Exercises: Average rate of change of a function

Calculate the average rate of change of the function $f(x) = 3x - 5$ on the interval $[1, 4]$. Does this value match the gradient of the line representing the function?

Find the average rate of change of the quadratic function $g(x) = x^2 - 4x + 1$ on the intervals $[0, 2]$ and $[2, 4]$.

Given the function $h(t) = t^3 + 2t$, determine its average rate of change on the interval $[-1, 2]$.

Calculate the average rate of change of the rational function $f(x) = \frac{2x + 1}{x - 1}$ on the interval $[2, 5]$.

Find the average rate of change of the radical function $p(x) = \sqrt{x + 3}$ on the interval $[1, 6]$.

Look at the graph of the function $f(x)$ shown in the figure. Calculate the average rate of change on the interval $[-2, 2]$. What geometrical interpretation does this value have with respect to the secant line drawn?

The distance in metres travelled by a cyclist as a function of time in seconds is given by the expression $d(t) = 2t^2 + 5t$. What was their average speed between the times $t = 2$ and $t = 5$?

The temperature in degrees Celsius of an industrial oven evolves according to the function $T(t) = 20 + 5t - t^2$, where $t$ is time in minutes. Calculate the average rate of change of temperature during the first $3$ minutes.

Aoife is trying to calculate the average rate of change of the function $f(x) = x^2 - 3x$ on the interval $[2, 5]$. On her exam, she writes the following steps:

1. a) Identify the conceptual error made by Aoife and explain in your own words why the average rate of change is not equivalent to evaluating the function at the width of the interval.
2. b) Calculate $\text{ARC}[2, 5]$ correctly, showing all your steps.
3. c) Based on the sign of the result obtained in the previous part, describe the slope of the secant line passing through the points with abscissas $x = 2$ and $x = 5$.

On a summer day in Cork, weather records indicate that the temperature $T$ (in degrees Celsius) between $10:00$ ($t = 0$) and $16:00$ ($t = 6$) fits a piecewise linear model:

1. a) Calculate the average rate of change of the temperature on the interval $[0, 4]$ and on the interval $[4, 6]$. Do not forget to state the units of measurement for the results.
2. b) Without performing any additional numerical calculations, deduce the value of $\text{ARC}[1, 3]$. Justify your answer using the properties of the average rate of change in affine functions.
3. c) Calculate $\text{ARC}[0, 6]$ over the entire period and interpret its physical meaning.

Determine the value of the parameter $k$ so that the average rate of change of the function $f(x) = kx^2 - 3x + 2$ on the interval $[1, 3]$ is equal to $5$.

Compare the average rate of change of the functions $f(x) = 2^x$ and $g(x) = x^3$ on the interval $[2, 4]$. Which of the two functions experiences greater average growth on that interval?

The number of bacteria in a laboratory culture after $t$ hours is modelled by the function $N(t) = 500 \times e^{0.2t}$. Calculate the average rate of change of the population in the time interval between $t = 0$ and $t = 5$ hours. Express the result in terms of the number $e$.

The monthly profits of a company, in thousands of euros, are given by the function $B(x) = -2x^2 + 24x - 10$, where $x$ is the amount spent on advertising in thousands of euros. Calculate the average rate of change of profit if the advertising investment increases from $3$ to $6$ thousand euros, and interpret the result.

Given the function $f(x) = -x^2 + 6x - 5$, find the equation of the secant line passing through the points on the graph with x-coordinates $x = 1$ and $x = 4$. How is the gradient of this line related to the average rate of change on that interval?

Calculate the average rate of change of the function $f(x) = \frac{1}{x}$ on the general interval $[a, a+h]$, simplifying the resulting expression as much as possible.

A distress flare is launched vertically upwards from the top of a cliff. Its height in metres above sea level after $t$ seconds is given by $h(t) = 80 + 30t - 5t^2$. Determine the average velocity of the flare on the intervals $[1, 3]$ and $[3, 5]$. What does the sign of the results obtained indicate?

The following graph shows the volume of water $V(t)$, in litres, contained in a tank over $8$ hours. From the highlighted points on the graph, calculate the average rate of change of the volume on the intervals $[0, 4]$ and $[4, 8]$. In which period was the emptying faster on average?

Consider the exponential function $f(x) = 2^x$.

1. a) Before carrying out any numerical calculations, observe the general behaviour of an exponential function and estimate, with reasoning, which of the two rates will be greater: $\text{ARC}[0, 2]$ or $\text{ARC}[2, 4]$. Do you expect both to be positive?
2. b) Calculate $\text{ARC}[0, 2]$ and $\text{ARC}[2, 4]$ exactly. Check whether the numerical result confirms your previous estimation.
3. c) Can there be any non-constant polynomial function for which $\text{ARC}[0, 2] = \text{ARC}[2, 4]$ holds? Justify what type of function would satisfy this equality and why this does not happen for $f(x) = 2^x$.

We want to study the average rate of change of the quadratic function $f(x) = 2x^2 - 3x + 1$ on three intervals starting at the same point $a = 1$: $[1, 2]$, $[1; 1.5]$, and $[1; 1.1]$. Two classmates propose different strategies:

• Clara's method: calculate $f(b)$ and $f(1)$ separately for each endpoint and apply the formula $\text{ARC} = \frac{f(b) - f(1)}{b - 1}$ three times independently.

