Exercises: Identifying the appropriate functional model

The graph shows the monthly user data for a podcast platform during its first year.

Which of these functional models best describes the trend: linear, quadratic, exponential or logarithmic? Justify your choice by ruling out at least two of the other models.

The temperature of a cup of coffee is measured every $5$ minutes after serving:

• a) Plot the points and explain why a linear model is not suitable.

• b) Given that the room temperature is $22\ ^\circ$C, reason which type of model (with which asymptote) best describes the cooling.

You can do it on paper and send a photo with your answer.

True or false? Reason your answer with a counterexample: "Any dataset that grows fits a linear model well."

These data show the cost of printing posters depending on the number of units:

• a) Check that the differences between consecutive costs are constant.

• b) Which functional model is suitable? Determine its expression and calculate how much $100$ posters would cost.

A municipal mobility technician, Hugo, analyzes the rollout of charging points for electric vehicles in his city over four consecutive quarters. The recorded data are:

Hugo states in his technical report: "Given that between the first and the second quarter the number of chargers exactly doubles (from $12$ to $24$), the phenomenon undoubtedly follows an exponential growth of the type $y = 12 \cdot 2^{x-1}$".

1. a) Identify the conceptual error in Hugo's deduction. Why is checking a ratio or quotient in a single interval not enough to conclude that a function is exponential?
2. b) Calculate the first differences $(\Delta y)$ between consecutive quarters and identify with reasoning which functional model is appropriate, indicating its analytical equation.
3. c) Evaluate what value Hugo's erroneous model would predict for quarter $4$ and compare it with the actual data to illustrate the magnitude of his error.

To calibrate a business management simulator, you need to construct a test data table for three independent variables ($A$, $B$, and $C$) evaluated at the discrete instants $x = 0$, $x = 1$, $x = 2$, and $x = 3$. The design conditions are the following:

• Variable $A$: starts at $y(0) = 15$ and its consecutive first differences must be constant and equal to $-3$.

• Variable $B$: starts at $y(0) = 8$ and the ratio between each value and the preceding one, $\frac{y(x+1)}{y(x)}$, must remain constant and equal to $1{,}5$.

• Variable $C$: has an absolute minimum at the point $(1, 2)$, has constant second differences $\Delta^2 y = 4$, and satisfies $y(0) = 4$.

1. a) Construct the table with the numerical values of the three variables for $x \in \{0, 1, 2, 3\}$.
2. b) Deduce the explicit analytical formula $y = f(x)$ that generates the data for variable $A$ and variable $B$.

The figure shows three scatter plots corresponding to three different phenomena, as shown in the figure.

Match each graph with one of these phenomena and with the functional model that best suits it, justifying each choice:

• 1. The average production cost of a company as a function of the number of units manufactured.
• 1. The followers of an account that grows through recommendations between users.
• 1. A family's monthly transportation expenses as a function of the number of trips made.

The population of a small village with limited access to housing has gone from $800$ to $1.600$ inhabitants in ten years, but the rate of growth is slowing down as available housing is occupied (estimated maximum capacity: $2.000$ inhabitants).

• a) Explain why a pure exponential model would not be realistic in the long term.

• b) Which model with a horizontal asymptote would be more suitable? Describe its shape and its qualitative parameters.

• c) What $30$-year prediction would each model give? Comment on the difference.

A 2nd-year Bachillerato student records the cumulative number of specific Economics terms ($P$) that she manages to retain in long-term memory as a function of the total hours of intensive study ($t$):

1. a) Before performing formal calculations, observe the data and qualitatively estimate whether the marginal return on study (terms learned per additional hour invested) increases, remains constant, or decreases as time progresses. According to this initial estimate, which models would you immediately rule out between linear and exponential?
2. b) Check your estimate by calculating the average rate of change (ARC) of learning in the intervals $[1, 2]$, $[2, 4]$, and $[4, 8]$. What does this calculation confirm about the linearity of the process?
3. c) Observe what happens to the increment of terms each time the cumulative study time doubles. Reason why a logarithmic model of the type $P(t) = a + b \ln t$ describes this cognitive phenomenon better than a second-degree polynomial model, and reflect on what would happen to the validity of the model if $t = 100$ hours.

A cooperative oil mill in Jaén has compiled data from its recent seasons to analyze how the production volume of extra virgin olive oil influences its monthly net profit. The following table shows the profit $B(x)$ (in thousands of euros) according to the volume of bottled oil $x$ (in tens of thousands of liters):

1. a) Prepare the table of first differences ($\Delta B$) and second differences ($\Delta^2 B$) for the observed data. Justify from these results why the observed behavior corresponds to a quadratic model and formally rule out the linear and exponential models.
2. b) Knowing that for a constant step size $h=1$ the second difference satisfies $\Delta^2 B = 2a$, determine the coefficients of the parabola $B(x) = ax^2 + bx + c$ that exactly interpolates the data.
3. c) Interpret the economic meaning of the vertex of the parabola: starting from which bottled volume does the cooperative's total profit begin to decline, and what is the maximum profit it can aspire to achieve?

