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 $[2, 5]$.

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

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 $[1, 4]$.

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 secant line to the graph of $f(x)$ passing through the points with x-coordinates $x = 2$ and $x = 5$.

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?

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.

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 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 $V(t) = 480\,000 \times 0.85^t$, 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 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.

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.

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.

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 $f(6)$.

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 $14$.

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 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.

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 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.

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?

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?

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.

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

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 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.

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.

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.