Average Rate of Change and Percent Change
Emma runs a short-video channel. In January it had 1,000 followers. By March it had 1,400. Did it grow fast? You could say "200 new followers each month, on average." You could also say "40% more followers." Both statements are true, and they answer different questions. In this section you learn to compute both and decide which one fits the situation.
Net Change and Average Rate of Change
When a quantity changes, there are two natural questions. How much did it change in total? How fast did it change, on average?
Why divide by $b - a$? The net change is the total. Dividing by the length of the interval spreads that total evenly over each unit of $x$, like finding miles per hour from miles and hours.
Given the linear function $f(x) = 3x - 5$, calculate the net change on the interval $[2, 5]$.
Given the linear function $f(x) = 3x - 5$, calculate the net change on the interval $[2, 5]$.
Calculate the average rate of change of the quadratic function $f(x) = x^2 - 4x$ on the interval $[1, 4]$.
Calculate the average rate of change of the quadratic function $f(x) = x^2 - 4x$ on the interval $[1, 4]$.
Given the function $f(x) = 2x^2 + 1$, find the net change and the average rate of change on the interval $[-1, 2]$.
Given the function $f(x) = 2x^2 + 1$, find the net change and the average rate of change on the interval $[-1, 2]$.
From Tables and Graphs: The Secant Line
You do not always get a formula. Often you get a table or a graph. The recipe is the same: find $f(a)$ and $f(b)$, then compute $\dfrac{f(b) - f(a)}{b - a}$.
With a graph, read the $y$-values at the two ends of the interval. Choose points that sit exactly on grid intersections so your readings are accurate. Connect the two points with a straight line, called a secant line. Notice what its slope is: rise over run, $\dfrac{\Delta y}{\Delta x}$.
From the following table of values, which shows the distance $d$ (in miles) covered by a cyclist as a function of time $t$ (in hours), calculate the average speed (which…
From the following table of values, which shows the distance $d$ (in miles) covered by a cyclist as a function of time $t$ (in hours), calculate the average speed (which is equivalent to the average rate of change) between $t = 1$ and $t = 3$ hours.
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$t$ (h): $0, 1, 2, 3, 4$
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$d$ (mi): $0, 15, 32, 45, 60$
The following graph shows the monthly revenue of an online store (in thousands of dollars) during the first 6 months of the year. Calculate and compare the average rate…
The following graph shows the monthly revenue of an online store (in thousands of dollars) during the first 6 months of the year. Calculate and compare the average rate of change on the intervals $[1, 3]$ and $[4, 6]$.
Look at the graph representing the number of infected people during an epidemic (in thousands of people) over 8 weeks. Visually identify and then calculate exactly the…
Look at the graph representing the number of infected people during an epidemic (in thousands of people) over 8 weeks. Visually identify and then calculate exactly the average rate of change on the two-consecutive-week interval where this rate reaches its maximum value.
Interpreting the Sign and Comparing Rates
A number alone is not an answer. In a word problem, say what it means, with units.
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A positive rate means the quantity increased on average.
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A negative rate means it decreased on average.
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A rate of zero means the quantity ended where it started.
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Units are "units of $y$ per unit of $x$": dollars per month, feet per second, degrees per hour.
To compare how fast two changes happen, compare their absolute values. The sign only tells the direction.
The temperature $T$ (in $^\circ\text{C}$) of an industrial oven as a function of time $t$ (in hours) is given by the expression $T(t) = -2t^2 + 12t + 20$. Calculate the…
The temperature $T$ (in $^\circ\text{C}$) of an industrial oven as a function of time $t$ (in hours) is given by the expression $T(t) = -2t^2 + 12t + 20$. Calculate the average rate of change of the temperature on the interval $[1, 4]$ and interpret the meaning of the result in the context of the problem.
The profit of a construction company (in thousands of dollars) is modeled by the function $G(x) = -x^2 + 10x - 9$, where $x$ is the month of the year. Calculate the…
The profit of a construction company (in thousands of dollars) is modeled by the function $G(x) = -x^2 + 10x - 9$, where $x$ is the month of the year. Calculate the average rate of change on the interval $[2, 5]$ and on the interval $[6, 8]$. What do the signs obtained in each case indicate about the progress of the company?
Percent Change
Back to Emma's channel. Going from 1,000 to 1,400 followers is a net change of $400$. Is that a lot? It depends on where you started. Gaining 400 followers is huge for a new channel and tiny for a channel with a million followers. To compare the change with the starting value, use percent change.
Why divide by $f(a)$? A percent always measures "out of the starting amount." Going from 80 to 100 is $\dfrac{20}{80} = 25\%$ more than you began with.
The number of active users of a mobile app went from $2000$ to $2500$ over the course of a month. Calculate the relative growth rate expressed as a percentage.
The number of active users of a mobile app went from $2000$ to $2500$ over the course of a month. Calculate the relative growth rate expressed as a percentage.
The water level of a reservoir (in cubic feet) during the summer months follows the function $N(t) = \frac{120}{t+1}$, where $t$ is the number of months elapsed ($t=0$…
The water level of a reservoir (in cubic feet) during the summer months follows the function $N(t) = \frac{120}{t+1}$, where $t$ is the number of months elapsed ($t=0$ is the beginning of June, $t=1$ is the beginning of July, etc.). Calculate the average rate of change between the beginning of June and the beginning of August ($t=2$) and the percentage growth rate in that same period.
