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 meters traveled 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.

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 pesos, are given by the function $B(x) = -2x^2 + 24x - 10$, where $x$ is the amount spent on advertising in thousands of pesos. Calculate the average rate of change of profit if the advertising investment increases from $3$ to $6$ thousand pesos, 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 upward from the top of a cliff. Its height in meters 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 liters, 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?

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 center 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.