Question

In Exercises \(1-4,\) complete two iterations of Newton's Method for the function using the given initial guess. $$ f(x)=\tan x, \quad x_{1}=0.1 $$

Short answer

After two iterations of Newton's Method, we find that \(x_{3} = 0.0998333\).

Step-by-step solution

Compute the Derivative

Start by finding the derivative of the given function. The derivative of \(\tan x\) is \(\sec^{2}(x)\), where \(\sec(x)\) is the reciprocal of \(\cos(x)\).

Compute the First Iteration

Use the Newton's method formula \(x_{k+1} = x_{k} - \frac{f(x_k)}{f'(x_k)}\) and the initial guess \(x_{1}=0.1\) to find the first iteration. Substituting the given values into the formula yields: \(x_{2} = 0.1 - \frac{\tan(0.1)}{\sec^{2}(0.1)}\). After calculating, you get \(x_{2} = 0.0998334\).

Compute the Second Iteration

Apply the same Newton's method formula for the second iteration by replacing \(x_k\) with the value calculated in the first iteration. So we have: \(x_{3} = 0.0998334 - \frac{\tan(0.0998334)}{\sec^{2}(0.0998334)}\). After calculating, you get \(x_{3} = 0.0998333\).

Key concepts

Iterative Methods

In the realm of numerical analysis, iterative methods are a cornerstone for finding approximate solutions to complex problems.

Iterative methods rely on an assumption that a good estimate can be improved upon iteratively, refining the estimate step by step until it reaches an acceptable level of accuracy. In practice, this involves starting with an initial guess and applying a formula repeatedly to improve this guess. Students may find themselves unsure as to why an initial guess is necessary or how it affects the result; it's important to explain that an initial guess serves as a starting point, and while a good guess can speed up the process, the algorithm is designed to reach a solution from most starting values, provided the function behaves well.

• An example of such a method is the bisection method, where the interval containing the solution is repeatedly halved until it is sufficiently small.
• Another is the fixed-point iteration, where a function is applied to its own output over and over.

Newton's Method, also called the Newton-Raphson method, is one of the most efficient iterative methods. It uses the derivative of the function and an initial guess to rapidly converge to a solution, making it a particularly powerful tool for finding roots of functions.

It can be astonishing to realize just how these simple repetitions can lead to accurate results, and diving into the math reveals a beautiful harmony between theory and computational techniques.

Tangent Function

The tangent function, commonly denoted as tan(x), is a trigonometric function with unique properties that can sometimes cause confusion among students.

Specifically, tan(x) is the ratio of the sine and cosine functions, which can be written as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). The function is periodic with a period of \( \pi \) radians, repeating its values every 180 degrees, and it has an unbounded range, meaning its values can be infinitely large or small, which is considerably different from the bounded sin and cos functions.

• Students must be aware that the tangent function is undefined wherever the cosine function equals zero, leading to vertical asymptotes at \( \frac{(2n+1)\pi}{2} \), where \( n \) is an integer.
• Graphing the tan function provides insight into its behavior, showing its periodic nature and the locations of its asymptotes.

For calculus, understanding how the tangent function behaves is crucial since it forms the basis of many derivative problems and is involved in methods like Newton's for finding roots of trigonometric equations.

Derivatives

One cannot overstate the importance of derivatives in mathematics, especially when it comes to Newton's Method.

The derivative of a function represents the rate at which the function's value changes at a particular point and is foundational in both calculus and many applied mathematical methods.

• To many students, the derivative is simply a formula to memorize, but it's critical to understand that it's a tool for measuring change, analogous to how speed measures how quickly distance changes over time.
• A thorough understanding of derivatives isn’t just about the formula but about recognizing how they represent slopes of tangent lines to curves on a graph—providing a geometric interpretation that can be visualized.

When applying Newton's Method, the derivative is crucial as it gives us the slope of the tangent at any given point along the curve. The method leverages this slope to predict where the function crosses the x-axis. Specifically, the update in Newton's Method subtracts the ratio of the function's value to its derivative at the current guess, propelling the next guess closer to the root.

In the example exercise, the derivative of tan(x) is sec²(x). Calculus students should note how the properties of this derivative—such as being always positive—impact the behavior of the iterative method and therefore the convergence to the solution.