Rate of change, average rate of change and growth rate
A speedometer does not measure the average speed of a whole journey, but the speed at each particular instant. Until now you have only known how to calculate an average: how much a function changes, on average, between two points. In this section that idea is made precise with its geometric interpretation (the gradient of a secant line) and we begin to answer a finer question: what happens to the function exactly at one point, rather than on average over a stretch? That question is the entry point to the concept of the derivative, which will be formalised later on.
Rate of change and average rate of change
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 $t = 1$, $T = 18$…
The following table shows the temperature $T$ (in $^{\circ}C$) of a chemical reaction as a function of time $t$ (in minutes).
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For $t = 0$, $T = 15$
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For $t = 1$, $T = 18$
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For $t = 2$, $T = 24$
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For $t = 3$, $T = 32$
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For $t = 4$, $T = 45$
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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…
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 $[1, 4]$.
Calculate the average rate of change of the quadratic function $f(x) = x^2 - 6x + 8$ on the interval $[1, 4]$.
Geometric interpretation: the secant line
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 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 $[1, 4]$.
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 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.
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…
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 $x = 5$.
Growth rate: relative and percentage change
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.
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.
The value of a car depreciates according to the function $V(t) = 24000 \cdot 0,85^t$, where $t$ is the number of years 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 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 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…
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.
Instantaneous rate of change: narrowing the interval
The idea is simple: you fix the endpoint $a$ and let the other endpoint $b$ get closer and closer to $a$. The secant joining $A(a,f(a))$ with $B(b,f(b))$ keeps rotating and, when $b$ is very close to $a$, it approaches a line that "touches" the graph exactly at $A$ without crossing it: the tangent line.
The views of a viral video grow according to the mathematical model $N(t) = 100 \cdot 2^t$, where $N$ is the number of views in thousands and $t$ is measured in days.…
The views of a viral video grow according to the mathematical model $N(t) = 100 \cdot 2^t$, 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.
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 $h > 0$.
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 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 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.
Comparing rates of change over different stretches
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…
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?
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 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?
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…
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.
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…
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.
Back to the beginning: The speedometer measures the instantaneous rate of change: the average speed calculated over such a tiny time interval that it practically coincides with the gradient of the tangent line at that exact second.