Question

How do you find the average rate of change of $f\left( x \right)=-5x+2$ from $\left[ 2,4 \right]$

Answer 1

Hint: Now we want to find the average rate of change of the given function. The rate of change of function is nothing but the slope of the function. Now to find the average rate of change we will find the slope of the line joining the end points of the function in the interval $\left[ 2,4 \right]$ . We know that the line joining the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ hence we can easily find the average rate of change of the function in $\left[ 2,4 \right]$

Complete step by step solution:
Now let us first understand the concept of rate of change.
Rate of change of the function is nothing but the slope of the function.
Now to find the instantaneous rate of change of function at a point we can differentiate the function at the point.
Now since we want to find the average rate of change we will consider the end points of the function and find the slope.
Now the interval is given $\left[ 2,4 \right]$ .
Let us first find the values $f\left( 2 \right)$ and $f\left( 4 \right)$ .
Now we have $f\left( 2 \right)=-5\left( 2 \right)+2=-8$ and $f\left( 4 \right)=-5\left( 4 \right)+2=-18$
Hence now we have the points $\left( 2,-8 \right)$ and $\left( 4,-18 \right)$ on the function.
Now we want to find the slope of the line joining $\left( 2,-8 \right)$ and $\left( 4,-18 \right)$
Now we know that the slope of line joining $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Hence the slope of line joining $\left( 2,-8 \right)$ and $\left( 4,-18 \right)$ is $\dfrac{-18-\left( -8 \right)}{4-2}=\dfrac{-10}{2}=-5$
Hence the slope of the line is – 5.
Hence the average rate of change of the function $f\left( x \right)=-5x+2$ in $\left[ 2,4 \right]$ is – 5.

Note: Now the average rate of change is the total change in function divided by the length of interval. Hence we can directly write the average rate of change as $\dfrac{f\left( b \right)-f\left( a \right)}{b-a}$ . Note that the average rate of change is nothing but the slope of the line joining $\left( a,f\left( a \right) \right)$ and $\left( b,f\left( b \right) \right)$ .