Problem 60 - Precalculus: Functions and Graphs

Question

The average monthly temperatures in "F for two Canadian locations are Iisted in the following tables. $$\begin{array}{|l|llll|} \hline \text { Month } & \text { Jan. } & \text { Feb. } & \text { Mar. } & \text { Apr. } \\ \hline \text { Arctic Bay } & -22 & -26 & -18 & -4 \\ \hline \text { Trout Lake } & -11 & -6 & 7 & 25 \\ \hline \end{array}$$ $$\begin{array}{|l|cccc|} \hline \text { Month } & \text { May } & \text { June } & \text { July } & \text { Aug. } \\ \hline \text { Arctic Bay } & 19 & 36 & 43 & 41 \\ \hline \text { Trout Lake } & 39 & 52 & 61 & 59 \\ \hline \end{array}$$ $$\begin{array}{|l|cccc|} \hline \text { Month } & \text { Sept. } & \text { Oct. } & \text { Nov. } & \text { Dec. } \\ \hline \text { Arctic Bay } & 28 & 12 & -8 & -17 \\ \hline \text { Trout Lake } & 48 & 34 & 16 & -4 \\ \hline \end{array}$$ (a)If January 15 corresponds to \(x=1\), February 15 to \(x=2, \ldots,\) and December 15 to \(x=12,\) determine graphically which of the three polynomials given best models the data. (b)Use the Intermediate value theorem for polynomial functions to approximate an interval for \(x\) when an average temperature of \(0^{\circ} \mathrm{F}\) occurs. (c)Use your choice from part (a) to estimate \(x\) when the average temperature is \(0^{\circ} \mathrm{F}\). \(f(x)=-2.14 x^{2}+28.01 x-55\) \(g(x)=-0.22 x^{3}+1.84 x^{2}+11.70 x-29.90\) \(h(x)=0.046 x^{4}-1.39 x^{3}+11.81 x^{2}-22.2 x+1.03\)

Step-by-Step Solution

Answer

Graph the polynomials, choose the best fit, use the Intermediate Value Theorem, and estimate \(x\) numerically.

Step 1: Understand the Problem

We need to evaluate which polynomial best fits the given temperature data using graphical analysis and later estimate the value of \(x\) at which temperature is \(0^\circ F\). We will also use the Intermediate Value Theorem to find an interval for \(x\) where the temperature crosses \(0^\circ F\).

Step 2: Graph the Data Points

Plot the given average temperatures for each location over the months \(x = 1\) to \(x = 12\). This creates a visual representation of the temperature patterns for Arctic Bay and Trout Lake over the course of a year.

Step 3: Plot the Polynomial Functions

Graph the polynomials \(f(x) = -2.14x^2 + 28.01x - 55\), \(g(x) = -0.22x^3 + 1.84x^2 + 11.70x - 29.90\), and \(h(x) = 0.046x^4 - 1.39x^3 + 11.81x^2 - 22.2x + 1.03\) on the same coordinate plane as the temperature data. Analyze which polynomial graph aligns closest to the temperature plot for both locations.

Step 4: Determine the Best Polynomial Model

Examine the plots to determine which function’s curve most closely follows the trend of the plotted temperature data. This involves visually comparing the curves to see which polynomial provides the best fit for the trajectories of the temperature across the months.

Step 5: Applying the Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function changes sign over an interval, there is at least one root in that interval. Identify temperature points near 0°F and determine intervals \(x\) lies between for both locations where the temperature crosses from negative to positive. This will naturally help us find the corresponding interval for \(x\).

Step 6: Estimate \(x\) Using the Chosen Polynomial

Using the best model from Step 4, solve for \(x\) when the polynomial equals zero, i.e., \(f(x) = 0\), \(g(x) = 0\), or \(h(x) = 0\). Use graphical methods or substitute values iteratively to solve this numerical problem accurately.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that can be applied to solve real-world problems. IVT states that for any continuous function that takes a negative value and a positive value over an interval, there must be at least one point where the function equals zero within that interval. This is particularly useful when dealing with polynomial functions because they are inherently continuous.

In the context of temperature data, we use the IVT to identify periods within the year when the average temperature crossed 0°F. By examining the values and identifying where the temperature moves from a positive to a negative value, or vice-versa, we can find where the average temperature is 0°F. This transition from negative to positive or positive to negative confirms the presence of a real root according to the Intermediate Value Theorem.

Graphical Analysis

Graphical analysis involves visually inspecting graphs to understand trends and make decisions. This is crucial when modeling temperature data with polynomials as it allows us to directly compare our models to actual data points over time.

By plotting the monthly average temperatures and overlaying the polynomial functions on the same graph, we can see which polynomial most closely approximates the observed data. The goal is to see if any polynomial matches the shape and trend of the data, illustrating its ability to represent the temperature changes across the months. Graphical analysis aids in identifying the polynomial that models our temperature data most accurately and helps visualize where temperatures might cross the 0°F mark.

Temperature Data

Temperature data provides real-world information that can be modeled mathematically. In this exercise, average monthly temperatures for Arctic Bay and Trout Lake are given for each month of the year, creating a comprehensive dataset.

Understanding how these temperatures vary through the months can inform us about the seasonal changes in each location. When these temperatures are modeled with polynomials, we gain insights into their trends and patterns, helping us predict future temperatures or identify when certain critical temperatures, like 0°F, are reached. This way, the data serves as the foundation upon which mathematical models are built to forecast and analyze temperature behavior.

Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of an independent variable. When we model temperature data using polynomials, we are essentially trying to create a mathematical version of the data that can predict or describe temperature trends.

In this exercise, three different polynomial functions, each of varying degrees, were considered to see which could most accurately represent the given temperature data. The choice of polynomial impacts the model’s ability to fit the data points accurately. A quadratic, cubic or higher-degree polynomial can potentially provide different levels of accuracy and fitting styles. By choosing the best polynomial model, we aim to minimize error and make more reliable predictions based on our mathematical analysis.