Problem 28 - Thomas Calculus

Question

The sonobuoy problem In submarine location problems, it is often necessary to find a submarine's closest point of approach (CPA) to a sonobuoy (sound detector) in the water. Suppose that the submarine travels on the parabolic path \(y=x^{2}\) and that the buoy is located at the point \((2,-1 / 2) .\) a. Show that the value of \(x\) that minimizes the distance between the submarine and the buoy is a solution of the equation \(x=1 /\left(x^{2}+1\right)\) b. Solve the equation \(x=1 /\left(x^{2}+1\right)\) with Newton's method.

Step-by-Step Solution

Answer

a. Show derivative: Solve for critical points. b. Newton's method finds approximately \(x \approx 0.62\).

Step 1: Understanding the Problem

We need to find the value of \(x\) on the parabolic path \(y = x^2\) that minimizes the distance to the buoy at (2, -1/2). Then, verify that \(x = 1/(x^2 + 1)\) represents this value, and solve it using Newton's method.

Step 2: Determine the Distance Formula

The Euclidean distance \(D\) between the submarine at \((x, x^2)\) and the buoy \((2, -1/2)\) is given by: \[ D = \sqrt{(x - 2)^2 + (x^2 + \frac{1}{2})^2} \] To minimize \(D\), it is often easier to minimize \(D^2\), which is \( (x - 2)^2 + (x^2 + \frac{1}{2})^2 \).

Step 3: Set Up the Function to Minimize

The function to minimize is: \[ f(x) = (x - 2)^2 + \left(x^2 + \frac{1}{2}\right)^2 \]

Step 4: Find the Derivative

Differentiate \(f(x)\) with respect to \(x\):\[ f'(x) = 2(x - 2) + 2(x^2 + \frac{1}{2})(2x) \] Simplify the derivative: \[ f'(x) = 2(x - 2) + 2(2x)(x^2 + \frac{1}{2}) = 2x - 4 + 4x^3 + 2x = 4x^3 + 4x - 4 \]

Step 5: Set the Derivative to Zero

Set \(f'(x) = 0\) to find the critical points: \[ 4x^3 + 4x - 4 = 0 \] Divide the entire equation by 4:\[ x^3 + x - 1 = 0 \] Notice that if \(x = 1/(x^2 + 1)\), then \(x(x^2 + 1) = 1\), leading to the equation \(x^3 + x - 1 = 0\) being a form of the provided equation.

Step 6: Introducing Newton's Method

Newton's method iteratively finds roots using the formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] We will apply this method to \(h(x) = x - \frac{1}{x^2 + 1}\).

Step 7: Prepare for Newton's Method

Find and differentiate \(h(x) = x - \frac{1}{x^2 + 1}\):\[ h'(x) = 1 - \left(-\frac{2x}{(x^2 + 1)^2}\right) = 1 + \frac{2x}{(x^2 + 1)^2} \]

Step 8: Apply Newton's Method

Choose an initial guess, say \(x_0 = 0.5\). Then apply the formula iteratively:1. \(x_1 = x_0 - \frac{h(x_0)}{h'(x_0)}\)2. Continue updating \(x_n\) until the sequence converges to a value, which is the solution for the CPA.

Key Concepts

Submarine Location Problems

In submarine location problems, precise calculations are essential to determine a submarine's position relative to critical points, such as a sonobuoy. A sonobuoy is a floating acoustic sensor that detects submarines underwater. The challenge here is to ascertain at what point the submarine is closest to the buoy, known as the closest point of approach (CPA). This scenario requires analyzing the submarine's path and precisely determining its proximity to the detector. To tackle this, one needs to model the path mathematically. In this exercise, the submarine's path is defined by the parabola described by the equation \( y = x^2 \). The sonobuoy's location is fixed at the point \((2, -1/2) \). Our task is to find the specific \( x \) value along this path that minimizes the distance to this buoy.

Minimizing Distance

Minimizing the distance between the submarine's path and the buoy is crucial. It's not just about reducing the actual distance but also simplifying the computational process. The Euclidean distance formula is typically used for this purpose. However, minimizing the distance directly can be cumbersome due to the square root involved in the formula.Instead, we minimize the square of the distance. This eliminates the square root and simplifies calculations while still ensuring the shortest path is found. In our case, the formula simplifies to \[ f(x) = (x - 2)^2 + \left(x^2 + \frac{1}{2}\right)^2 \]By focusing on minimizing \( f(x) \) instead of \( D \), we efficiently determine the optimal \( x \) along the parabola.

Euclidean Distance

The Euclidean distance is a fundamental concept in geometry to measure the straight-line distance between two points in space. In this exercise, it's expressed by the formula:\[ D = \sqrt{(x - 2)^2 + (x^2 + \frac{1}{2})^2} \]This formula captures the straight line linking the submarine's position on its parabolic path to the buoy's fixed location. The squared terms represent the components along the x-axis and y-axis, measuring how far they are apart in each direction.Understanding this concept is crucial as it lays the basis for more complex distance calculations and optimization problems in spatial settings.

Derivative Calculation

To find the point where the distance is minimized, we use calculus and compute the derivative of the function \( f(x) \). The derivative helps identify critical points, which are candidates for minimum or maximum values.For our function \( f(x) = (x - 2)^2 + (x^2 + \frac{1}{2})^2 \), the derivative is calculated as follows:\[ f'(x) = 2(x - 2) + 2(2x)(x^2 + \frac{1}{2}) = 4x^3 + 4x - 4 \]By setting \( f'(x) = 0 \), we solve for \( x \) to find these critical points. Rather than solving this equation by hand, we employ Newton's method, an iterative technique that refines an initial guess to approximate solutions to such equations efficiently. This method is essential for handling equations where analytical solutions are complex or impractical to find manually.