• Daniel's method: express the upper endpoint as $1+h$, deduce a simplified general algebraic formula for $\text{ARC}[1, 1+h] = \frac{f(1+h) - f(1)}{h}$ in terms of the increment $h$, and then specialise by substituting $h = 1$, $h = 0.5$, and $h = 0.1$.

1. a) Apply Daniel's method: expand and simplify as much as possible the algebraic expression for $\text{ARC}[1, 1+h]$, and find the three required values.
2. b) Apply Clara's method only for the interval $[1; 1.1]$ and verify that both methods lead to the exact same result.
3. c) Argue which of the two methods is more advantageous when many intervals sharing the origin $a = 1$ must be evaluated, and explain how the expression obtained by Daniel allows you to deduce the trend of the rate as the increment $h$ approaches zero.

The average rate of change of the function $f(x) = x^3 - 3x^2 + c$ on the interval $[1, b]$ is equal to $6$. Given that $b > 1$, determine the value of $b$. Does the result depend on the value of the constant $c$?

A quadratic function of the form $f(x) = ax^2 + bx + 5$ is known to have an average rate of change of $1$ on the interval $[0, 2]$ and $9$ on the interval $[2, 4]$. Set up a system of equations and find the values of the parameters $a$ and $b$.

Prove analytically that the average rate of change of any linear function $f(x) = mx + n$ on an arbitrary interval $[a, b]$ is always constant and equal to $m$.

Newton's law of cooling states that the temperature of an object approaches ambient temperature exponentially. An engineer models the temperature of a metal component in degrees Celsius after $t$ minutes using $T(t) = 25 + 150 \times (0,8)^t$. Calculate the average rate of change on the intervals $[0, 5]$ and $[5, 10]$. What conclusion do you draw about the rate of cooling over time?

Consider the function $f(x) = \frac{4}{x}$. The secant line to the graph of $f(x)$ passing through the points with x-coordinates $x = 1$ and $x = 4$ forms a right-angled triangle with the positive coordinate axes. Calculate the area of this triangle.

Given the piecewise function:

$f(x) = \begin{cases} x^2 - 2x & \text{if } x \le 2 \\ \frac{4}{x} - 2 & \text{if } x > 2 \end{cases}$

Calculate the average rate of change on the interval $[0, 4]$. Check whether the function is continuous at $x=2$ and explain whether this affects the calculation of the average rate of change.

Calculate the average rate of change of the function $f(x) = \sqrt{x+2}$ on the interval $[2, 2+h]$. Then, determine which value this rate of change approaches as $h$ tends to $0$.

The force of gravitational attraction between the Earth and a satellite varies as a function of the distance $r$ (in thousands of km) from the centre of the Earth according to the expression $F(r) = \frac{4 \times 10^5}{r^2}$. Calculate the average rate of change of the force when the distance increases from $10$ to $20$ thousand km. Interpret the meaning of the sign of the result obtained.

The goal is to determine a quadratic function of the form $f(x) = ax^2 + bx$ (without a constant term) that simultaneously satisfies two conditions on its rate of change:

1. Its average rate of change on the interval $[0, 2]$ is equal to $5$.

2. Its average rate of change on the interval $[1, 4]$ is equal to $11$.

1. a) Set up a system of linear equations from the definition of ARC and calculate the exact values of the coefficients $a$ and $b$.
2. b) For the function obtained, take a symmetric interval of width $2h$ centred at any abscissa $x_0$, that is, $[x_0 - h, x_0 + h]$. Prove algebraically that $\text{ARC}[x_0 - h, x_0 + h]$ depends exclusively on the centre $x_0$ and does not depend at all on the width $h$.
3. c) Check whether this property holds for the cubic function $g(x) = x^3$ by evaluating and comparing $\text{ARC}[-1, 1]$ and $\text{ARC}[-2, 2]$ (both intervals centred at $x_0 = 0$). Conclude with justification whether this independence from the width is a general characteristic or specific to certain families of functions.

The Road Safety Authority (RSA) monitors a $6\text{ km}$ section on the M6 motorway using an average speed camera. In this section, the maximum speed limit is $90\text{ km/h}$. A vehicle enters the section at $12:00:00$ ($t = 0\text{ h}$), and its position $s(t)$ in kilometres within the section is described by the function $s(t) = 96t - 288t^2$, where time $t$ is measured in hours.

1. a) Set up the corresponding equation to calculate the exact time at which the vehicle reaches $6\text{ km}$ and exits the section. Express the elapsed time in exact minutes.
2. b) Calculate the average rate of change of position over the entire journey through the section, $\text{ARC}[0, t_{\text{exit}}]$, and interpret your result as the vehicle's average speed in $\text{km/h}$. Will the average speed camera issue a fine to this driver?
3. c) A preliminary report states: "Given that the average speed in the section was below $90\text{ km/h}$, the vehicle respected the speed limit throughout the entire journey". Refute this statement mathematically: calculate $\text{ARC}[0; 1/60]$ corresponding to the first minute of the journey, compare this value with the legal limit of $90\text{ km/h}$, and reflect on the fundamental difference between the average rate of change over an interval and the actual instantaneous speed.