Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Use your technology to find a fourth degree polynomial close to the data in the table below taken from the graph of average solar intensity at Eugene, Oregon in Figure 9.1 .3 on page 405 . $$ \begin{array}{lrrrrrrr} \text { Day } & 1 & 60 & 120 & 180 & 240 & 300 & 360 \\ \text { Day minus 180 } & -179 & -120 & -60 & 0 & 60 & 120 & 180 \\ \text { Solar intensity, kw-hr/m }^{2} & 1.0 & 2.0 & 4.5 & 6.7 & 5.6 & 1.8 & 1.0 \end{array} $$ Your technology will object that the equations to compute the coefficients of a fourth degree polynomial to the Day - Solar Intensity data are ill conditioned (subject to roundoff error); the equations to fit Day minus 180 - Solar Intensity are OK. Therefore, fit a fourth degree polynomial to the Day minus 180 - Solar Intensity data, and interpret the polynomial according for plotting the data and for integration. You should get $$ \begin{array}{c} P(t)=8.733910^{-9}(t-180)^{4}-1.222510^{-7}(t-180)^{3}-4.558710^{-4}(t-180)^{2} \\\ +3.414910^{-3}(t-180)+6.63072 \end{array} $$ a. Plot the data and a graph of your polynomial. b. Compute the integral of \(\int_{0}^{365} P_{2}(t) d t\). c. Compare your answers to the estimate of 1324 computed using the trapezoid rule on 12 intervals of length 30 and one interval of length \(5,\) in Exercise 9.1 .3 on page 404 .

Step-by-Step Solution

Answer

Plot the polynomial and calculate the integral; compare the result to 1324.

Step 1: Enter Data into the Technology Tool

First, input the given data into a technology tool capable of polynomial regression (such as a graphing calculator or software like Excel, Python, or MATLAB). Enter the modified Day data (`Day minus 180`) against the Solar Intensity.

Step 2: Fit the Fourth Degree Polynomial

Use the tool to fit a fourth-degree polynomial to the `Day minus 180` data against the solar intensity data. The polynomial equation you get is expected to be close to \( P(t)=8.7339 \times 10^{-9}(t-180)^{4} - 1.2225 \times 10^{-7}(t-180)^{3} - 4.5587 \times 10^{-4}(t-180)^{2} + 3.4149 \times 10^{-3}(t-180) + 6.63072 \).

Step 3: Plot Data and Polynomial

Plot both the given data points and the polynomial function over the Day range to visually inspect how well the polynomial fits the data. This can be done using graph plotting tools within the software.

Step 4: Set Up the Integral for Calculation

Set up the integral \( \int_{0}^{365} P(t) \, dt \), where \( P(t) \) is the polynomial obtained. Ensure that you are integrating the polynomial over the specified interval from 0 to 365.

Step 5: Calculate the Integral

Use numerical integration methods available in your tool (like definite integral calculation) to compute the integral of the polynomial function over the interval from 0 to 365.

Step 6: Compare with Trapezoid Rule Estimate

The computed integral should be compared to 1324, which is the estimate obtained using the trapezoid rule over 12 intervals of length 30 and one interval of length 5. This comparison helps determine the accuracy of the polynomial fit and integration.

Key Concepts

Numerical Integration

Numerical integration involves finding the approximate value of an integral, especially when the integral of a particular function cannot be solved analytically. In the context of polynomial regression, numerical integration can help determine the accumulated value or area under a curve represented by a polynomial equation. This can be significant in various fields, such as physics and engineering.

To perform numerical integration, one typically uses methods like the trapezoid rule, Simpson's rule, or software tools with built-in capabilities for integration. These methods are particularly useful when working with complex polynomial functions where manual calculation is unfeasible.

The integral \[ \int_{0}^{365} P(t) \, dt \] involves integrating the obtained polynomial over the interval from 0 to 365, which represents the number of days in a year. By performing this integration, one can determine the total solar intensity over the specified period. The accuracy of this result can be compared with estimates derived from simpler numerical techniques like the trapezoid rule.

Data Fitting

Data fitting is a process used to formulate a curve or mathematical function that best represents a series of data points. In this exercise, the goal is to find a fourth-degree polynomial that fits the data on solar intensity across days. This process involves adjusting the coefficients of the polynomial so that the curve aligns closely with the given data points.

To achieve this, one could use various technology tools that perform polynomial regression. These tools generate a polynomial equation where the coefficients are calculated to minimize the deviation between the data points and the curve. The result is a polynomial function, \[ P(t) \], that models the solar intensity data as accurately as possible.

Fitting a polynomial to transform 'Day minus 180' data offers a stable option that minimizes ill-conditioning. It reduces roundoff errors, as it centralizes the data, improving the fitting accuracy and leading to a better representation of the change in solar intensity.

Graphing Tools

Graphing tools are invaluable for visualizing mathematical data and its relationship to fitted polynomial functions. Once a polynomial has been determined via data fitting, it is plotted alongside the original data points. This visual representation allows users to discern how well the polynomial fits the dataset.

Various software applications and graphing calculators can be used for this task. They allow you to input data, calculate the polynomial, and display the graph simultaneously. Visual inspection helps in ensuring that the polynomial correctly models trends in the data, such as peaks and troughs in solar intensity over time.

Moreover, interactive graphing tools provide additional features like zooming and adjusting viewports, which facilitate a detailed examination of the polynomial’s accuracy in different segments of the dataset. These tools are essential for both educational purposes and advanced scientific analysis.

Trapezoid Rule

The trapezoid rule is a method of numerical integration used to approximate the value of a definite integral. It works by dividing the area under a curve into trapezoidal sections rather than using simpler rectangles, which can lead to more accurate results.

For the solar intensity data and its fourth-degree polynomial representation, the trapezoid rule can serve as a comparison method. By constructing trapezoids between data points and summing their areas, we arrive at an estimated value for the integral.\[ \text{Estimate using trapezoid rule} = 1324 \]

This rule is particularly useful when the function under consideration is complex. Breaking the range of integration (0 to 365 days) into smaller intervals makes the calculation of areas more manageable and the estimation more accurate. In this specific context, using 12 intervals of length 30 and one of length 5 provides a robust estimate against which the polynomial's integral can be compared, thereby validating the performance of the polynomial model.