Problem 49 - Calculus III

Question

Find the radius of curvature of \(6 y=x^{3}\) at the point \(\left(2, \frac{4}{3}\right)\).

Step-by-Step Solution

Answer

The radius of curvature at the point is \( \frac{5\sqrt{5}}{2} \).

Step 1: Understand the Formula for Radius of Curvature

The radius of curvature \( R \) at a point on a curve is given by \[ R = \frac{(1 + (y')^2)^{3/2}}{|y''|} \] where \( y' \) is the first derivative and \( y'' \) is the second derivative of the function.

Step 2: Rewrite the Equation

The given equation is \( 6y = x^3 \). First, solve for \( y \): \[ y = \frac{x^3}{6} \].

Step 3: Calculate the First Derivative

Differentiate \( y = \frac{x^3}{6} \) with respect to \( x \) to find \( y' \): \[ y' = \frac{d}{dx}\left(\frac{x^3}{6}\right) = \frac{1}{6} \cdot 3x^2 = \frac{x^2}{2} \].

Step 4: Calculate the Second Derivative

Differentiate \( y' = \frac{x^2}{2} \) to find \( y'' \): \[ y'' = \frac{d}{dx}\left(\frac{x^2}{2}\right) = 2 \cdot \frac{x}{2} = x \].

Step 5: Evaluate Derivatives at the Given Point

Plug \( x = 2 \) into the derivatives. Calculate \( y' \) at \( x = 2 \): \[ y' = \frac{2^2}{2} = 2 \]. Calculate \( y'' \) at \( x = 2 \): \[ y'' = 2 \].

Step 6: Calculate the Radius of Curvature

Substitute \( y' = 2 \) and \( y'' = 2 \) into the formula for \( R \): \[ R = \frac{(1 + (2)^2)^{3/2}}{|2|} = \frac{(1 + 4)^{3/2}}{2} = \frac{5^{3/2}}{2} \].

Step 7: Simplify the Result

Calculate \( 5^{3/2} \). It equals \( \sqrt{5^3} = \sqrt{125} \), which simplifies to \( 5\sqrt{5} \), thus, \[ R = \frac{5\sqrt{5}}{2} \].

Key Concepts

Differentiation

To deeply understand how the radius of curvature works, it's essential to first grasp the concept of differentiation. Differentiation is a fundamental concept in calculus and helps us understand the rate at which one quantity changes with respect to another. When dealing with a function like \( y = \frac{x^3}{6} \), differentiation allows us to find the derivatives, which essentially represent the slope of the function at any given point. This is crucial when calculating the radius of curvature, as it requires both first and second derivatives.

At its core, differentiation involves a few key rules:

• Power Rule: If you have \( x^n \), its derivative is \( nx^{n-1} \).
• Constant Factor Rule: If a constant is multiplied by a function, the constant remains outside the differentiation operation.
• Sum Rule: The derivative of a sum of functions is the sum of their derivatives.

Understanding these basic rules can help simplify the process of finding derivatives and getting to the correct result when dealing with more complex functions.

First Derivative

The first derivative, denoted as \( y' \) or \( \frac{dy}{dx} \), gives us information about the rate of change of a function. In simple terms, it's the slope of the tangent line to the curve at any specific point. For the function \( y = \frac{x^3}{6} \), finding the first derivative involves applying basic differentiation rules.

In our example, differentiating results in:

• We first apply the power rule to \( x^3 \), obtaining \( 3x^2 \).
• Then, using the constant factor rule, since it is already divided by 6, the derivative becomes \( \frac{3x^2}{6} \), simplifying it to \( \frac{x^2}{2} \).

The first derivative is crucial for understanding how the curve behaves at different points. It is directly used in the formula for the radius of curvature, which involves calculating \((1 + y'^2)^{3/2}\). Calculating the first derivative correctly ensures we know how steep or flat the curve is at our point of interest.

Second Derivative

After determining the first derivative, the next step is to find the second derivative, represented as \( y'' \). The second derivative provides insight into how the slope of the function changes. It reveals information about the curve's concavity—whether the curve is bending upwards or downwards.

For the function \( y = \frac{x^3}{6} \), the process to find the second derivative is:

• Start with the first derivative \( y' = \frac{x^2}{2} \).
• Differentiate again, applying the power rule to \( x^2 \) and using constant factor rule: \( \frac{d}{dx}(x^2) = 2x \).
• Thus, the second derivative simplifies to \( y'' = x \).

Understanding the second derivative is vital as it is used to calculate the denominator of the radius of curvature formula, \(|y''|\). Calculating it accurately allows us to determine exactly how the curvature behaves. Specifically, at our given point, this curvature value influences the size of the radius, showcasing how sharply or loosely the curve turns at that point. This knowledge is critical for solving geometric problems involving curves and ensuring precise solutions.