The weekly cases of a flu epidemic in a district were:

• a) Plot the data and reason why neither an exponential nor a logistic model describes the whole process.

• b) Propose what type of model (or combination of piecewise models) could fit each phase of the phenomenon, justifying your choice.

• c) What reasonable prediction would you make for week $10$?

You can do it on paper and send a photo with your answer.

An analyst fits the monthly electricity consumption data of a municipality (which is high in winter, low in spring and rises again in summer due to air conditioning) with a second-degree polynomial.

• a) Reason why a quadratic model can yield absurd medium-term predictions for this phenomenon.

• b) Which family of functions would be more suitable for a phenomenon that repeats every year? Write down the general form of that model and explain the meaning of its parameters.

• c) What general precaution must be taken when extrapolating any model fitted to data?

Following the liberalization of high-speed rail transport in Spain, a team of analysts from the National Commission on Markets and Competition (CNMC) monitors the market share of train versus airplane on a new corridor. The data for the first four semesters after the entry of new operators are:

To predict the medium- and long-term evolution (for example, in semester $t = 10$), three team members propose different approaches:

• Method 1 (Exponential Model): Proposes $C(t) = 12 \cdot (1{,}4)^{t-1}$, arguing that in the early periods the relative growth ratio is high and constant.

• Method 2 (Parabolic Model): Proposes fitting a second-degree polynomial $C(t) = at^2 + bt + c$, arguing that the simple differences increase from semester to semester.

• Method 3 (Logistic or Capped Model): Proposes a function of the type $C(t) = \dfrac{M}{1 + a e^{-kt}}$, where $M$ represents the theoretical maximum achievable share on the corridor (for example, $M = 85\%$).

1. a) Analytically demonstrate the unfeasibility of Method 1 in the medium term by calculating the share it would predict for semester $t = 8$ and explaining the inconsistency that arises in the context of the problem.
2. b) Explain why Method 2, despite being able to fit the first four points well, will conceptually fail both in the long term if the branches of the parabola tend toward $+\infty$, and if it is forced to have a maximum vertex shortly after $t=4$.
3. c) Reasonably defend why Method 3 is the only one of the three with a structural foundation to model the penetration of a closed market. In your justification, analyze the change in concavity (inflection point) and the presence of the horizontal asymptote in relation to demand saturation.

The National Statistics Institute (INE) collects census population data in a municipality in the Sayago region (Castilla y León), intensely affected by rural depopulation over the last decades:

(The value for $2020$ is rounded to the nearest whole number of people).

1. a) Rigorously analyze whether the population loss fits a linear, a quadratic, or a decreasing exponential rate. To do so, calculate and compare both the consecutive differences and the consecutive ratios between regular 5-year periods.
2. b) Obtain the analytical expression of the functional model $P(t) = P_0 \cdot b^t$, where $t$ is the number of years elapsed since the year $2000$.
3. c) During the 2020 municipal plenary session, a councilor argues: "Between the year 2000 and 2020 we have lost an average of about $118$ people per year; if we continue losing exactly that same number each year, in the year $2034$ the village will have reached $0$ inhabitants (complete depopulation)". Identify which mathematical model the councilor is implicitly using, explain why his hypothesis is inconsistent with the real trend in the data, and deduce what your exponential model actually predicts for the year $2034$ and for the long-term limit ($t \to +\infty$).

Topic challenge. Can a function predict the behaviour of the energy market? As a socioeconomic analyst, you will model electrical power demand according to the time of day using real data from Red Eléctrica de España (REE).

1. Sampling and estimation: Visit the REE portal and take a simple random sample of $n=10$ hourly consumption records on a weekday. In a spreadsheet, calculate the sample mean $\bar{x}$ and the sample variance $s^2_{n-1}$, explaining why $\bar{x}$ is an unbiased estimator of daily average demand.

2. Model identification: Enter the data into GeoGebra and select the most consistent function $f(t)$ (for example, a simple rational function $f(t)=\frac{a\cdot t}{t^2+b}$ or a polynomial function) to describe the observed trend.

3. Analytical study: Determine in your notebook the contextualised domain of definition, axis intercepts, symmetry, intervals of monotonicity with their relative extrema, and infinite branches (asymptotes) of $f(t)$.

4. Final report: Prepare a one-page report that compares the analytical curve with the scatter plot and assesses the validity of the asymptotic model against the estimated socioeconomic reality.

Assessment criteria:

• Correctness in calculating sample estimators, bias, and sample variance.

• Reasoned justification of the functional model chosen with respect to the sample.

• Mathematical rigour in the analytical study (domain, intercepts, monotonicity, and asymptotes).

• Clarity and critical sense in the interpretation of the final report.