Exercises: Rate of change, average rate of change and growth rate

The following table shows the temperature $T$ (in $^{\circ}C$) of a chemical reaction as a function of time $t$ (in minutes).

• For $t = 0$, $T = 15$

• For $t = 1$, $T = 18$

• For $t = 2$, $T = 24$

• For $t = 3$, $T = 32$

• For $t = 4$, $T = 45$

• For $t = 5$, $T = 60$

Calculate the Rate of Change (RC), that is, the total change in temperature $T(5) - T(2)$, and the Average Rate of Change (ARC) on the interval $[2, 5]$.

Given the linear function $f(x) = 4x - 5$, calculate its average rate of change on the intervals $[-2, 1]$ and $[3, 6]$. What conclusion can you draw about the average rate of change of linear functions?

Calculate the average rate of change of the quadratic function $f(x) = x^2 - 6x + 8$ on the interval $[1, 4]$.

The graph shows the growth of a plant, where $h(t)$ is the height in centimetres and $t$ is the time in weeks. From the graph, calculate the average rate of change of the height on the interval $[1, 4]$.

A cyclist travels a distance given by the function $d(t) = 3t^2 + 2t$, where $d$ is measured in metres and $t$ in seconds. Find the average speed of the cyclist, which is equivalent to the average rate of change of position, between the times $t = 2$ and $t = 5$.

Knowing that the average rate of change of a function $f(x)$ on the interval $[2, 6]$ is equal to $5$, and that $f(2) = 7$, determine the value of $f(6)$.

Calculate the average rate of change of the inverse proportion function $f(x) = \frac{24}{x}$ on the interval $[2, 8]$.

The balance $S$ (in euros) of a savings account as a function of time $t$ (in months) is given by the expression $S(t) = 150 + 25t$. Determine the average rate of change in the interval $[0, 6]$ and interpret the numerical result in the context of the problem.

The number of active users of a mobile app went from $2000$ to $2500$ over the course of a month. Calculate the percentage growth rate.

Cian is analyzing the population change of a town in County Galway between the years 2015 ($t = 0$) and 2023 ($t = 8$). The number of registered inhabitants increased from $2\,400$ to $3\,000$. To calculate the average annual rate of change, Cian writes on his exam:

And states in his conclusion: “The town grew at an average rate of $0.25$ inhabitants per year”.

1. a) Identify the conceptual error that Cian made and explain what quantity he actually calculated instead of the average rate of change (ARC).
2. b) Correct his solution by calculating the true average rate of change (ARC) over the interval $[0, 8]$, stating and interpreting its units.
3. c) Take Cian's erroneous result ($0.25$) and interpret what information it provides if expressed as a percentage growth rate (GR).

The Met Éireann weather station in Cork records the following temperatures during a summer day: at $11:00$ ($t = 11$) the thermometer reads $30\,^{\circ}\text{C}$ and at $16:00$ ($t = 16$) it reaches the daily maximum temperature at $42.5\,^{\circ}\text{C}$.

1. a) Before carrying out any formal calculations, make an intuitive and reasoned estimate: do you think the temperature will have risen more or less than $2\,^{\circ}\text{C}$ per hour on average? Justify your answer in one sentence using the raw data.
2. b) Calculate the change (TV) and the average rate of change (TVM) of the temperature over the interval $[11, 16]$, specifying the units for each result.
3. c) Check whether your initial estimate was correct by comparing it with the exact value of the TVM, and reason: does this average value mean that the temperature rose by exactly that amount in each of the five hours of the interval?

The views of a viral video grow according to the mathematical model $N(t) = 100 \times 2^t$, where $N$ is the number of views in thousands and $t$ is measured in days. Calculate the average rate of change in the intervals $[1, 3]$ and $[3, 5]$. Compare the results and explain what they indicate about the video's rate of growth.

Determine the value of the parameter $a$ such that the average rate of change of the quadratic function $f(x) = ax^2 - 4x + 3$ in the interval $[1, 5]$ is exactly equal to $14$.

The following graph shows the number of viewers $V(t)$, in thousands, of the broadcasts of an e-sports tournament during the first $5$ months of the season. Calculate the average rate of change in the intervals $[1, 3]$ and $[3, 5]$. Interpret the meaning of the signs obtained in each case.

A stone is dropped from a high bridge. The height $h$ (in metres) above the water surface as a function of time $t$ (in seconds) is given by the equation $h(t) = 45 - 5t^2$. Calculate the average rate of change in the interval $[1; 2.5]$. What physical quantity does this value represent, and why does it have a negative sign?

Given the radical function $f(x) = \sqrt{2x - 3}$, find its average rate of change in the intervals $[2, 6]$ and $[6, 14]$.

