Average rate of change measures how much a function changes over an interval compared to how much the input changes. It is found using the slope between two points on a function, which is also called the secant line slope. These problems include polynomial, radical, logarithmic, exponential, trigonometric, absolute value, and symbolic interval examples.
Notes
Average Rate of Change over \([a,b]\)
\(ARC=\displaystyle \frac{f(b)-f(a)}{b-a}\)
Questions & Solutions
\(\textbf{1)}\) Find the average rate of change of \(f(x)=x^2\) over the interval \([2,5]\).
\(\textbf{2)}\) Find the average rate of change of \(f(x)=2x^3\) over the interval \([0,4]\).
\(\textbf{3)}\) Find the average rate of change of \(f(x)=\cos{x}\) over the interval \([0,2\pi]\).
\(\textbf{4)}\) Find the average rate of change of \(f(x)=3x\) over the interval \([3,8]\).
\(\textbf{5)}\) Find the average rate of change of \(f(x)=|x|\) over the interval \([-3,2]\).
\(\textbf{6)}\) Find the average rate of change of \(f(x)=x^3\) over the interval \([1,3]\).
\(\textbf{7)}\) Find the average rate of change of \(f(x)=\sqrt{x}\) over the interval \([1,9]\).
\(\textbf{8)}\) Find the average rate of change of \(f(x)=\ln{x}\) over the interval \([1,4]\).
\(\textbf{9)}\) Find the average rate of change of \(f(x)=e^x\) over the interval \([0,2]\).
\(\textbf{10)}\) Find the average rate of change of \(f(x)=x^2 – 4x\) over the interval \([2,6]\).
\(\textbf{11)}\) Find the average rate of change of \(f(x)=\sin{x}\) over the interval \([0,\pi]\).
\(\textbf{12)}\) Find the average rate of change of \(f(x)=2x^2+3x\) over the interval \([1,4]\).
\(\textbf{13)}\) Find the average rate of change of \(f(x)=\frac{1}{x}\) over the interval \([2,5]\).
\(\textbf{14)}\) Find the average rate of change of \(f(x)=x^2+1\) over the interval \([-2,3]\).
\(\textbf{15)}\) Find the average rate of change of \(f(x)=\sqrt{x+1}\) over the interval \([3,8]\).
\(\textbf{16)}\) Find the average rate of change of \(f(x)=x^3-3x\) over the interval \([-1,2]\).
\(\textbf{17)}\) Find the average rate of change of \(f(x)=2^x\) over the interval \([1,4]\).
\(\textbf{18)}\) Find the average rate of change of \(f(x)=\tan{x}\) over the interval \([0,\frac{\pi}{4}]\).
\(\textbf{19)}\) A car travels \(120\) miles in \(3\) hours. Find the average rate of change of distance with respect to time.
\(\textbf{20)}\) The temperature of a cup of coffee changes from \(180^\circ\)F to \(120^\circ\)F over \(15\) minutes. Find the average rate of change of the temperature.
Challenge Questions
\(\textbf{21)}\) Find the average rate of change of \(f(x)=x^2\) over the interval \([a,b]\).
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
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\(\bullet\text{ Horizontal Tangent Line}\)
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\(\bullet\text{ Mean Value Theorem}\)
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\(\bullet\text{ Related Rates}\)
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\(\bullet\text{ Increasing and Decreasing Intervals}\)
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\(\bullet\text{ Intervals of concave up and down}\)
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\(\bullet\text{ Inflection Points}\)
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\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
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\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)