Problem 103 - A Graphical Approach to Precalculus with Limits

Question

Atmospheric Carbon Dioxide The carbon dioxide content in the atmosphere at Barrow, Alaska, in parts per million (ppm) can be modeled with the function $$C(x)=0.04 x^{2}+0.6 x+330+7.5 \sin (2 \pi x)$$ where \(x\) is in years and where \(x=0\) corresponds to \(1970 .\) (Source: Zeilik, M., S. Gregory, and E. Smith. Introductory Astronomy and Astrophysics, Fourth Edition, Saunders College Publishers.) (a) Graph \(C\) for \(5 \leq x \leq 25\). (Hint: For the range, use \(320 \leq y \leq 380\) (b) Define a new function \(C\) that is valid if \(x\) represents the actual year, where \(1970 \leq x \leq 1995\)

Step-by-Step Solution

Answer

(a) Graph with 5 ≤ x ≤ 25 and 320 ≤ y ≤ 380. (b) New function: C(x - 1970) with the updated variable.

Step 1: Understand the Model Function

The given function for carbon dioxide content is \(C(x)=0.04 x^{2}+0.6 x+330+7.5 \sin(2 \pi x)\). Here, \(x\) represents the number of years since 1970. This means \(x = 0\) corresponds to the year 1970, \(x = 1\) corresponds to 1971, and so on.

Step 2: Graph the Function for Given Range

To graph the function \(C(x)\), consider the range \(5 \leq x \leq 25\), which corresponds to the years 1975 to 1995. The graph will be plotted with \(x\) on the horizontal axis ranging from 5 to 25, and \(y\) on the vertical axis ranging from 320 ppm to 380 ppm. Use a graphing calculator or software to plot points and connect them, considering the trigonometric component that introduces oscillations to the model.

Step 3: Redefine the Function by Actual Year

To redefine the function \(C\) based on the actual year \(x\), where 1970 \leq x \leq 1995, substitute \(x\) with \(x - 1970\). This changes our function to \(C(x - 1970) = 0.04(x - 1970)^2 + 0.6(x - 1970) + 330 + 7.5 \sin(2 \pi (x - 1970))\). Simplifying this gives the new function in terms of the actual year \(x\).

Key Concepts

Mathematical Modeling

In precalculus, mathematical modeling is a vital tool for representing real-world scenarios using mathematical expressions. The function given in this exercise models the carbon dioxide content in the atmosphere over time. Here, the model is a combination of both polynomial and trigonometric components, each representing different aspects of the atmospheric data.

• The polynomial part, \(0.04x^2 + 0.6x + 330\), suggests a general trend or growth in the atmospheric carbon dioxide levels over time.
• The trigonometric part, \(7.5 \sin(2 \pi x)\), introduces periodic oscillations, which mimic the regular annual fluctuations in carbon dioxide levels.

This model is essential as it transforms complex, time-dependent phenomena into a simplified mathematical form, allowing for easier analysis and predictions.

Graphing Functions

Graphing functions is an effective way to visualize mathematical relationships. In this case, the function \(C(x)\) is graphed over the range \(5 \leq x \leq 25\), which corresponds to the years 1975 to 1995.

• To begin, set up the graph with years on the horizontal axis (\(x\)) and carbon dioxide levels (in ppm) on the vertical axis (\(y\)).
• Plot key points within this range by calculating \(C(x)\) at several values of \(x\).
• Notice the wave-like pattern in the graph due to the trigonometric function, which represents expected seasonal fluctuations.

This visualization helps in understanding both the steady increase and seasonal variations in carbon dioxide over the chosen timeframe.

Trigonometric Functions

In our model, trigonometric functions such as the sine function play a crucial role in representing periodic phenomena.

• The term \(7.5 \sin(2 \pi x)\) adds periodicity, with a cycle repeated every year, owing to the \(2 \pi\) factor.
• The amplitude of 7.5 parts per million captures the magnitude of variation seen throughout each cycle.
• This periodic component aligns with the natural cyclic patterns like the Earth's seasonal changes, affecting carbon dioxide concentrations.

Understanding trigonometric functions and their application in modeling cyclical processes is essential in fields ranging from meteorology to economics.

Polynomial Functions

Polynomial functions provide a foundation for many mathematical models. In the given function, the polynomial terms \(0.04x^2 + 0.6x + 330\) indicate an underlying trend of carbon dioxide increase over time.

• The quadratic term \(0.04x^2\) implies an accelerating rise, suggesting that the rate of increase is not constant.
• The linear term \(0.6x\) gives a base annual increase in carbon dioxide concentration.
• The constant term \(330\) sets the initial level of carbon dioxide concentration observed in 1970.

Polynomial functions like these are easy to manipulate and adjust when trying to fit data trends, making them powerful tools for initial approximations in mathematical modeling.