Problem 3 - Advanced Calculus

Question

(a) Determine the number of real roots of the equation $$ \sin x+\frac{1}{6} x=1 $$ (b) Find each root, accurate to 0001 .

Step-by-Step Solution

Answer

The equation \(\sin x + \frac{1}{6}x = 1\) has multiple real roots. The roots can be found by using numerical methods such as the Newton-Raphson method to the desired accuracy of 0.0001.

Step 1: Visualize Problem

The equation is \(\sin x + \frac{1}{6}x = 1\). Plot this equation to visualize the number of real roots.

Step 2: Determine Number of Real Roots

By observing the graph, the function \(\sin x + \frac{1}{6}x\) intersects with the line \(y = 1\) at more than one point. Hence, it can be determined that the equation has multiple real roots.

Step 3: Find Each Root

Use a numerical method such as the Newton-Raphson method to find roots of this equation. This can be done by setting up an iteration equation \[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\] where \(f(x) = \sin x + \frac{1}{6}x - 1\) and \(f'(x) = \cos x + \frac{1}{6}\). Continue the process until the desired accuracy is achieved.

Key Concepts

Root Finding

Root finding is a crucial numerical method used to identify where a particular function equals zero. In the given problem, finding the roots of the equation \( sin x + \frac{1}{6}x = 1 \) involves first rewriting it in the form of a standard root-finding problem: \(f(x) = \sin x + \frac{1}{6}x - 1 = 0\). The aim here is to determine the values of \(x\) for which this function equals zero. Finding these roots is vital as they represent the solutions of the equation. Understanding this concept involves comprehending the behavior of the function over a specific interval and efficiently identifying where it crosses the x-axis through numerical techniques.

Root finding often employs graphical methods to get an initial visual estimate of the number and location of roots. This is done by plotting the function and observing its intersections with the x-axis, which represents the points of solution. However, graphical approaches alone may not provide precise solutions, necessitating the use of numerical methods.

Newton-Raphson Method

The Newton-Raphson method is an effective numerical technique for root finding. It is iterative, meaning it refines an initial guess through successive approximations to find increasingly precise solutions. For the equation \(f(x) = \sin x + \frac{1}{6}x - 1\), once you establish an initial guess \(x_0\), the Newton-Raphson formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) is employed.

Here, \(f'(x) = \cos x + \frac{1}{6}\) is the derivative of \(f(x)\), indicating the slope of the tangent to the function at any point. The method utilizes this tangent slope to approximate the root more closely. Although the method converges quickly under ideal conditions, an initial guess near the actual root is crucial for achieving desired precision.

• Advantages: Fast convergence for functions with continuous derivatives.
• Considerations: Requires a reasonably close initial guess to avoid divergence.

The Newton-Raphson method is particularly powerful when analytic solutions are cumbersome or impossible to compute, making it an essential tool for solving real-world equations.

Trigonometry

Trigonometry plays a pivotal role in solving the given equation. Here, the function incorporates \(\sin x\), a trigonometric expression with well-known fluctuating behavior between -1 and 1. Such behavior affects the function's intersection with horizontal lines like \(y = 1\).

Understanding the periodic nature of trigonometric functions like \(\sin x\) is vital when finding roots through numerical methods and graphing. These functions repeat values over regular intervals, known as periods. The characteristics of \(\sin x\) include:

• Periodicity of \(2\pi\): The function repeats its cycle every \(2\pi\) radians.
• Range: \([-1, 1]\), impacting the potential intersections with other linear functions.

When coupled with the linear term \(\frac{1}{6}x\), trigonometry influences the overall shape of the graph, thus guiding the choice of numerical techniques like the Newton-Raphson method for accurate root determination.

Function Visualization

Function visualization is a practical approach to understanding complex equations before precise calculation. By plotting the given function \(f(x) = \sin x + \frac{1}{6}x - 1\), we gain insights into its behavior and possible intersections with the x-axis, representing real roots.

Creating a graph helps detect:

• Overall shape and trends of the function over a specified domain.
• Approximate locations of roots and number of intersections with horizontal lines.
• Regions where numerical methods like Newton-Raphson can be applied effectively.

Visualization aids in selecting suitable intervals for numerical techniques. It can also highlight symmetries or discontinuities that could affect the choice of the initial guesses required for iterative methods. While not providing precise solutions, function visualization is invaluable for preparing a comprehensive numerical analysis.