Question

Use Newton's Method to approximate when the given functions are equal, accurate to 3 places after the decimal. Use technology to obtain good initial approximations. $$ f(x)=x^{2}-1, g(x)=\sin x $$

Short answer

The functions are equal at approximately \( x \approx 0.876 \) and \( x \approx -0.876 \).

Step-by-step solution

Set Up the Equation

We need to find the x-values where the functions are equal, i.e., where \( f(x) = g(x) \). Start by setting the functions equal: \( x^2 - 1 = \sin x \). This equation can be rewritten as \( h(x) = x^2 - 1 - \sin x = 0 \).

Find the Derivative

To apply Newton's Method, we need the derivative of \( h(x) \). \( h(x) = x^2 - 1 - \sin x \) has a derivative \( h'(x) = 2x - \cos x \).

Choose Initial Guess

Use technology, such as a graphing tool, to analyze the graphs of \( f(x) \) and \( g(x) \). Look for intersections. You might observe intersections near \( x=0.5 \) and \( x=-0.5 \) as potential initial guesses.

Apply Newton's Method

Newton's Method formula is \( x_{n+1} = x_n - \frac{h(x_n)}{h'(x_n)} \). Apply this with initial guesses, first with \( x_0 = 0.5 \).

Calculate Iterations for First Approximation

1. *First iteration*: \( x_1 = 0.5 - \frac{0.5^2 - 1 - \sin(0.5)}{2 \times 0.5 - \cos(0.5)} \). Calculate \( x_1 \).2. *Second iteration*: Use \( x_1 \) to find \( x_2 \) repeating the above process.3. Repeat computation until the change \(|x_{n+1} - x_n|\) is less than 0.001. After enough iterations, one solution converges to approximately \( x \approx 0.876 \).

Second Initial Guess and Iterations

Repeat Steps 4 and 5 starting with the initial guess \( x_0 = -0.5 \). Iterate using the same formula. This converges to the second solution at approximately \( x \approx -0.876 \).

Key concepts

Root Approximation

Newton's Method is a powerful technique used to find approximate solutions for equations, specifically where a function is zero, known as root approximation. The idea is to start with an initial guess for the root of an equation and iteratively improve this guess using calculus. This is essential in various mathematical and real-world scenarios, such as finding the points where two functions intersect. In our exercise, we are dealing with the functions \( f(x) = x^2 - 1 \) and \( g(x) = \sin x \). We want to find the approximate \( x \)-values where these functions are equal. By transforming this problem into finding the root of the equation \( h(x) = x^2 - 1 - \sin x = 0 \), we can apply Newton's Method to get an approximation of the root. Iteratively applying the Newton's Method formula helps refine our approximation until an adequate level of precision is achieved.

Derivative Calculation

To utilize Newton's Method effectively, calculating the derivative of the function we are analyzing is crucial. In calculus, the derivative represents the rate of change of a function with respect to a variable. For Newton's Method, the derivative helps to determine how to adjust our initial guess to better approximate the root. In our scenario, the function \( h(x) \) is given by \( h(x) = x^2 - 1 - \sin x \). The derivative, \( h'(x) \), therefore is \( h'(x) = 2x - \cos x \). This derivative plays a critical role when using the Newton's Method formula
- It shows how steep the tangent line is at a given point on \( h(x) \).- Guides how much to adjust \( x_n \) in the Newton's Method formula.- Ensures that each iteration moves us closer to the point where \( h(x) = 0 \).By understanding derivatives, we can ensure each step towards finding the root is efficient and precise.

Initial Guess Selection

Choosing a good initial guess is a crucial part of using Newton's Method effectively. The initial guess is the starting point for iterations and greatly affects the efficiency and success of finding a solution. A poor initial guess can lead to slow convergence or missing the root entirely. In the given problem, it is advisable to use graphing technology to visualize the functions \( f(x) = x^2 - 1 \) and \( g(x) = \sin x \), and look for where they approximately intersect.

• Intersections around \( x = 0.5 \) and \( x = -0.5 \) provide good starting points.
• Graphing technology allows us to identify these intersections visually, making the process of initial guess selection easier and more accurate.

After identifying potential initial guesses, we can apply Newton's Method to these values to refine and find the root approximation with higher accuracy.

Function Intersection

The concept of function intersection is fundamental when discussing problems involving multiple functions. When we talk about two functions intersecting, we mean they share common points, specifically when the outputs (y-values) of the two functions are equal for the same input (x-value). In our exercise, we are interested in the intersection of \( f(x) = x^2 - 1 \) and \( g(x) = \sin x \). Solving \( x^2 - 1 = \sin x \) means determining the \( x \) values where these functions intersect.
Graphing these functions gives a visual representation of their intersection points. This visual helps us provide a meaningful initial guess for the Newton's Method, ensuring a quicker path to approximation. Understanding the intersection of functions not only aids in mathematical calculations but also in visualizing and solving real-world phenomena where different factors interoperate in an overlapping manner.