Question

How do you find the average rate of change of y with respect to x on the interval [1,4] where $ y={{x}^{2}}+x+1 $ ?

Answer 1

Hint: In this question, we need to find average rate of change of y with respect to x for given interval [1,4] if $ y={{x}^{2}}+x+1 $ . For this, we will first take value of $ {{x}_{1}}\text{ and }{{x}_{2}} $ as 1 and 4 respectively. Then we will find the value of y for particular values x as $ {{y}_{1}}\text{ and }{{y}_{2}} $ respectively. At last we will apply the formula $ \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} $ which will give us the value of the rate of change.

Complete step by step answer:
Here we are given the interval as [1,4] and $ y={{x}^{2}}+x+1 $ .
We need to find the rate of change of y with respect to x. For this, let us suppose $ {{x}_{1}} $ as 1 and $ {{x}_{2}} $ as 4. Now let us find the value of y as $ {{y}_{1}},{{y}_{2}} $ for values of $ {{x}_{1}},{{x}_{2}} $ respectively.
Putting $ {{x}_{1}}=1 $ in y we get, $ y={{\left( 1 \right)}^{2}}+1+1=1+1+1=3 $ .
Therefore, we get $ {{y}_{1}}=3 $ .
Putting $ {{x}_{2}}=4 $ in y we get $ y={{\left( 4 \right)}^{2}}+4+1=16+4+1=21 $ .
Therefore we get $ {{y}_{2}}=21 $ .
We know that the average rate of change of y over an interval between 2 points (a,f(a)) and (b,f(b)) is the slope of the secant line connecting between two points. This average rate of change can be calculated using the formula, $ \dfrac{f\left( b \right)-f\left( a \right)}{b-a} $ .
For this, we will have the formula for 2 points $ \left( {{x}_{1}},{{y}_{1}} \right)\text{ and }\left( {{x}_{2}},{{y}_{2}} \right) $ as $ \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} $ .
Putting the values of $ {{x}_{1}},{{x}_{2}} $ , $ {{y}_{1}},{{y}_{2}} $ as found above we get $ \dfrac{21-3}{4-1}=\dfrac{18}{3} $ .
As we know, $ 6\times 3=18 $ so $ \dfrac{18}{3} $ will be 6.
Hence the rate of change of y with respect to x is the interval [1,4] with $ y={{x}^{2}}+x+1 $ is equal to 6.

Note:
Students should note that, if we are taking y first then we need to take $ {{x}_{2}} $ first in the denominator. We can use formula as $ \dfrac{{{y}_{1}}-{{y}_{2}}}{{{x}_{1}}-{{x}_{2}}} $ also. Take care while finding the values of y for both the values of x.