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 centimeters 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 meters 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 pesos) of a savings account as a function of time $t$ (in months) is given by the expression $S(t) = 3\,000 + 500t$. 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.

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 meters) 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 liters) 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 $V(t) = 480\,000 \times 0.85^t$, where $t$ is the number of years since its purchase and $V(t)$ is the price in pesos. 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 hectometers) 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 $16\,000$ pesos at launch. Two months later, its price dropped to $12\,800$ pesos. Assuming a linear decrease model, calculate the monthly average rate of change and the total percentage growth rate in those first two months.

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 meters) of a vehicle as a function of its speed $v$ (in km/h) is modeled 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 meters 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.