How to Find the Center of Curvature for the Curve y sin(2x) / x^2

The concept of the center of curvature is essential in understanding the local behavior of a curve at a specific point. This article provides a detailed guide on how to compute the center of curvature for the curve defined by the equation y sin(2x) / x^2. This includes the first and second derivatives, curvature, and the calculation of the center of curvature.

1. Computing the First and Second Derivatives

To begin with, the first step is to find the first and second derivatives of the function y sin(2x) / x^2. These derivatives are crucial for the subsequent steps.

1.1 First Derivative

The first derivative, denoted as dy/dx, can be found using the quotient rule. The function is given by y sin(2x) / x^2, so we apply the quotient rule:

[f'(x) frac{2 sin(2x) cos(2x) cdot x^2 - 2x sin^2(2x)}{x^4}]

This simplifies to:

[f'(x) frac{2 sin(2x) cos(2x) cdot x - sin^2(2x)}{x^3}]

1.2 Second Derivative

For the second derivative, we need to differentiate the first derivative again. This process can be intricate, but it involves the application of the quotient rule and the product rule:

[f''(x) frac{d}{dx} left(frac{2 sin(2x) cos(2x) cdot x - sin^2(2x)}{x^3}right)]

The exact form of the second derivative, after performing the algebra, is complex and not shown here for brevity. However, it can be used for the curvature calculation.

2. Calculating the Curvature

The curvature kappa; of the curve at a specific point is calculated using the formula:

[kappa frac{f'(x)}{1 (f'(x))^2^{3/2}}]

Substituting the value of the first derivative from the previous steps will give you the curvature. This step involves some algebraic manipulation, but it is a straightforward application of the given formula.

3. Finding the Center of Curvature

The center of curvature can be found using the radius of curvature, which is calculated as the reciprocal of the curvature:

[R frac{1}{kappa}]

The coordinates of the center of curvature are given by:

[(x - frac{f'(x) f''(x)}{kappa}, y - frac{1}{kappa})]

For the specific curve y sin(2x) / x^2, you need to substitute the values of the first and second derivatives into these formulas. Let's consider an example where you want to find the center of curvature at x 1: Calculate f(1) and f'(1). Substitute these values into the curvature formula to find kappa;. Use these values to find the radius of curvature R and the coordinates of the center of curvature.

Summary

The process involves the following steps:

Find the first and second derivatives of the function. Calculate the curvature using these derivatives. Translate the curvature and the first derivative into the coordinates of the center of curvature.

If you need help with the specific calculations, feel free to reach out!

Example:

Given the function y sin(2x) / x^2, calculate the first and second derivatives:

dy/dx (2 sin(2x) cos(2x) cdot x - sin^2(2x)) / x^3 d^2y/dx^2 complex expression (see above)

Then, using a specific point x 1, compute the curvature and the coordinates of the center of curvature.