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 $t = 1$, $T = 18$

• For $t = 2$, $T = 24$

• For $t = 3$, $T = 32$

• For $t = 4$, $T = 45$

• For $t = 5$, $T = 60$

Calculate the net change and the average rate of change 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 inches 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 illustrates the value of a share of stock $V(t)$ in dollars during the first $5$ months of the year. Calculate the average rate of change on the intervals $[1, 3]$ and $[3, 5]$. Interpret the economic meaning of the signs obtained in each case.

Consider the quadratic function $f(x) = x^2 - 3x + 2$. Calculate the average rate of change on the interval $[2, 5]$ and use this value to write the equation of the secant line to the graph of $f(x)$ that passes through the points with x-coordinates $x = 2$ and $x = 5$.

A cyclist covers a distance given by the function $d(t) = 3t^2 + 2t$, where $d$ is measured in feet and $t$ in seconds. Find the average speed of the cyclist, which is equivalent to the average rate of change of position, between times $t = 2$ and $t = 5$.

The balance $S$ (in dollars) of a savings account as a function of time $t$ (in months) is given by the expression $S(t) = 150 + 25t$. Determine the average rate of change on the interval $[0, 6]$ and interpret the numerical result in the context of the problem.

A stone is dropped from a high bridge. The height $h$ (in feet) relative to the surface of the water as a function of time $t$ (in seconds) is given by the equation $h(t) = 45 - 5t^2$. Calculate the average rate of change on the interval $[1, 2.5]$. What physical quantity does this value represent and why does it have a negative sign?

A population of bacteria grows following the mathematical model $N(t) = 100 \cdot 2^t$, where $N$ is the number of bacteria and $t$ is measured in hours. Calculate the average rate of change on the intervals $[1, 3]$ and $[3, 5]$. Compare the results and explain what they indicate about the growth rate of this population.

The volume $V$ (in gallons) of water in an industrial tank that is 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 on the intervals $[0, 5]$ and $[5, 10]$. In which time interval does the tank empty faster?

The braking distance $d$ (in feet) of a vehicle as a function of its speed $v$ (in mph) is modeled by the function $d(v) = 0.005v^2 + 0.2v$. Calculate the average rate of change on the intervals $[40, 60]$ and $[100, 120]$. Explain with reasoning why the average increase in braking distance is much greater at high speeds.

Calculate the average rate of change of the inverse variation function $f(x) = \frac{24}{x}$ on the interval $[2, 8]$.

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

Given the following piecewise function, calculate the average rate of change on the interval $[1, 4]$.

$f(x) = \begin{cases} 3x + 1 & \text{if } x \le 2 \\ x^2 + 3 & \text{if } x > 2 \end{cases}$

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 $a$ so that the average rate of change of the quadratic function $f(x) = ax^2 - 4x + 3$ on 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 $f(x)$ on the interval $[1, 5]$ is equal to $4$.

Obtain the simplified algebraic expression for the average rate of change of the function $f(x) = 2x^2 - 5x + 1$ on the generic interval $[a, a+h]$, where $h > 0$.

The graph shows the distance to the starting point $d(t)$ in miles of a cyclist throughout a training session of $5$ hours. Calculate the average rate of change on the full interval $[0, 5]$ and justify, by analyzing the graph, in which one-hour interval the cyclist reached the highest instantaneous speed.

A function $f(x)$ has the property that its average rate of change on any interval $[a, b]$ is always constant and equal to $-4$. If it is known that the graph of the function passes through the point $(3, 10)$, determine the algebraic expression of $f(x)$ and calculate the exact value of $f(8)$.

The profit $B$ (in thousands of dollars) generated by an advertising campaign as a function of the investment $x$ (in thousands of dollars) follows the model $B(x) = \frac{120x}{x + 10}$. Calculate the average rate of change on the intervals $[10, 20]$ and $[40, 50]$. Interpret the economic phenomenon of diminishing returns based on the results obtained.

A rectangle is inscribed in a right triangle with a base of $10$ in and a height of $8$ in, with two of its sides resting on the perpendicular legs. 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 on 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 on the intervals $[1, 2]$ and $[-2, -1]$. Justify analytically why the same result is obtained using the fact that the function is odd.