Problem 133 - Calculus Volume 3

Question

Find the curvature \(\kappa\) for the curve \(y=x-\frac{1}{4} x^{2}\) at the point \(x=2\).

Step-by-Step Solution

Answer

The curvature \(\kappa\) at \(x=2\) is \(\frac{1}{2}\).

Step 1: Find the First Derivative

The curvature of a curve is based on its derivatives. First, find the first derivative of the function \(y = x - \frac{1}{4} x^2\). Differentiating it with respect to \(x\), we get: \[y' = 1 - \frac{1}{2}x\].

Step 2: Evaluate the First Derivative at x = 2

Substitute \(x = 2\) into the first derivative to find the slope at that point. \[y'(2) = 1 - \frac{1}{2}(2) = 1 - 1 = 0\].

Step 3: Find the Second Derivative

Next, find the second derivative of the function to assess the rate of change of the slope. Differentiating \(y' = 1 - \frac{1}{2}x\), we obtain: \[y'' = -\frac{1}{2}\].

Step 4: Use the Curvature Formula

The curvature for a function \(y = f(x)\) is given by the formula: \(\kappa = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}\). Substitute \(f'(2) = 0\) and \(f''(2) = -\frac{1}{2}\) into this formula: \[\kappa = \frac{|-\frac{1}{2}|}{(1+0^2)^{3/2}} = \frac{\frac{1}{2}}{1} = \frac{1}{2}\].

Key Concepts

First Derivative

The first derivative of a function, denoted as \(y'\) or \(f'(x)\), plays a crucial role in calculus, especially when it comes to understanding the properties of a curve. This derivative represents the rate of change of the function at any given point along its curve.

To find the first derivative of a function like \(y = x - \frac{1}{4} x^2\), you apply the basic rules of differentiation. For this particular function, taking the derivative with respect to \(x\), we get:

• For \(x\), the derivative is 1, because the derivative of \(ax\) where \(a\) is a constant is \(a\).
• For \(-\frac{1}{4}x^2\), you follow the power rule \(nx^{n-1}\), which yields \(-\frac{1}{2}x\).

Thus, the first derivative becomes \(y' = 1 - \frac{1}{2}x\).

The first derivative helps in determining the slope of the tangent line to the curve at a specific point. For example, plugging \(x = 2\) into the first derivative equation gives \(y'(2) = 0\), which indicates that at \(x = 2\), the tangent line is horizontal, implying a potential maximum or minimum point.

Second Derivative

The second derivative, denoted as \(y''\) or \(f''(x)\), gives further insight into a function's behavior, especially concerning its concavity and the nature of its curvature.

Once you have the first derivative \(y' = 1 - \frac{1}{2}x\), you can find the second derivative by differentiating it again with respect to \(x\). Applying the basic rule for differentiation to the linear component \(-\frac{1}{2}x\), we find:

• The derivative of \(-\frac{1}{2}x\) is \(-\frac{1}{2}\), since it's a constant multiplied by \(x\).

This results in the second derivative \(y'' = -\frac{1}{2}\).

The sign of the second derivative tells us about the curve's concavity:

• If \(y'' > 0\), the curve is concave up (like a cup) at that interval.
• If \(y'' < 0\), the curve is concave down (like a frown).

In our function, \(y'' = -\frac{1}{2}\) suggests that the curve is concave down at \(x = 2\). Understanding the second derivative is crucial for calculating curvature and analyzing the overall shape of the graph.

Curvature Formula

Curvature \(\kappa\) is a measure of how sharply a curve bends at a given point. The curvature formula is expressed as:\[\kappa = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}\]

In this formula, the absolute value of the second derivative \(|f''(x)|\) represents the magnitude of curvature, and the denominator accounts for the potential influence of the slope on the curvature.

To find the curvature at a specific point, such as \(x = 2\), you substitute the value of the first and second derivatives at this point into the formula:

• The first derivative at \(x = 2\) is \(f'(2) = 0\).
• The second derivative is \(f''(2) = -\frac{1}{2}\).

Plugging these into the formula gives:\[\kappa = \frac{| -\frac{1}{2} |}{(1 + 0^2)^{3/2}} = \frac{\frac{1}{2}}{1} = \frac{1}{2}\]

This result indicates that the curvature of the curve at \(x = 2\) is \(\frac{1}{2}\), representing the rate and direction of the curve's bending at that point. Understanding curvature is essential for fields like physics and engineering, where it helps describe how objects move along paths and affects structural stability.