Problem 153 - Calculus Volume 3

Question

The surface of a large cup is formed by revolving the graph of the function \(y=0.25 x^{1.6}\) from \(x=0\) to \(x=5\) about the \(y\) -axis (measured in centimeters). Find the curvature \(\kappa\) of the generating curve as a function of \(x\).

Step-by-Step Solution

Answer

The curvature \(\kappa(x) = \frac{0.24x^{-0.4}}{(1 + 0.16x^{1.2})^{3/2}}\).

Step 1: Differentiate the function

First, we need to find the derivative of the function \(y = 0.25x^{1.6}\). Using the power rule \(\frac{d}{dx}[x^n] = nx^{n-1}\), the derivative is \( y' = \frac{dy}{dx} = 0.25 \times 1.6x^{1.6 - 1} = 0.4x^{0.6}\).

Step 2: Differentiate the derivative

Now, find the second derivative of \(y\). Differentiating \(y' = 0.4x^{0.6}\) using the power rule again, we get \( \frac{d^2y}{dx^2} = 0.4 \times 0.6x^{0.6-1} = 0.24x^{-0.4}\).

Step 3: Apply the curvature formula

The curvature \(\kappa\) is given by the formula:\[\kappa = \frac{|\frac{d^2y}{dx^2}|}{(1 + (\frac{dy}{dx})^2)^{3/2}}\]Substitute \(\frac{dy}{dx} = 0.4x^{0.6}\) and \(\frac{d^2y}{dx^2} = 0.24x^{-0.4}\) into the formula:\[\kappa = \frac{|0.24x^{-0.4}|}{(1 + (0.4x^{0.6})^2)^{3/2}}\]This simplifies to:\[\kappa = \frac{0.24x^{-0.4}}{(1 + 0.16x^{1.2})^{3/2}}\]

Key Concepts

Derivative

The derivative is a fundamental concept in calculus. It measures how a function changes as its input changes. In simple terms, the derivative of a function at a point gives the slope or rate of change at that instant. For the function \( y = 0.25x^{1.6} \), finding the derivative involves identifying how \( y \) changes as \( x \) changes.
To differentiate \( y = 0.25x^{1.6} \), we use the power rule. This rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Here, \( n \) is 1.6, so when applying the power rule:

• Multiply the exponent by the coefficient: \( 0.25 \times 1.6 = 0.4 \)


• Decrease the exponent by 1: \( 1.6 - 1 = 0.6 \)

This gives us the first derivative: \( y' = 0.4x^{0.6} \).
Knowing how to find a derivative is crucial, as it is used to find the curvature of a given function, which is essential in understanding the shape and behavior of curves in geometry and physics.

Curvature Formula

Curvature measures how sharply a curve bends at any given point. In this exercise, we are interested in calculating the curvature of the curve generated by \( y = 0.25x^{1.6} \). The curvature formula we use is:
\[ \kappa = \frac{|\frac{d^2y}{dx^2}|}{(1 + (\frac{dy}{dx})^2)^{3/2}} \]
This formula requires two derivatives:

• The first derivative \( y' \), which measures the rate of change of \( y \) with respect to \( x \), already calculated as \( 0.4x^{0.6} \).


• The second derivative \( \frac{d^2y}{dx^2} \), which gives the rate of change of the rate of change, found to be \( 0.24x^{-0.4} \).

Using these, the curvature is computed by plugging these derivatives into the formula. This tells us how the curve bends. Knowing the curvature helps in understanding the geometric properties of the curve, which is especially useful in designing roads, paths, and other structures that incorporate curves.

Power Rule

The power rule is a simple yet powerful technique in calculus for finding the derivative of functions of the form \( x^n \). It states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). This rule is instrumental because it provides a quick way to differentiate polynomial functions.
In this problem, we used the power rule twice:

• First, to find the first derivative \( y' \) of \( y = 0.25x^{1.6} \). By applying the rule, we derived \( 0.4x^{0.6} \).


• Second, to differentiate \( y' \) to get the second derivative \( 0.24x^{-0.4} \).

The power rule is easy to apply. Multiply the coefficient by the exponent, then reduce the exponent by one.
This simplicity is why the power rule is one of the first differentiation techniques taught in calculus, emphasizing its essential role in solving real-world problems that involve rate of change and curvature.