The volume $V$ (in litres) of water in an industrial tank being emptied is given by the function $V(t) = 500 - 10t - t^2$, where $t$ is measured in minutes. Calculate the average rate of change in the intervals $[0, 5]$ and $[5, 10]$. In which time interval does the tank empty faster?

Consider the quadratic function $f(x) = x^2 - 3x + 2$. Calculate the average rate of change in the interval $[2, 5]$ and use this value to write the equation of the secant line to the graph of $f(x)$ passing through the points with x-coordinates $x = 2$ and $x = 5$.

Given the following piecewise function, calculate the average rate of change in the interval $[1, 4]$.

$f(x) = \begin{cases} 3x + 1 & \text{if } x \le 2 \\ x^2 + 3 & \text{if } x > 2 \end{cases}$

The value of a car depreciates according to the function $V(t) = 24000 \times 0.85^t$, where $t$ is the number of years since its purchase and $V(t)$ is the price in euros. Calculate the percentage growth rate in the first year (interval $[0, 1]$) and explain the practical meaning of its sign.

The water level of a reservoir (in cubic hectometres) during the summer months follows the function $N(t) = \frac{120}{t+1}$, where $t$ is the number of elapsed months ($t=0$ 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 price of a tech product was $800$ € at launch. Two months later, its price dropped to $640$ €. Assuming a linear decrease model, calculate the monthly average rate of change and the total percentage growth rate in those first two months.

The total transport cost (in euros) charged by a logistics company as a function of the delivery distance $x$ (measured in tens of kilometers, with $x \ge 0$) is given by the function $C(x) = 0.5x^2 + 2x + 10$.

1. a) A transport starts from headquarters ($x = 0$) heading to a location at an unknown distance of $b$ tens of kilometers. Knowing that the average rate of change of the cost along that segment was $5$ euros per ten kilometers (that is, $\text{ARC}[0, b] = 5$), set up the corresponding equation and calculate the exact value of $b$.
2. b) Another customer requests a shipment between two warehouses located at positions $x = 3$ and $x = k$ (with $k > 3$). The customer claims that along that segment the average rate of change was only $3$ euros per ten kilometers ($\text{ARC}[3, k] = 3$). Prove algebraically that it is impossible to find a value $k > 3$ that satisfies this condition and explain geometrically why the shape of the parabola prevents such a low slope in that region.

Based on data from the Central Statistics Office (CSO), the average price per square meter of housing in two counties evolved as follows between the years 2014 ($t = 0$) and 2024 ($t = 10$):

• In Leitrim, it went from $800$ €/m$^2$ to $1\,200$ €/m$^2$.

• In County Dublin, it went from $2\,200$ €/m$^2$ to $3\,000$ €/m$^2$.

1. a) Model the evolution by calculating the average rate of change (ARC) of the price in each county over the period $[0, 10]$. In which of the two did the price grow at a higher absolute annual rate?
2. b) Calculate the percentage growth rate (GR) over that same decade for both counties. Which one experienced a greater relative price increase compared to its starting level?
3. c) Two media outlets headline the news differently: Newspaper A publishes “Dublin doubles Leitrim's annual rate of housing price increase”, while Newspaper B headlines “Housing in Leitrim becomes more expensive over the last decade at a percentage rate far higher than Dublin's”. Explain which mathematical indicator (ARC or GR) supports each headline and argue why neither newspaper is lying.

Obtain the simplified algebraic expression for the average rate of change of the function $f(x) = 2x^2 - 5x + 1$ in the generic interval $[a, a+h]$, where $h > 0$.

The stopping distance $d$ (in metres) of a vehicle as a function of its speed $v$ (in km/h) is modelled by the function $d(v) = 0.005v^2 + 0.2v$. Calculate the average rate of change in the intervals $[40, 60]$ and $[100, 120]$. Give reasons why the average increase in stopping distance is much greater at high speeds.

The graph shows the distance from the starting point $d(t)$ in km of a cyclist over the course of a $5$-hour training session. Calculate the average rate of change over the complete interval $[0, 5]$ and justify, by analysing the graph, in which one-hour interval the cyclist reached the highest instantaneous speed.

Consider the piecewise function:

$f(x) = \begin{cases} x^2 - kx & \text{if } x < 3 \\ 2x + k & \text{if } x \ge 3 \end{cases}$

Determine the value of the parameter $k$ knowing that the average rate of change of $f(x)$ in the interval $[1, 5]$ is equal to $4$.

The number of followers $S$ (in thousands) of a video channel as a function of the number of published videos $x$ follows the model $S(x) = \frac{120x}{x + 10}$. Calculate the average rate of change in the intervals $[10, 20]$ and $[40, 50]$. Using the results obtained, interpret the phenomenon that each new video brings fewer and fewer new followers.

