The average change

Given the function $f(x)=x^2+x$,

find the average change (AC) in the interval $[0, 10]$ and $[0,2]$.

Using the definition $$AC=\dfrac{y}{x}=\dfrac{f(a+x)-f(a)}{x}$$ Given the interval $[0,10]$,

$$\Delta y=f(10)-f(0)=(10^2+10)-0=110$$

$$\Delta x=10-0=10$$

$$AC=\dfrac{\Delta y}{\Delta x}=\dfrac{110}{10}=11$$

Given the interval $[0,2]$

$$\Delta y=f(2)-f(0)=(2^2+2)-0=6$$

$$\Delta x=2-0=2$$

$$AC=\dfrac{\Delta y}{\Delta x}=\dfrac{6}{2}=3$$

Interval $[0,10]: \ AC=11$

Interval $[0,2]: \ AC=3$