Problem 18 - Thomas Calculus in SI Units

Question

Estimate \(\pi\) to as many decimal places as your calculator will display by using Newton's method to solve the equation \(\tan x=0\) with \(x_{0}=3\).

Step-by-Step Solution

Answer

Use Newton's method starting with 3 and iterate until convergence to estimate \(\pi\).

Step 1: Understand Newton's Method

Newton's method, or the Newton-Raphson method, is an iterative method for finding the roots of a function. It involves using the formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]where \( f(x) \) is the function for which we are finding the root and \( f'(x) \) is its derivative.

Step 2: Define the Function and Its Derivative

Given \( \tan x = 0 \), find the root closest to \( x_0 = 3 \). Thus, the function is \( f(x) = \tan x \). The derivative is \( f'(x) = \sec^2 x = 1 + \tan^2 x \).

Step 3: Initiate the Iteration

Start with the initial guess \( x_0 = 3 \). The iterative formula becomes:\[ x_{n+1} = x_n - \frac{\tan x_n}{1 + \tan^2 x_n} \]

Step 4: First Iteration

Calculate the value of \( x_1 \) with \( x_0 = 3 \): \[ x_1 = 3 - \frac{\tan(3)}{1 + \tan^2(3)} \] Use a calculator to find \( \tan(3) \) and \( \tan^2(3) \), then substitute to get \( x_1 \).

Step 5: Subsequent Iterations

Continue iterating until the value of \( x_n \) does not change significantly. It typically requires a few iterations for the approximation to stabilize and converge to the nearest root of \( \tan x = 0 \).

Step 6: Conclude the Iteration

After several iterations, the value of \( x_n \) will converge to \( \pi \) or a close multiple of \( \pi \), since \( \tan\pi = 0 \). At this point, your calculator will show \( \pi \) to its precision capability.

Key Concepts

Iterative Method

An iterative method is a process that repeatedly applies a specific calculation in order to reach a desired outcome. It starts with an initial estimate and continually improves this estimate through successive approximations.
The Newton's method is a popular iterative technique used to find roots. The core idea is to use the function and its derivative to make a sequence of improving estimations.

• Start with an initial guess, called the seed, close to the expected root.
• Use the formula \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] where \( f(x) \) is the function whose root you seek.
• Compute the next estimate by applying the formula iteratively until the results stabilize.

Newton's method is prized for its efficiency and speed, but it primarily works well when starting near a true root, making the choice of the initial guess crucial.

Roots of a Function

In mathematics, the roots of a function are the values that make the function equal to zero. Finding roots is a common goal in calculus and algebra because they often represent meaningful quantities.

• If \( f(x) = 0 \), then \( x \) is a root of the function \( f \).
• For trigonometric functions like \( \tan x \), the roots are points where the function's graph intersects the x-axis.
• Understanding the characteristics of the function helps in estimating where the roots might lie.

Finding accurate roots involves various methods and Newton's method is one efficient way, especially for differentiable functions.

Trigonometric Functions

Trigonometric functions, like \( \tan x \), are fundamental in mathematics with broad applications in various fields including physics and engineering.

• The function \( \tan x \) is defined as the ratio of the sine and cosine functions: \( \tan x = \frac{\sin x}{\cos x} \).
• \( \tan x \) is periodic, with roots at \( x = k\pi \) where \( k \) is an integer, because \( \tan(k\pi) = 0 \).
• The derivative, \( \sec^2 x = 1 + \tan^2 x \), plays a key role in applying Newton's method to find the roots.

These trigonometric properties are essential for identifying where these intersections, or roots, occur on the graph.

Calculus

Calculus, a field in mathematics, is all about change and is foundational for techniques like Newton's method.

• Calculus involves differentiation and integration which are used to analyze and describe behaviors of functions.
• The derivative, \( f'(x) \), is at the heart of Newton's method, providing a linear approximation to the function at any given point.
• This derivative information helps iterate closer estimates of the root for functions like \( \tan x \).

Through calculus, complex functions can be handled more simply, enabling us to solve practical problems such as calculating \( \pi \) efficiently.