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 \([2,5]\).

Show Answer

The answer is \( 7 \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{5^2-2^2}{5-2}\)

\(\,\,\,\,\,\,\displaystyle\frac{25-4}{5-2}\)

\(\,\,\,\,\,\,\displaystyle\frac{21}{3}\)

\(\,\,\,\,\,\,\)The answer is \( 7 \)

\(\textbf{2)}\) Find the average rate of change of \(f(x)=2x^3\) over the interval \([0,4]\).

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The answer is \( 32 \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{2\cdot 4^3-2\cdot 0^3}{4-0}\)

\(\,\,\,\,\,\,\displaystyle\frac{2 \cdot 64 -2 \cdot 0}{4-0}\)

\(\,\,\,\,\,\,\displaystyle\frac{128}{4}\)

\(\,\,\,\,\,\,\)The answer is \( 32 \)

\(\textbf{3)}\) Find the average rate of change of \(f(x)=\cos{x}\) over the interval \([0,2\pi]\).

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The answer is \( 0 \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{\cos{(2\pi)}-\cos{(0)}}{2\pi-0}\)

\(\,\,\,\,\,\,\displaystyle\frac{1-1}{2\pi}\)

\(\,\,\,\,\,\,\displaystyle\frac{0}{2\pi}\)

\(\,\,\,\,\,\,\)The answer is \( 0 \)

\(\textbf{4)}\) Find the average rate of change of \(f(x)=3x\) over the interval \([3,8]\).

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The answer is \( 3 \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{3(8)-3(3)}{8-3}\)

\(\,\,\,\,\,\,\displaystyle\frac{24-9}{8-3}\)

\(\,\,\,\,\,\,\displaystyle\frac{15}{5}\)

\(\,\,\,\,\,\,\)The answer is \( 3 \)

\(\textbf{5)}\) Find the average rate of change of \(f(x)=|x|\) over the interval \([-3,2]\).

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The answer is \( -\frac{1}{5} \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{|2|-|-3|}{2-(-3)}\)

\(\,\,\,\,\,\,\displaystyle\frac{2-3}{2+3}\)

\(\,\,\,\,\,\,\displaystyle\frac{-1}{5}\)

\(\,\,\,\,\,\,\)The answer is \( -\frac{1}{5} \)

\(\textbf{6)}\) Find the average rate of change of \(f(x)=x^3\) over the interval \([1,3]\).

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The answer is \( 13 \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{3^3-1^3}{3-1}\)

\(\,\,\,\,\,\,\displaystyle\frac{27-1}{2}\)

\(\,\,\,\,\,\,\displaystyle\frac{26}{2}\)

\(\,\,\,\,\,\,\)The answer is \( 13 \)

\(\textbf{7)}\) Find the average rate of change of \(f(x)=\sqrt{x}\) over the interval \([1,9]\).

Show Answer

The answer is \( \frac{1}{4} \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{\sqrt{9}-\sqrt{1}}{9-1}\)

\(\,\,\,\,\,\,\displaystyle\frac{3-1}{8}\)

\(\,\,\,\,\,\,\displaystyle\frac{2}{8}\)

\(\,\,\,\,\,\,\)The answer is \( \frac{1}{4} \)

\(\textbf{8)}\) Find the average rate of change of \(f(x)=\ln{x}\) over the interval \([1,4]\).

Show Answer

The answer is \( \frac{\ln{4}}{3} \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{\ln{4}-\ln{1}}{4-1}\)

\(\,\,\,\,\,\,\displaystyle\frac{\ln{4}-0}{4-1}\)

\(\,\,\,\,\,\,\displaystyle\frac{\ln{4}}{3}\)

\(\,\,\,\,\,\,\)The answer is \( \frac{\ln{4}}{3} \)

\(\textbf{9)}\) Find the average rate of change of \(f(x)=e^x\) over the interval \([0,2]\).

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The answer is \( \displaystyle \frac{e^2-1}{2} \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{e^2-e^0}{2-0}\)

\(\,\,\,\,\,\,\displaystyle\frac{e^2-1}{2}\)

\(\,\,\,\,\,\,\)The answer is \( \displaystyle \frac{e^2-1}{2} \)

\(\textbf{10)}\) Find the average rate of change of \(f(x)=x^2 – 4x\) over the interval \([2,6]\).

Show Answer

The answer is \( 4 \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{(6^2 – 4 \cdot 6) – (2^2 – 4 \cdot 2)}{6-2}\)

\(\,\,\,\,\,\,\displaystyle\frac{(36 – 24) – (4 – 8)}{4}\)

\(\,\,\,\,\,\,\displaystyle\frac{12 + 4}{4}\)

\(\,\,\,\,\,\,\displaystyle\frac{16}{4}\)

\(\,\,\,\,\,\,\)The answer is \( 4 \)

\(\textbf{11)}\) Find the average rate of change of \(f(x)=\sin{x}\) over the interval \([0,\pi]\).

