Problem 104 - A Graphical Approach to Precalculus with Limits

Question

At Mauna Loa, Hawaii, atmospheric carbon dioxide levels in parts per million (ppm) have been measured regularly since 1958 . The function $$L(x)=0.022 x^{2}+0.55 x+316+3.5 \sin (2 \pi x)$$ can be used to model these levels, where \(x\) is in years and \(x=0\) corresponds to \(1960 .\) (Source: Nilsson, A., Greenhouse Earth. John Wiley and Sons.) (a) Graph \(L\) for \(15 \leq x \leq 35 .\) (Hint: For the range, use \(325 \leq y \leq 365\) (b) When do the seasonal maximum and minimum carbon dioxide levels occur? (c) \(L\) is the sum of a quadratic function and a sine function. What is the significance of each of these functions? Discuss what physical phenomena may be responsible for each function.

Step-by-Step Solution

Answer

Part (a): Graph for \(15 \leq x \leq 35\) and \(325 \leq y \leq 365\). Part (b): Seasonal maxima/minima occur at \(x = n + 0.25\) and \(x = n + 0.75\). Part (c): Quadratic term models long-term rise; sine models seasonal variation.

Step 1: Define the Function Components

The given function is \(L(x) = 0.022x^2 + 0.55x + 316 + 3.5 \sin(2\pi x)\). This function is composed of a quadratic term \(0.022x^2 + 0.55x + 316\) and a sinusoidal term \(3.5 \sin(2\pi x)\). The quadratic function models the overall increasing trend in carbon dioxide levels, while the sinusoidal function models the seasonal fluctuations.

Step 2: Set the Range for Graphing

For part (a), we need to graph the function for \(15 \leq x \leq 35\), with \(325 \leq y \leq 365\) as the range. This corresponds to the years between 1975 and 1995, and the \(y\)-range captures expected carbon dioxide levels in parts per million (ppm). Deploy graphing software or use a graphing calculator to visualize this range.

Step 3: Analyze the Sine Function for Maxima and Minima

The maximum value of \(3.5 \sin(2\pi x)\) occurs when \(\sin(2\pi x) = 1\), and the minimum when \(\sin(2\pi x) = -1\). Thus, the maxima of the function occur when \(x = n + 0.25\) and minima when \(x = n + 0.75\), where \(n\) is an integer. These correspond to seasonal changes.

Step 4: Determine the Physical Significance

The quadratic part, \(0.022x^2 + 0.55x + 316\), accounts for the gradual increase in atmospheric carbon dioxide over the years due to factors like fossil fuel combustion and deforestation. The sine function, \(3.5 \sin(2\pi x)\), represents seasonal variations, possibly due to plant growth cycles that cause carbon dioxide levels to rise and fall.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function with the general form of \(ax^2 + bx + c\). In the context of the modeled equation \(L(x) = 0.022x^2 + 0.55x + 316 + 3.5 \sin(2\pi x)\), the quadratic portion is \(0.022x^2 + 0.55x + 316\). This part of the function is crucial for representing long-term trends. It forms a parabola when graphed.The quadratic function in this scenario signifies the gradual increase in CO2 levels over time.
An upward opening parabola indicates a steady increase, suggesting that atmospheric levels are rising.The coefficients \(a\), \(b\), and \(c\) tell us about the direction and shape of the parabola. Here:

• \(a = 0.022\), indicating a gentle upward curve.
• \(b = 0.55\), guiding the slope of the tangent to the maximum or minimum points.
• \(c = 316\), representing the y-intercept or starting value in our case.

Understanding these aspects can help predict future CO2 levels and plan effective environmental policies.

Sine Functions

Sine functions, often in the form \(A \sin(Bx + C) + D\), represent periodic or oscillating behavior, making them vital for modeling seasonal changes. In \(L(x) = 0.022x^2 + 0.55x + 316 + 3.5 \sin(2\pi x)\), the sine component \(3.5 \sin(2\pi x)\) captures this periodicity.Here, the sine function indicates yearly fluctuations in CO2 levels due to natural processes. This is equivalent to the Earth’s seasonal cycles impacting CO2 concentrations via fluctuations in plant growth—more plants grow and absorb CO2 in spring and summer, leading to a decrease in CO2 levels.
This only to increase again in fall and winter when plants die off.Key parameters include:

• Amplitude \(A = 3.5\), showing the maximum variation from the average CO2 level.
• Frequency modulation \(2\pi x\), indicating the number of cycles per unit time, which is one complete oscillation per year since \(2\pi\) is the period of sine wave.

This understanding helps track seasonal variations and adapt agricultural and environmental strategies.

Graphing Functions

Graphing functions involve plotting equations on a coordinate system to visualize their behavior. For the function \(L(x)\), graphing helps us see trends and variations over time.For part (a) of the exercise, the task was to graph \(L(x)\) between the years 1975 to 1995 or \(15 \leq x \leq 35\). The \(y\)-axis represents CO2 levels in ppm with a range of \(325 \leq y \leq 365\).
This provides a clear visualization.Using graphing software or a calculator allows you to:

• Accurately draw the combination of quadratic and sine components.
• Identify intersections, peaks, and troughs which correspond to changes in CO2 levels.

The intersection points and oscillations offer insight into when CO2 levels rise and fall within a year and show the long-term growth due to human activity. This graphical representation brings clarity to complex algebraic functions and their real-world impacts.

CO2 Emissions

Carbon dioxide (CO2) emissions are a significant concern in modern climate science due to their impact on global warming and climate change. In the context of the equation given for atmospheric CO2 levels, understanding both human-induced and natural causes is fundamental.The quadratic component \(0.022x^2 + 0.55x + 316\) models the rise in CO2 due to anthropogenic activities. This includes:

• Burning fossil fuels for energy and transportation.
• Deforestation, reducing CO2 absorption.

These activities contribute to the long-term upward trend seen in the graph.On the other hand, the sine function \(3.5 \sin(2\pi x)\) suggests natural seasonal fluctuations in CO2 levels. For example:

• Increased photosynthesis during warm months.
• Reduced plant activity in colder months.

Both components together form a comprehensive model, accounting for the complex dynamics of atmospheric CO2, guiding environmental policies and research.