91Ó°ÊÓ

Taking the derivative of a function is an important skill in calculus. This process helps us understand how a function changes at any given point. For the exercise provided, we start with the function \[ y = \frac{x^4}{8} + \frac{1}{4x^2} \] To find the derivative \( \frac{dy}{dx} \), we differentiate each term individually with respect to \(x\).

1. Differentiate \( \frac{x^4}{8} \) using the power rule: the power rule tells us to multiply by the exponent and reduce the power by one, so it becomes \( \frac{1}{2} x^3 \).
2. For \( \frac{1}{4x^2} \), recall it can be written as \( \frac{1}{4} x^{-2} \). Applying the power rule gives us \( -\frac{1}{2x^3} \).

Combining these, the derivative \( \frac{dy}{dx} \) is then:\[ \frac{dy}{dx} = \frac{1}{2} x^3 - \frac{1}{2x^3} \]This derivative captures the rate of change of the function, laying groundwork for calculating the surface area by revolution.

Numerical integration is a critical computational technique used to approximate the value of integrals, especially when they can't be solved analytically. In the given exercise, we have an integral:\[ 2\pi\int_{1}^{2} (\frac{x^4}{8} + \frac{1}{4x^2})\sqrt{\frac{1}{4}x^6 + \frac{1}{4x^6}} \, dx \]Due to its complexity, direct analytical integration is impractical. Thus, numerical methods such as Simpson's rule or adaptive quadrature are employed. Numerical methods work by approximating the area under a curve through summing the areas of shapes, like rectangles or trapezoids:

• Simpson's Rule: Uses parabolic segments to approximate the area, providing better accuracy with fewer segments compared to trapezoidal methods.
• Adaptive Quadrature: Adjusts the segment sizes based on the curve's shape, ensuring more precise results where the function changes rapidly.

These techniques involve computational power, often using software tools. Such tools calculate a close approximation of the integral, which in this task contributes to determining the surface area from the curve's revolution.

The power rule in calculus is one of the simplest, yet most powerful tools for differentiation. It's used to find derivatives of polynomial functions, which are functions with terms like \(x^n\). Employing this rule makes solving problems involving derivatives straightforward. The power rule states: For any term \(x^n\), the derivative with respect to \(x\) is given by:\[ \frac{d}{dx} x^n = nx^{n-1} \]In the exercise's function \( y = \frac{x^4}{8} + \frac{1}{4x^2} \), we apply the power rule as follows:

• For \( \frac{x^4}{8} \), its derivative is \( \frac{4}{8} x^{4-1} = \frac{1}{2} x^3 \).
• For \( \frac{1}{4x^2} \), rewrite it as \( \frac{1}{4}x^{-2} \), where the derivative gives \(-\frac{2}{4}x^{-2-1} = -\frac{1}{2x^3} \).

The power rule helps simplify differentiation, enabling more complex calculations such as those required for surface areas of revolved curves.