Question

The normal daily maximum temperatures \(T\) (in degrees Fahrenheit) for Denver, Colorado, are shown in the table. (Source: National Oceanic and Atmospheric Administration). $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Month } & \text { Jan } & \text { Feb } & \text { Mar } & \text { Apr } & \text { May } & \text { Jun } \\ \hline \text { Temperature } & 43.2 & 47.2 & 53.7 & 60.9 & 70.5 & 82.1 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Month } & \text { Jul } & \text { Aug } & \text { Sep } & \text { Oct } & \text { Nov } & \text { Dec } \\ \hline \text { Temperature } & 88.0 & 86.0 & 77.4 & 66.0 & 51.5 & 44.1 \\ \hline \end{array} \end{aligned} $$(a) Use a graphing utility to plot the data and find a model for the data of the form \(T(t)=a+b \sin (\pi t / 6-c)\) where \(T\) is the temperature and \(t\) is the time in months, with \(t=1\) corresponding to January. (b) Use a graphing utility to graph the model. How well does the model fit the data? (c) Find \(T^{\prime}\) and use a graphing utility to graph the derivative. (d) Based on the graph of the derivative, during what times does the temperature change most rapidly? Most slowly? Do your answers agree with your observations of the temperature changes? Explain.

Short answer

The function \(T(t)\) is a sinusoidal model fitted to the temperature data. Its derivative, \(T^{\prime}(t)\), indicates the rate of temperature change, allowing determination of the points of highest and lowest temperature change. Its validity should be confirmed by checking adequacy with original data. The exact model and its fit will depend on the specific regression analysis performed.

Step-by-step solution

Plotting the Temperature Data

Using a graphing utility, plot the temperature against the time (in months). Observing the plot will let you know it takes a sinusoidal form, which validates choosing a model in the form of \(T(t)=a+b \sin (\pi t / 6-c)\) where T is the temperature and t is the time in months.

Determining the Sinusoidal Model

By looking at the plotted data and applying regression analysis, find the best-fit model in the form of \(T(t)=a+b \sin (\pi t / 6-c)\). Set \(t=1\) to correspond to January. This step will likely involve the use of a graphing calculator or software for the regression analysis and to determine the coefficients a, b, and c.

Evaluating the Fit of the Model

Plot the obtained model from Step 2 alongside the original data and examine how well it fits the data. This can be done by visually assessing how closely the regression line follows the original temperature values and also determining the residual errors.

Find the Derivative of the Temperature Function

Calculate the derivative function, \(T^{\prime}(t)\), of the temperature function. This represents the rate of change of temperature over time. Use the graphing calculator or software to plot the derivative function.

Analyzing the Temperature Change

Interpret the plot of \(T^{\prime}(t)\) to determine when the temperature changes most rapidly and when it changes most slowly. This involves locating peaks (for rapid change) and troughs (for slow change) in the derivative function. These correspond to times when temperature changes most rapidly and most slowly respectively.

Comparison with Initial Observations

Compare these results with the observational understanding of temperature changes from the original data set. Identify if any discrepancies exist and if they do, provide a reasonable explanation for them.