The total cost of producing $x$ units of an item is given by the function $C(x) = 0,1x^2 + 5x + 200$. If the production volume in the factory goes from $40$ to $60$…
The total cost of producing $x$ units of an item is given by the function $C(x) = 0,1x^2 + 5x + 200$. If the production volume in the factory goes from $40$ to $60$ units, calculate the increase in total cost (net change), the average cost for each additional unit produced (average rate of change), and the percentage growth rate of the total cost on that interval.
Exponential Functions: Constant Percent Change
Look at $f(x) = 2^x$. Its values double each time $x$ increases by 1.
Each step is a 100% increase, but the net change grows: $+2$, then $+4$, then $+8$. The average rate of change keeps getting bigger, while the percent change stays the same. That is the signature of an exponential function.
For longer intervals, compute directly from the values. Two 5% raises are not a 10% raise, as the next example shows.
Try $b < 1$ with the sliders. The curve falls, and each step is a constant percent decrease.
The growth of a population of bacteria in a culture is given by the function $P(t) = 100 \cdot 2^t$, where $t$ is measured in hours. Calculate the percentage growth rate…
The growth of a population of bacteria in a culture is given by the function $P(t) = 100 \cdot 2^t$, where $t$ is measured in hours. Calculate the percentage growth rate of the population on the time interval $[0, 2]$.
The value of a car depreciates according to the function $V(t) = 24000 \cdot 0,85^t$, where $t$ is the years elapsed since its purchase and $V(t)$ is the price in…
The value of a car depreciates according to the function $V(t) = 24000 \cdot 0,85^t$, where $t$ is the years elapsed since its purchase and $V(t)$ is the price in dollars. Calculate the percentage growth rate in the first year (interval $[0, 1]$) and explain the practical meaning of its sign.
Consider the exponential function $f(x) = 2^x$. Calculate the percentage growth rate on the intervals $[1, 2]$, $[2, 3]$, and $[3, 4]$. From the results obtained, prove…
Consider the exponential function $f(x) = 2^x$. Calculate the percentage growth rate on the intervals $[1, 2]$, $[2, 3]$, and $[3, 4]$. From the results obtained, prove algebraically that for any interval of the form $[n, n+1]$ (with $n \in \mathbb{N}$), the percentage growth rate of this function is always constant.
Linear Functions: Constant Rate of Change
For a linear function $f(x) = mx + b$, the average rate of change is the same on every interval, and it equals the slope $m$.
The percent change of a linear function, however, is not constant. It depends on the starting value.
This gives you a way to tell the two growth types apart. A linear function has a constant average rate of change. An exponential function has a constant percent change for equal-length intervals.
The price of a tech product was $\$800$ at its launch. Two months later, its price dropped to $\$640$. Assuming a linear decay model, calculate the monthly average rate…
The price of a tech product was $\$800$ at its launch. Two months later, its price dropped to $\$640$. Assuming a linear decay model, calculate the monthly average rate of change and the total percentage growth rate over those first two months.
Prove algebraically that the average rate of change of any linear function $f(x) = mx + n$ on any interval $[a, b]$ (with $a \neq b$) is always constant and equal to its…
Prove algebraically that the average rate of change of any linear function $f(x) = mx + n$ on any interval $[a, b]$ (with $a \neq b$) is always constant and equal to its slope $m$.
A marketing company analyzes the performance of two advertising campaigns. Campaign A generates visits to a website according to the linear model $V_A(t) = 100t + 500$,…
A marketing company analyzes the performance of two advertising campaigns. Campaign A generates visits to a website according to the linear model $V_A(t) = 100t + 500$, and campaign B according to the exponential model $V_B(t) = 100 \cdot 1,5^t$ (where $t$ is measured in days). Calculate the percentage growth rate of both campaigns on the interval $[2, 4]$ and determine which one experiences a greater relative growth in that period.
Finding Unknowns
Some problems give you a rate and ask for a missing number. Write the average rate of change as an expression, set it equal to the given rate, and solve.
Knowing that the average rate of change of the function $f(x) = ax^2 - 3x$ on the interval $[1, 3]$ is equal to $5$, determine the numerical value of the parameter $a$.
Knowing that the average rate of change of the function $f(x) = ax^2 - 3x$ on the interval $[1, 3]$ is equal to $5$, determine the numerical value of the parameter $a$.
The position of an object in free fall is given by $e(t) = 5t^2$ (where $e$ is measured in feet and $t$ in seconds). Find the exact instant of time $t > 1$ for which the…
The position of an object in free fall is given by $e(t) = 5t^2$ (where $e$ is measured in feet and $t$ in seconds). Find the exact instant of time $t > 1$ for which the average speed (which coincides with the average rate of change) on the interval $[1, t]$ is exactly $25$ ft/s.
Determine the value of the parameter $k$ so that the percentage growth rate of the function $f(x) = \frac{k}{x+2}$ on the interval $[2, 8]$ is exactly $-60\%$. Does the…
Determine the value of the parameter $k$ so that the percentage growth rate of the function $f(x) = \frac{k}{x+2}$ on the interval $[2, 8]$ is exactly $-60\%$. Does the final result depend on the initial value of $k$? Explain your reasoning algebraically.
Back to the start: Emma's channel grew by $\dfrac{1400 - 1000}{2} = 200$ followers per month on average, which is the average rate of change. It also grew by $\dfrac{400}{1000} = 40\%$, which is the percent change. The first tells you how fast, the second tells you how much compared with where she started.