Average Rate of Change and Percent Growth Rate
You drive from Chicago to St. Louis, about 300 miles, in 5 hours. Your average speed was 60 mph, but your speedometer never has to read exactly 60. How can one number summarize a whole trip, and what does it hide? In this section you will measure how fast a function changes over an interval, in two ways: in absolute terms (miles per hour) and in relative terms (percent).
Net change and average rate of change
Why divide by $b - a$? A long interval usually produces a larger net change just because it is long. Dividing by the width gives the change per one unit of $x$, so intervals of different lengths can be compared. The units of the answer are "units of $y$ per unit of $x$", such as miles per hour, dollars per month, or feet per second.
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 $[-1, 2]$.
Tables, graphs and secant lines
You do not always get a formula. Sometimes you get a table or a graph. The idea is the same: read off the two endpoint values, then compute $\dfrac{\Delta y}{\Delta x}$.
On a graph, the average rate of change has a picture. Join the two points $A(a, f(a))$ and $B(b, f(b))$ with a straight line, called a secant line.
Why? Slope is "rise over run", and here the rise is $\Delta y = f(b) - f(a)$ and the run is $\Delta x = b - a$. It is the same formula you used for lines in Algebra 1, now applied to two points of a curve.
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): $0, 15, 32, 45, 60$
Look at the graph of the function $f(x)$ and calculate the average rate of change on the interval $[0, 3]$.
Look at the graph of the function $f(x)$ and calculate the average rate of change on the interval $[0, 3]$.
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 $[4, 6]$.
Reading rates of change in context
A number like $-16$ is only useful if you can say what it means. Always answer with the sign, the units, and the interval.
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A positive rate means the output increases on average over the interval. A negative rate means it decreases.
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Compare how fast things change by comparing the absolute values of the rates. The sign only tells you the direction.
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The units are (units of $y$) per (unit of $x$).
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?
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.
Percent growth rate
Is a gain of 10 followers big news? It depends on where you started. The average rate of change measures change in absolute terms. The percent growth rate (many books call it the percent change) measures it relative to the starting value.
A negative percent growth rate means a decrease. For example, a drop from $50$ to $40$ is $\dfrac{40 - 50}{50} = -20\%$.
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 price of a share of stock went from $\$15$ to $\$18$. What was its percentage growth rate during that period?
The price of a share of stock went from $\$15$ to $\$18$. What was its percentage growth rate during that period?
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.
Linear functions: a constant rate
Different families of functions have different "signatures" when you look at their rates of change. Lines have the simplest one.
A line changes by the same amount per unit of $x$, but its percent change shrinks as the values get larger. For $L(t) = 40t + 100$: from $t = 0$ to $1$ the value goes $100 \to 140$ ($+40\%$), and from $t = 1$ to $2$ it goes $140 \to 180$ (about $+28.6\%$).
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 $m$.
Exponential functions: a constant percent rate
Exponential functions behave the opposite way. Let $P(t) = A \cdot b^t$ with $A > 0$ and $b > 0$.
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.
Comparing linear and exponential models
| Linear $mx + b$ | Exponential $A b^x$ | |
|---|---|---|
| Average rate of change | constant ($= m$) | grows or shrinks |
| Percent growth rate (equal-width intervals) | changes | constant ($= b - 1$ per unit) |
When two models are given, compute the percent growth rate of each on the same interval, then compare.
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.
Inverse variation and other functions
The same two formulas work for every function. Just evaluate at the endpoints.
Calculate the average rate of change of the inverse variation function $f(x) = \frac{12}{x}$ on the interval $[2, 6]$.
Calculate the average rate of change of the inverse variation function $f(x) = \frac{12}{x}$ on the interval $[2, 6]$.
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.
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.
Cost and profit models
Quadratic models appear whenever cost or profit curves bend. The method does not change.
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.
Working backward: finding unknown values
Sometimes you are told the rate and have to find a parameter or an endpoint. Write the average rate of change as an expression, set it equal to the given value, 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.
Knowing that a quadratic function of the form $f(x) = ax^2 + bx + c$ has its vertex exactly at the point $(2, -1)$ and has a y-intercept at $(0, 3)$, set up the…
Knowing that a quadratic function of the form $f(x) = ax^2 + bx + c$ has its vertex exactly at the point $(2, -1)$ and has a y-intercept at $(0, 3)$, set up the corresponding system to find its expression, and then calculate its average rate of change on the interval $[1, 4]$.
Back to the road trip: the average speed of 60 mph is the average rate of change of your distance over $[0, 5]$ hours. It is the constant speed that would cover the same 300 miles in the same time, but your speedometer shows the rate over a tiny interval around one moment, and that can be higher or lower.