here $a$ and $b$ are interval $\left[ a,b \right]$ or the average rate of change on this interval should be approximately equal to the value of the derivative (rate of change) of the function halfway through the interval.

Complete step-by-step solution:
According to the question we have to calculate the average rate of change of the function $f\left( x \right)=2{{x}^{2}}-3x-1$ on the interval $\left[ 2,2.1 \right]$. We can calculate the average rate of change by it’s formula, which is given by, $\dfrac{f\left( b \right)-f\left( a \right)}{b-a}$, where $a$ and $b$ are interval $\left[ a,b \right]$. If we see it by an example then it will be much better. So, if $f\left( x \right)=4{{x}^{2}}-2x+2$ and we have to find its average rate of change at interval $\left[ 1,1.2 \right]$. For finding the average rate of change of the function we use the formula, $\dfrac{f\left( b \right)-f\left( a \right)}{b-a}$. So, here $b=1.2$ and $a=1$. So, the average rate of change will be,
$\dfrac{5.36-4}{0.2}=\dfrac{1.36}{0.2}=6.8$
So, the average rate of change is 6.8.
Now if we take our question then, $f\left( x \right)=2{{x}^{2}}-3x-1$ and $a=2$ and $b=2.1$. So, the average rate of change will be, $\dfrac{f\left( b \right)-f\left( a \right)}{b-a}$,
$\Rightarrow$ average rate of change $=\dfrac{f\left( 2.1 \right)-f\left( 2 \right)}{2.1-2}$
$\begin{aligned} & \because f\left( 2.1 \right)=1.52,f\left( 2 \right)=1 \\\ & \Rightarrow \dfrac{1.52-1}{0.1}=\dfrac{0.52}{0.1}=5.2 \\\ \end{aligned}$
So, the average rate of change is 5.2.

Note: For calculating the rate of change, it is the difference between the values of the function at interval with the ratio of difference of same interval values. We have to be careful when taking the values from the interval because if anyone will be wrong then it will affect all our function and this will affect our answer and that will be wrong.