Question

How do you find the average rate of change of \[f\left( x \right) = {x^2} - 6x + 8\] over the interval \[\left[ {4,9} \right]\] ?

Answer 1

Hint: Here we are given with a function and an interval. We need to find the average rate of change of that function over the given interval. This change is given by finding the values of that function for lower and higher intervals. Then taking the ratio of difference of value of function for higher interval and lower interval to the difference of intervals. This is given by the formula,
\[\dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]
Where b and a are the higher and lower intervals respectively.

Complete step by step solution:
Here we are given that \[f\left( x \right) = {x^2} - 6x + 8\] and \[\left[ {4,9} \right]\] gives the interval as \[\left[ {a,b} \right]\].
Now let’s find the values \[f\left( b \right)\& f\left( a \right)\].
\[f\left( b \right) = f\left( 9 \right) = {9^2} - 6 \times 9 + 8\]
\[f\left( b \right) = f\left( 9 \right) = 81 - 54 + 8\]
On calculating we get,
\[f\left( b \right) = f\left( 9 \right) = 35\]
Now find the value of other interval.
\[f\left( a \right) = f\left( 4 \right) = {4^2} - 6 \times 4 + 8\]
\[f\left( a \right) = f\left( 4 \right) = 16 - 24 + 8\]
On calculating we get,
\[f\left( a \right) = f\left( 4 \right) = 0\]
Now we will place the values in the formula to find the value of average rate of change of function
\[\dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}} = \dfrac{{35 - 0}}{{9 - 4}}\]
On further calculations we get,
\[\dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}} = \dfrac{{35}}{5}\]
On dividing we get,
\[\dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}} = 7\]
This is the correct answer.

Note: Here the point to be noted is only the values of the function for the given interval. Because the average rate is just the difference between the interval values for the given function. Sometimes in some cases the rate is also related to the differentiation also. We call it a derivative also. Rates of change of growth or rate of change of temperature are given by this way.