Problems from The average change

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

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

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

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