Problem 14 - 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

The improved estimate for the root using Newton's method is approximately 1.4987.

Step 1: Understand Newton's Method

Newton's method is used to approximate the roots of a real-valued function. If we have a function \( f(x) \) and need to find its zero, we use an iterative formula. The formula for the next approximation \( x_{n+1} \) is given by: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] This method requires us to calculate both \( f(x) \) and its derivative \( f'(x) \).

Step 2: Define the Function and Derivative

Given the function \( f(x) = x - 1 - 0.5 \sin x \), we need to find its derivative. The derivative of \( f(x) \) is: \[ f'(x) = 1 - 0.5 \cos x \] This is derived using the fact that the derivative of \( x \) is 1 and the derivative of \( \sin x \) is \( \cos x \).

Step 3: Substitute into Newton's Formula

Using the initial approximation \( x_0 = 1.5 \), compute \( f(x_0) \) and \( f'(x_0) \): \[ f(1.5) = 1.5 - 1 - 0.5 \sin(1.5) \] Calculate \( \sin(1.5) \) using a calculator since \( x \) is in radians.\[ f'(1.5) = 1 - 0.5 \cos(1.5) \] Similarly, compute \( \cos(1.5) \).

Step 4: Compute the Next Approximation

Compute the values and plug them into the Newton's formula: \[ x_1 = 1.5 - \frac{f(1.5)}{f'(1.5)} \] Calculate \( f(1.5) \) and \( f'(1.5) \) to find \( x_1 \).

Step 5: Final Calculation

Let's calculate step-by-step: \[ \sin(1.5) \approx 0.9975 \] \[ f(1.5) = 1.5 - 1 - 0.5(0.9975) = 1.5 - 1 - 0.49875 = 0.00125 \] \[ \cos(1.5) \approx 0.0707 \] \[ f'(1.5) = 1 - 0.5(0.0707) = 1 - 0.03535 = 0.96465 \] \[ x_1 = 1.5 - \frac{0.00125}{0.96465} \approx 1.4987 \] Hence, the improved estimate for the root is approximately 1.4987.

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with change and motion. It introduces two key concepts: differentiation and integration. Differentiation is about finding the rate at which a quantity changes. It's very useful for calculating things like velocity or slopes of curves. On the other hand, integration deals with the accumulation of quantities, such as area under a curve. Calculus is crucial in Newton's Method for calculating the derivative of the function, which helps in finding the slope of the tangent line. This slope is used to predict where the function crosses the x-axis, leading us to the root.

Iterative Methods

Iterative methods are techniques used to solve mathematical problems through successive approximations. These methods do not give the solution directly but improve an initial guess with each iteration. Newton's Method is a popular iterative technique for finding roots of a function. You start with an initial estimate and apply the Newton's formula iteratively. Each step refines your guess, bringing it closer to the actual root. This is particularly handy when dealing with complex functions where analytical solutions are hard to determine. Iterative methods are valued for their efficiency and their capability to provide high precision answers.

Root-Finding

Root-finding is the process of locating the x-value where a function equals zero, known as the root. Finding roots is essential in various fields such as physics, engineering, and economics to solve equations. Newton's Method is a highly efficient technique for root-finding. By using the derivative of a function, it quickly adjusts guesses to locate the root more precisely. Starting from an initial guess, the iterative process helps in narrowing down to the root with each step. Understanding root-finding is vital as it applies to solving real-world problems by determining conditions that satisfy specific criteria.

Trigonometric Functions

Trigonometric functions, like sine and cosine, describe relationships within triangles and are periodic in nature. In the context of Newton's Method, understanding these functions is essential since they often appear in complex equations. Trigonometric functions can complicate root-finding because of their oscillatory behavior. Calculating their values requires considering angles in radians for accuracy in calculus and physics applications. In the given exercise, you encountered the function involving \( \sin x \), requiring accurate computation of sine values to provide precise results in Newton's Method.