A rectangle is inscribed in a right-angled triangle with base $10$ cm and height $8$ cm, with two of its sides resting on the perpendicular sides. The area of the rectangle as a function of its base $x$ is given by the expression $A(x) = x(8 - 0.8x)$. Calculate the average rate of change of the area in the intervals $[2, 4]$ and $[6, 8]$. Interpret the change of sign of the average rate of change in this geometric context.

Given the rational function $f(x) = \frac{x^2 + 4}{x}$, calculate and compare the average rate of change in the intervals $[1, 2]$ and $[-2, -1]$. Justify analytically why the same result is obtained by using the odd symmetry of the function.

A function $f(x)$ has the property that its average rate of change in any interval $[a, b]$ is always constant and equal to $-4$. If the graph of the function is known to pass through the point $(3, 10)$, determine the algebraic expression for $f(x)$ and calculate the exact value of $f(8)$.

The position of an object in free fall is given by $e(t) = 5t^2$ (where $e$ is measured in metres and $t$ in seconds). Find the exact instant in time $t > 1$ for which the average velocity (which coincides with the average rate of change) in the interval $[1, t]$ is exactly $25$ m/s.

Determine the value of the parameter $k$ such that the percentage growth rate of the function $f(x) = \frac{k}{x+2}$ in the interval $[2, 8]$ is exactly $-60\%$. Does the final result depend on the initial value of $k$? Reason your answer algebraically.

Two friends compare the growth of their video channels. Channel A gains followers according to the linear model $S_A(t) = 100t + 500$, and channel B according to the exponential model $S_B(t) = 100 \times 1.5^t$ (where $t$ is measured in days and $S$ in followers). Calculate the percentage growth rate of both channels in the interval $[2, 4]$ and determine which experiences a greater relative growth in that period.

Consider the exponential function $f(x) = 2^x$. Calculate the percentage growth rate in 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.

The volume of water remaining (in cubic meters) in an agricultural irrigation pond near Limerick after opening a drainage gate is modeled by the function $V(t) = 100 - 5t - t^2$, where $t$ is the time in hours elapsed since the opening ($0 \le t \le 7$). It is necessary to determine the instantaneous rate at which water flows out right at the instant $t = 2$ hours, for which two procedures are proposed:

• Method A (Numerical approximation via successive ARCs): Calculate the average rate of change (ARC) over increasingly smaller time intervals starting from $t = 2$, specifically $[2; 2.5]$ and $[2; 2.1]$, and observe the trend.

• Method B (Generic algebraic calculation): Set up the expression $\text{ARC}[2, 2+h] = \dfrac{V(2+h) - V(2)}{h}$ with a generic increment $h > 0$, simplify the fraction algebraically, and deduce the value as $h$ approaches zero ($h \to 0$).

1. a) Apply Method A: calculate the two numerical ARCs and formulate a conjecture about what the instantaneous emptying speed is at $2$ hours.
2. b) Apply Method B: expand $V(2+h)$, simplify the fraction by eliminating the indeterminate form caused by the variable $h$, and obtain the exact value as $h \to 0$.
3. c) Compare both methods and justify which one provides a rigorous mathematical proof as opposed to intuition, and which one would be more efficient if an engineer had to program the calculation of the instantaneous speed for many different instants.

The power generated (in megawatts, MW) during the first hours of a storm by a wind farm follows the linear model $P_1(t) = 2t + 4$, while an adjustable hydroelectric plant that starts up during the same storm follows the quadratic model $P_2(t) = t^2 - 2t + 4$, where $t$ is measured in hours ($0 \le t \le 6$).

1. a) Verify that over the overall interval $[0, 4]$ both facilities have exactly the same average rate of change ($\text{ARC}[0, 4] = 2\text{ MW/h}$). Reason whether this means that both increased their output at the same rate at all times; to do so, analyze the actual behavior of $P_2(t)$ between $t = 0$ and $t = 1$ and compare it with that of the secant line.
2. b) Determine algebraically for which instant $b > 2$ the growth rate of the hydroelectric plant over the interval $[0, b]$ is exactly $200\,\%$ ($\text{GR}_2[0, b] = 2$). Calculate the ARC over that segment and explain how the ARC, the GR, and the initial value $P_2(0)$ are related.
3. c) We want to find out from what moment the hourly increase of the hydroelectric plant clearly exceeds that of the wind farm. Set up and solve an inequality with the average rates of change over a generic one-hour interval $[a, a+1]$ to determine from which hour $a$ it holds that $\text{ARC}_2[a, a+1] - \text{ARC}_1[a, a+1] > 3\text{ MW/h}$.