Question

Plot Taylor polynomial approximations to \(\sin x\). The sine function can be approximated by a polynomial according to the following formula: $$ \sin x \approx S(x ; n)=\sum_{j=0}^{n}(-1)^{j} \frac{x^{2 j+1}}{(2 j+1) !} $$ The expression \((2 j+1) !\) is the factorial (see Exercise 3.14). The error in the approximation \(S(x ; n)\) decreases as \(n\) increases and in the limit we have that \(\lim _{n \rightarrow \infty} S(x ; n)=\sin x\). The purpose of this exercise is to visualize the quality of various approximations \(S(x ; n)\) as \(n\) increases. The first part of the exercise is to write a Python function \(\mathrm{S}(\mathrm{x}\), (n) that computes \(S(x ; n)\). Use a straightforward approach where you compute each term as it stands in the formula, i.e., \((-1)^{j} x^{2 j+1}\) divided by the factorial \((2 j+1) !\). (We remark that Exercise A.16 outlines a much more efficient computation of the terms in the series.) The next part of the exercise is to plot \(\sin x\) on \([0,4 \pi]\) together with the approximations \(S(x ; 1), S(x ; 2), S(x ; 3), S(x ; 6)\), and \(S(x ; 12)\) Name of program file: plot_Taylor_sin.py.

Short answer

Plot the sine function and its Taylor approximations using different orders, comparing them visually.

Step-by-step solution

Define the Function to Compute Taylor Series

Create a Python function that computes the Taylor series approximation for the sine function. Name the function `S(x, n)`, where `x` is the input angle in radians and `n` is the order of the Taylor polynomial approximation. Use a for loop to iterate through each term of the series up to `n`, and sum these terms to get the approximation. Each term is calculated as \((-1)^j\frac{x^{2j+1}}{(2j+1)!}\). Use the `math` library to calculate the power, factorial, and ensure proper evaluation of each term.

Set Up the Range and Plot

Use the `numpy` library to create an array of `x` values ranging from 0 to \(4\pi\). This is achieved by `numpy.linspace(0, 4*np.pi, num_points)` where `num_points` is the discretization level of your choice. Then, compute the actual sine values using `numpy.sin` for these `x` values.

Compute and Plot Approximations

For each specified order of approximation \(n = 1, 2, 3, 6, 12\), use the previously defined function `S(x, n)` to compute the Taylor series approximations for the sine function over the defined range. Store each result in a separate array.

Plotting with Matplotlib

Utilize the `matplotlib.pyplot` library to create the plots. Plot the true sine values as a reference. Then, plot each Taylor series approximation on the same graph. Use different line styles or colors for each approximation for distinction, and include a legend to identify each series order.

Observe and Compare

Set up the plot with appropriate labels, titles, and a grid. Make sure the x-axis is labeled as angle in radians, and include a legend to indicate which line corresponds to which polynomial approximation. Observations can be made on how the approximation accuracy improves as `n` increases by visual comparison with the true sine curve.

Key concepts

Sine Function

The sine function, denoted as \( \sin x \), is a fundamental trigonometric function that describes the y-coordinate of a point on the unit circle as it traverses around the origin. It is periodic, with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians. The sine wave is smooth and continuous, which can be mathematically defined for any real number. This function is essential in various fields such as physics, engineering, and signal processing.
The sine function is often approximated in computations because of its complex periodic nature. This is where Taylor series come into play, as they allow us to approximate \( \sin x \) using polynomials, which are simpler to compute. The quality of this approximation improves as more terms are added. Understanding this concept is crucial for both theoretical studies and practical applications.

Polynomial Approximation

Polynomial approximation involves approximating a function using a polynomial, which is a mathematical expression consisting of variables and coefficients. In the context of the sine function, a Taylor series is used to achieve this approximation. Given by the formula:

\[ \sin x \approx S(x ; n) = \sum_{j=0}^{n}(-1)^{j} \frac{x^{2j+1}}{(2j+1)!} \]

This series effectively creates a polynomial approximation of \( \sin x \). Each term in the series, \( (-1)^j \frac{x^{2j+1}}{(2j+1)!} \), represents a component of a polynomial that approximates the sine function more accurately as more terms (i.e., higher \( n \)) are included.

• The term \( (-1)^j \) ensures that the polynomial alternates in sign, mimicking the sine function's peaks and troughs.
• The factorial \( (2j+1)! \) in the denominator governs the rate of convergence, affecting the polynomial's steepness and curvature.

Increasing the order \( n \) of the polynomial increases its accuracy, allowing it to closely match \( \sin x \) over broader intervals.

Python Programming

Python, a versatile programming language, provides robust tools for computing Taylor series approximations. You can begin by defining a function, `S(x, n)`, that calculates these approximations by iterating through each term in the Taylor series.

• Use a loop to compute each term \( (-1)^j \frac{x^{2j+1}}{(2j+1)!} \) up to \( n \). Ensure you utilize Python's `math` library for functions like `factorial` and `pow` for precise calculations.
• The order of polynomial \( n \) dictates the number of terms, influencing the balance between computational cost and accuracy.

The complete Python script can incorporate the `numpy` library, which simplifies handling arrays and mathematical functions, such as generating an array of \( x \) values and computing the actual sine values with `numpy.sin`. Using Python in this way not only facilitates computations but also provides a foundation for further mathematical explorations and simulations.

Mathematical Visualization

Visualizing mathematical functions helps to understand their nature and behavior, and the approximation of sine using Taylor series is no exception. With Python, the `matplotlib.pyplot` library is often used for plotting, allowing you to visually compare the true \( \sin x \) curve with its polynomial approximations.
Create plots that display:

• The actual \( \sin x \) across a period, e.g., from 0 to \( 4\pi \).
• The Taylor series approximations \( S(x ; n) \) for different values of \( n \), such as 1, 2, 3, 6, and 12.

Each line in the plot can be differentiated using distinct colors or styles. A legend aids in identifying each approximation order. Observing these graphs reveals that as \( n \) increases, the polynomial curves more closely align with the sine wave, demonstrating visually how polynomial approximation works. Also, important features such as gridlines and axis labels enhance the plot's readability and interpretation, making it a valuable tool in educational settings for demonstrating the effectiveness of Taylor series.