Exercises: Growth rate

Given the linear function $f(x) = 3x - 5$, calculate the net change on the interval $[2, 5]$.

Calculate the average rate of change of the quadratic function $f(x) = x^2 - 4x$ on the interval $[1, 4]$.

The number of active users of a mobile app went from $2000$ to $2500$ over the course of a month. Calculate the relative growth rate expressed as a percentage.

From the following table of values, which shows the distance $d$ (in miles) covered by a cyclist as a function of time $t$ (in hours), calculate the average speed (which is equivalent to the average rate of change) between $t = 1$ and $t = 3$ hours.

• $t$ (h): $0, 1, 2, 3, 4$

• $d$ (mi): $0, 15, 32, 45, 60$

Calculate the average rate of change of the inverse variation function $f(x) = \frac{12}{x}$ on the interval $[2, 6]$.

The price of a share of stock went from $\$15$ to $\$18$. What was its percentage growth rate during that period?

Given the function $f(x) = 2x^2 + 1$, find the net change and the average rate of change on the interval $[-1, 2]$.

Look at the graph of the function $f(x)$ and calculate the average rate of change on the interval $[0, 3]$.

The temperature $T$ (in $^\circ\text{C}$) of an industrial oven as a function of time $t$ (in hours) is given by the expression $T(t) = -2t^2 + 12t + 20$. Calculate the average rate of change of the temperature on the interval $[1, 4]$ and interpret the meaning of the result in the context of the problem.

The growth of a population of bacteria in a culture is given by the function $P(t) = 100 \cdot 2^t$, where $t$ is measured in hours. Calculate the percentage growth rate of the population on the time interval $[0, 2]$.

Knowing that the average rate of change of the function $f(x) = ax^2 - 3x$ on the interval $[1, 3]$ is equal to $5$, determine the numerical value of the parameter $a$.

The following graph shows the monthly revenue of an online store (in thousands of dollars) during the first 6 months of the year. Calculate and compare the average rate of change on the intervals $[1, 3]$ and $[4, 6]$.

The value of a car depreciates according to the function $V(t) = 24000 \cdot 0,85^t$, where $t$ is the years elapsed since its purchase and $V(t)$ is the price in dollars. 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 feet) during the summer months follows the function $N(t) = \frac{120}{t+1}$, where $t$ is the number of months elapsed ($t=0$ 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 profit of a construction company (in thousands of dollars) is modeled by the function $G(x) = -x^2 + 10x - 9$, where $x$ is the month of the year. Calculate the average rate of change on the interval $[2, 5]$ and on the interval $[6, 8]$. What do the signs obtained in each case indicate about the progress of the company?

The price of a tech product was $\$800$ at its launch. Two months later, its price dropped to $\$640$. Assuming a linear decay model, calculate the monthly average rate of change and the total percentage growth rate over those first two months.

The position of an object in free fall is given by $e(t) = 5t^2$ (where $e$ is measured in feet and $t$ in seconds). Find the exact instant of time $t > 1$ for which the average speed (which coincides with the average rate of change) on the interval $[1, t]$ is exactly $25$ ft/s.

Prove algebraically that the average rate of change of any linear function $f(x) = mx + n$ on any interval $[a, b]$ (with $a \neq b$) is always constant and equal to its slope $m$.

Look at the graph representing the number of infected people during an epidemic (in thousands of people) over 8 weeks. Visually identify and then calculate exactly the average rate of change on the two-consecutive-week interval where this rate reaches its maximum value.

Determine the value of the parameter $k$ so that the percentage growth rate of the function $f(x) = \frac{k}{x+2}$ on the interval $[2, 8]$ is exactly $-60\%$. Does the final result depend on the initial value of $k$? Explain your reasoning algebraically.

A marketing company analyzes the performance of two advertising campaigns. Campaign A generates visits to a website according to the linear model $V_A(t) = 100t + 500$, and campaign B according to the exponential model $V_B(t) = 100 \cdot 1,5^t$ (where $t$ is measured in days). Calculate the percentage growth rate of both campaigns on the interval $[2, 4]$ and determine which one experiences a greater relative growth in that period.

Knowing that a quadratic function of the form $f(x) = ax^2 + bx + c$ has its vertex exactly at the point $(2, -1)$ and has a y-intercept at $(0, 3)$, set up the corresponding system to find its expression, and then calculate its average rate of change on the interval $[1, 4]$.

The total cost of producing $x$ units of an item is given by the function $C(x) = 0,1x^2 + 5x + 200$. If the production volume in the factory goes from $40$ to $60$ units, calculate the increase in total cost (net change), the average cost for each additional unit produced (average rate of change), and the percentage growth rate of the total cost on that interval.

Consider the exponential function $f(x) = 2^x$. Calculate the percentage growth rate on 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.