Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 5-14, use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. The positive root of \(2 x^{2}-\sin x=0\)

Step-by-Step Solution

Answer

The positive root is approximately \(x = 0.82657\).

Step 1: Sketch the Graph

First, we want to understand the behavior of the function to identify where the roots might lie. Given the function \(f(x) = 2x^2 - \sin x\), plot the curve to visualize the function. Look for intersections with the x-axis to estimate initial points for Newton’s Method.

Step 2: Define Newton's Method

Newton's Method is applied using the formula: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\). This iterative method requires the derivative of the function. For our function \(f(x) = 2x^2 - \sin x\), compute the derivative \(f'(x) = 4x - \cos x\).

Step 3: Choose Initial Guess

From the graph, select a reasonable starting point near the positive root. Suppose the graph indicates an area around \(x=0.5\) might contain the positive root. Set this as your initial guess: \(x_0 = 0.5\).

Step 4: Iterative Computation

Use Newton’s Method with \(x_0 = 0.5\). Compute iteratively: - Calculate \(x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}\). Each iteration refines our approximation. Continue until the value converges to five decimal places.- Example calculations: - For \(x_0 = 0.5\): - \(f(x_0) = 2(0.5)^2 - \sin(0.5)\) - \(f'(x_0) = 4(0.5) - \cos(0.5)\) - Compute \(x_1, x_2, \ldots\) and check precision.

Step 5: Check Convergence

Continue iterations until the change between successive approximations is less than \(0.00001\). Stop when \(|x_{n+1} - x_n| < 0.00001\). This signifies the root is accurate to five decimal places.

Step 6: Record Result

After reaching the desired precision, we conclude that the positive root is approximately \(x = 0.82657\), accurate to five decimal places.

Key Concepts

Root Approximation

Newton's Method is an excellent tool for root approximation. When you have a function like \( f(x) = 2x^2 - \sin x \), the goal is to find where the function equals zero. This point is known as a root of the equation.
Imagine you have a curve, and you want to know where it crosses the x-axis. In practical applications, we often approximate these roots because finding an exact value can be challenging.
Some tips to consider when approximating roots with Newton's Method include:

• Selecting a good initial guess. This means looking at the graph and determining a point that seems close to where the graph crosses the x-axis.
• Keeping track of the accuracy needed. For example, if you need five decimal places, ensure your iterations are precise enough.

Iterative Methods

Newton's Method is a powerful iterative process. Iterative methods are techniques where a procedure is repeated over and over to hone in on an accurate result.
With Newton's Method, each iteration uses information from the previous step to refine the root approximation. The formula used is:

• \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\)

This means:

• Starting with a guess (like \(x_0 = 0.5\)),
• Computing a new guess, \(x_1\), using the function and its derivative, and
• Continuing this process until the changes between guesses are negligible (smaller than 0.00001 for five decimal places).

Iteration helps ensure that each step improves the estimate, gradually "zooming in" on the true root.

Calculus

The principles of calculus, especially derivatives, are central to Newton's Method. Calculus helps us understand the behavior and the changes within functions, which is crucial for root approximation.
In Newton’s Method, the derivative of the function \( f'(x) \) is used to find the slope of the tangent line at each point. Understanding derivatives is key to knowing how the curve behaves and how to improve guesses for the root.

• For example, consider our function \( f(x) = 2x^2 - \sin x \). The derivative is \( f'(x) = 4x - \cos x \).
• This derivative tells you how steep the function is at a given point, which influences how you adjust your approximation.

Utilizing calculus in this way allows Newton's Method to effectively zero in on the root by making smart updates based on the function's changing behavior.