Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot on the same axes the given function along with the Maclaurin polynomials of orders \(1,2,3\), and \(4 .\) $$ \sin x $$

Step-by-Step Solution

Answer

Plot \( \sin x \) and the polynomials \( P_1(x) = x \), \( P_3(x) = x - \frac{x^3}{6} \), etc., on the same graph. Higher orders approximate better near zero.

Step 1: Understand Maclaurin Series

The Maclaurin series is a Taylor series expansion of a function about 0. For the function \( \sin x \), the Maclaurin series is:\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]This series is used to approximate \( \sin x \) with different orders of polynomials.

Step 2: Identify Maclaurin Polynomials

We need to find the Maclaurin polynomials of orders 1, 2, 3, and 4 for \( \sin x \).- Order 1: \( P_1(x) = x \)- Order 2: \( P_2(x) = x \)- Order 3: \( P_3(x) = x - \frac{x^3}{6} \)- Order 4: \( P_4(x) = x - \frac{x^3}{6} \)Note that polynomials of odd order have additional odd-degree terms and for even orders, the polynomial remains as for the lower odd order, because the next even coefficient in the series is zero.

Step 3: Set Up the Plot

Create a graph with common x and y axes. Choose a practical range for \( x \), e.g., \(-2\pi \) to \( 2\pi \), to visualize the approximations around \( x = 0 \).The range should be enough to show how the approximations differ from the actual \( \sin x \), especially as \( x \) moves away from 0.

Step 4: Plot the Functions

On the graph, plot the following:1. \( y = \sin x \) as a solid line (represent the actual function).2. \( y = P_1(x) = x \) as a dashed line.3. \( y = P_2(x) = x \) as a dotted line.4. \( y = P_3(x) = x - \frac{x^3}{6} \) as a dash-dot line.5. \( y = P_4(x) = x - \frac{x^3}{6} \) as a long-dash line.Ensure each line is distinguishable by line style or color and is clearly labeled.

Step 5: Analyze the Results

Observe how each polynomial follows the curve \( \sin x \) near \( x = 0 \). Notice that higher-order polynomials (like order 3 and 4) provide better approximations over a larger interval around \( x = 0 \). The order 1 and 2 polynomials deviate much sooner.

Key Concepts

Taylor Series

The Taylor series is a mathematical concept used to represent functions as infinite sums of terms. Each term is derived from the function's derivatives at a specific point. When we use a Taylor series to expand a function around the point zero, it becomes known as the "Maclaurin series." This approach is useful because it transforms complex functions into simpler polynomial forms, making them easier to analyze and compute.
For example, the Maclaurin series for the trigonometric function \( \sin x \) can be written as: \[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \] By studying Taylor series, we can understand how to approximate functions with various levels of precision by choosing the number of terms to consider in the polynomial approximation.

Polynomial Approximation

Polynomial approximation is the process of representing a complex function through a polynomial expression. This technique is crucial in numerical analysis and calculus, providing easier calculations and insights. With Maclaurin polynomials, we're essentially using polynomial approximation.
For instance, for \( \sin x \), the following are the Maclaurin polynomials:

• Order 1: \( P_1(x) = x \)
• Order 2: \( P_2(x) = x \)
• Order 3: \( P_3(x) = x - \frac{x^3}{6} \)
• Order 4: \( P_4(x) = x - \frac{x^3}{6} \)

These polynomials approximate \( \sin x \) around \( x = 0 \). You can observe how they capture different degrees of accuracy. As a rule, including more terms (higher orders) will yield polynomials that more closely resemble the original function, especially near the center point, here being zero.

Trigonometric Functions

Trigonometric functions, like \( \sin x \), play essential roles in mathematics, physics, and engineering. They describe relationships in cyclic systems, wave patterns, and oscillations. Approximating these functions allows for easier calculation and practical application in problem settings where precise calculation is needed.
The function \( \sin x \) is periodic and has unique properties, such as symmetrical cycles and oscillations between -1 and 1. While the exact evaluation of \( \sin x \)may be demanding depending on the context, using its expanded polynomial form through the Maclaurin series simplifies the process significantly.
Understanding and using such approximations is key when analytical solutions must be computed swiftly and with minimal error.

Plotting Functions

Plotting functions involves graphing them on coordinate axes to visually assess their behavior and relationships. In this exercise, we plot \( \sin x \) and its Maclaurin polynomials of orders 1 through 4.
Considerations for plotting include:

• The range for x-values, such as from -\( 2\pi \) to \( 2\pi \), which allows viewing several cycles of \( \sin x \).
• Distinct line styles (e.g., solid, dashed, dotted) or colors to differentiate plots of individual polynomials like \( P_1(x) \) from that of the actual \( \sin x \).
• Clear labeling for easy identification of each curve and understanding of their respective approximations.

By comparing these plots, we can visually evaluate how the approximations behave close to \( x = 0 \) and note deviations as \( x \) moves further away. This visual tool is especially useful in education and analysis, where understanding function behaviors intuitively and accurately is crucial.