Show Answer

The answer is \( 0 \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{\sin{(\pi)} – \sin{(0)}}{\pi – 0}\)

\(\,\,\,\,\,\,\displaystyle\frac{0 – 0}{\pi}\)

\(\,\,\,\,\,\,\)The answer is \( 0 \)

Challenge Questions

\(\textbf{12)}\) Find the average rate of change of \(f(x)=x^2\) over the interval \([a,b]\).

Show Answer

The answer is \( b+a \)

Show Work

\(\,\,\,\,\,\, \text{Average Rate of Change}=\displaystyle\frac{f(b)-f(a)}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{b^2-a^2}{b-a}\)

\(\,\,\,\,\,\,\displaystyle\frac{(b-a)(b+a)}{(b-a)}\)

\(\,\,\,\,\,\,\)The answer is \( b+a \)

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)

In Summary

Introduction to average rate of change: Average rate of change is a fundamental concept in calculus that measures the change in a function’s independent variable over the change in the dependent variable. It is an important tool for analyzing and predicting the behavior of various systems and it is used in a wide range of fields, including physics, engineering, economics, and biology.

The average rate of change of a function f(x) over the interval [a, b] is defined as

\(ARC=\displaystyle \frac{f(b)-f(a)}{b-a}\)

Average rate of change is typically taught in a calculus course, but it is closely tied to slope of a line, which shows up in earlier math classes. It is one of the foundational concepts that mixes with limits to give us derivatives.

One common mistake when working with average rate of change is confusing it with the instantaneous rate of change, which is the limit of the average rate of change as the time interval becomes infinitesimally small. Even though it occurs in a calculus class, the average rate of change doesn’t use calculus to find it.

Related topics to average rate of change: Average rate of change is closely related to several other concepts in calculus, including limits, derivatives, and integrals. It is also related to other mathematical concepts such as slope, linear approximation, and Taylor series. Understanding average rate of change is important for analyzing and predicting the behavior of various systems, and it has numerous applications in fields such as physics, engineering, economics, and biology.

Real world examples of average rate of change

The average rate of change of a stock price over a period of time can be calculated by dividing the change in the stock price by the length of time over which the change occurred. For example, if a stock price increases from $10 to $12 over a period of 2 days, the average rate of change is \(\frac{$12 – $10}{2} = $1\)/day.

The average rate of change of a company’s profits over a period of time can be calculated by dividing the change in profits by the length of time over which the change occurred. For example, if a company’s profits increase from $100,000 to $130,000 over a period of 3 years, the average rate of change is \(\frac{$130,000 – $100,000}{3 \text{ years}} = $10,000\)/year.

The average rate of change of a person’s weight over a period of time can be calculated by dividing the change in weight by the length of time over which the change occurred. For example, if a person’s weight increases from 150 pounds to 165 pounds over a period of 3 months, the average rate of change is \(\frac{165 – 150}{3 \text{ months}} = 5\) pounds/month.

The average rate of change of the temperature in a room over a period of time can be calculated by dividing the change in temperature by the length of time over which the change occurred. For example, if the temperature in a room increases from 72 degrees Fahrenheit to 78 degrees Fahrenheit over a period of 6 hours, the average rate of change is \(\frac{78 – 72}{6 \text{ hours}} = 1\) degree Fahrenheit/hour.

The average rate of change of the distance traveled by a car over a period of time can be calculated by dividing the change in distance traveled by the length of time over which the change occurred. For example, if a car travels a distance of 200 miles in 4 hours, the average rate of change is \(\frac{200 \text{ miles}}{4 \text{ hours}} = 50\) miles/hour.

Topics that use average rate of change

Economic modeling: Average rate of change is also used in economics to model the relationship between different variables, such as supply and demand, or the impact of changes in interest rates on the economy.

Population growth: In biology and demography, average rate of change is used to model the growth of a population over time. For example, ecologists might use average rate of change to predict the impact of a new predator on a population of prey animals.

Medicine: Average rate of change is used in medicine to study the effectiveness of different treatments over time. For example, a doctor might use average rate of change to compare the effectiveness of two different medications in reducing blood pressure.

Engineering: Average rate of change is also used in engineering to design and analyze systems that change over time, such as electrical circuits, mechanical systems, and control systems. Engineers might use average rate of change to determine the stability of a system or to optimize its performance.

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