Problem 14 - Essential Calculus Early Transcendentals

Question

Use Newton's method to approximate the indicated root of the equation correct to six decimal places. The positive root of \(3 \sin x=x\)

Step-by-Step Solution

Answer

The positive root is approximately 0.523599.

Step 1: Understand the function and initial guess

We need to approximate the positive root of the equation \(3 \sin x = x\). To apply Newton's method, we rewrite it as a function: \(f(x) = 3 \sin x - x\). An initial guess, close to the actual root, is necessary. Since \(x = 0\) gives \(f(0) = 0\), we try a slightly larger value as an initial guess, say \(x_0 = 0.5\).

Step 2: Derivative calculation

Calculate the derivative of the function \(f(x)\). Derivative of \(f(x) = 3 \sin x - x\) is \(f'(x) = 3 \cos x - 1\). This derivative will be used in Newton's formula for iteration.

Step 3: Apply Newton's Method Formula

Newton's Method formula is \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\). Substitute \(x_0 = 0.5\) into this formula to find \(x_1\).

Step 4: Iterative Calculation

Iterate using Newton's formula. Calculate: \[ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 0.5 - \frac{3 \sin(0.5) - 0.5}{3 \cos(0.5) - 1} \approx 0.5234 \] Continue calculating successive approximations \(x_2\), \(x_3\), etc., until the change between them is less than \(10^{-6}\).

Step 5: Stop When Desired Precision is Achieved

Continue iterations until the result converges to six decimal places. After performing a few iterations, you will find \(x_n\) doesn't change at the sixth decimal place. The approximate positive root is reached at this precision.

Key Concepts

Root Finding

Root finding is a fundamental problem in mathematics and numerical analysis. It involves pinpointing where a given function equals zero, or crosses the x-axis.
In other words, root finding helps us locate the values of \(x\) where \(f(x) = 0\).
There are various methods to find roots, with some suited for simpler equations and others for more complex ones:

• Analytical methods are exact, such as solving linear and quadratic equations using algebraic techniques.
• Numerical methods, like Newton's Method, approximate the root and are beneficial for more complicated or non-polynomial functions, such as trigonometric equations.

Newton's Method is particularly powerful for continuous and differentiable functions, making it useful in diverse scenarios like engineering and physics.

Iteration

Iteration is an essential process in numerical methods where a sequence of approximations converges toward a specific solution. It involves repeatedly applying a specific formula or operation until a desired level of accuracy is achieved.
In the case of Newton's Method, iteration helps refine our approximation of the root by leveraging initial estimates and continuously improving them.

• Start with an initial guess that is close to the expected root.
• Use the Newton's update formula: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\).
• Continue applying the formula to produce \(x_1, x_2, x_3, \ldots\) until changes are smaller than a predetermined threshold.

This iterative technique allows us to gradually hone in on the accurate root, increasing precision with each step.

Trigonometric Functions

Trigonometric functions like sinusoidal and cosine functions are fundamental in mathematics. They describe waves, oscillations, and periodic phenomena, making them critical in fields such as physics, engineering, and signal processing.
In our root-finding problem, the function \(3 \sin x - x\) involves the sine function, which oscillates between -1 and 1. This introduces unique characteristics:

• Periodic behavior can result in multiple roots within a limited interval.
• Special attention is required to select an appropriate initial guess, ensuring it lies near the root of interest.

Understanding the properties of trigonometric functions is essential for correctly applying Newton's method, particularly when dealing with periodic roots or oscillations.

Approximation Methods

Approximation methods are invaluable tools in mathematical computations, especially when calculating exact solutions is infeasible. They provide a way to derive usable solutions that are close enough to the truth for practical purposes.
Newton's Method is a classic example of an approximation method, which utilizes both function values and derivatives to find successively better approximations of roots.

• The advantage of Newton's Method lies in its rapid convergence, especially when the initial guess is close to the true root.
• Ensures that even complex equations lacking simple solutions can be tackled effectively within a certain degree of precision.

Overall, approximation methods help bridge the gap between theoretical equations and real-world applications, making complex problems more tractable.