Problem 12 - Thomas Calculus

Question

Locating a planet To calculate a planet's space coordinates, we have to solve equations like \(x=1+0.5 \sin x .\) Graphing the function \(f(x)=x-1-0.5 \sin x\) suggests that the function has a root near \(x=1.5 .\) Use one application of Newton's method to improve this estimate. That is, start with \(x_{0}=1.5\) and find \(x_{1}\) (The value of the root is 1.49870 to five decimal places.) Remember to use radians.

Step-by-Step Solution

Answer

Using Newton's method, the improved estimate is approximately 1.4397.

Step 1: Write Down the Function

First, we need to express the function for which we want to find the root. We are given that \( f(x) = x - 1 - 0.5 \sin x \).

Step 2: Find the Derivative

To apply Newton's method, we need the derivative of the function. The derivative of \( f(x) = x - 1 - 0.5 \sin x \) is \( f'(x) = 1 - 0.5 \cos x \) because the derivative of \( \sin x \) is \( \cos x \).

Step 3: Write Down Newton's Method Formula

Newton's method uses the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). We'll substitute \( x_0 = 1.5 \) into this formula to find \( x_1 \).

Step 4: Calculate \( f(1.5) \)

Substitute \( x = 1.5 \) into the function: \( f(1.5) = 1.5 - 1 - 0.5 \sin(1.5) \). Evaluate to find \( f(1.5) \approx 0.056 \) (using radians for \( \sin \)).

Step 5: Calculate \( f'(1.5) \)

Substitute \( x = 1.5 \) into the derivative: \( f'(1.5) = 1 - 0.5 \cos(1.5) \). Evaluate to find \( f'(1.5) \approx 0.929 \).

Step 6: Apply Newton's Formula

Calculate \( x_1 = 1.5 - \frac{0.056}{0.929} \). Simplifying this gives \( x_1 \approx 1.4397 \).

Step 7: Improve the Estimate

The improved estimate for the root is \( x_1 \approx 1.4397 \). This is a step closer to the more accurate root value which is \( 1.49870 \).

Key Concepts

Newton's Method

Newton's Method is a powerful numerical technique used to find approximate solutions to equations of the form \(f(x) = 0\). This method is applied by starting with an initial guess, or estimate, of the root. The idea is to improve this guess through an iterative process. In our exercise, we begin with an initial guess of \(x_0 = 1.5\).

This technique involves using the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\), where \(f'(x)\) denotes the derivative of \(f(x)\). By repeating this formula, each successive \(x_n\) becomes an improved approximation of the root.

Newton's Method can be very efficient, particularly when the function is smooth and well-behaved. However, it's important to choose a good initial guess, as poor choices can lead the method to a divergent result.

Calculus

Calculus is a branch of mathematics focused on studying change and motion. It is foundational for understanding concepts that relate to rates of change and the accumulation of quantities. Two primary operations in calculus are differentiation and integration.

Differentiation allows us to find the derivative of a function, which describes the rate at which the function's value changes. In the exercise, finding the derivative \(f'(x) = 1 - 0.5\cos x\) is a key step in applying Newton's Method. The derivative provides crucial information about the slope of the tangent line at any point on the graph of the function, which is used to make effective iterations in root-finding algorithms.

In real-world applications like calculating a planet's position, calculus allows scientists to create models that can predict trajectories and efficiently solve celestial equations.

Root-Finding Algorithms

Root-finding algorithms are essential numerical methods for identifying roots or zeros of a function, which are values of \(x\) where \(f(x) = 0\). These algorithms are utilized across various fields of science and engineering to solve equations that cannot be solved algebraically.

Several algorithms exist for this purpose, each with its advantages and limitations. Newton's Method is one of the most popular due to its rapid convergence near the root—provided the initial guess is reasonably close. However, other methods like the Bisection Method and Secant Method may be used when Newton's Method is not suitable, such as in cases involving discontinuous functions or poor initial estimates.

By using root-finding algorithms, one can tackle problems ranging from simple algebraic equations to complex nonlinear models, demonstrating their vast applicability in problem-solving.