Identifying the appropriate functional model

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.

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

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.

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?

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.

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.

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.