Question

(a) find the point on the curve at which the curvature \(K\) is a maximum and (b) find the limit of \(K\) as \(x \rightarrow \infty\)/. $$ y=\frac{1}{x} $$

Short answer

The maximum curvature occurs at \(x=1\) and \(x=-1\) and has the value of \(2\). As \(x \rightarrow \infty\), the curvature becomes \(0\).

Step-by-step solution

Compute the First Derivative

Start by finding the first derivative of the function \(y=\frac{1}{x}\) using the power rule. The derivative of \(y\) with respect to \(x\), \(y'\), is \(-\frac{1}{x^2}\).

Compute the Second Derivative

After finding the first derivative, find the second derivative, \(y''\), using the power rule again, and get \(\frac{2}{x^3}\).

Calculate the Curvature

Use the following formula for curvature: \(K = \frac{|y''|}{(1 + (y')^2)^{3/2}}\). Substituting the first and second derivatives into the formula, get: \(K = \frac{|\frac{2}{x^3}|}{(1 + (\frac{-1}{x^2})^2)^{3/2}}\). Simplifying the formula, get: \(K = \frac{2}{|x^3|} (1 + \frac{1}{x^4})^{-3/2}\).

Find the Maximum Curvature

To find for which \(x\) the curvature reaches a maximum, differentiate the curvature, set it equal to zero and solve for \(x\). Due to the complexity of the derivative, however, it's easier to note the curvature is defined only for \(x!=0\), and as \(x\) moves away from zero in either direction, the curvature decreases (since it's a reciprocal function), so the maximum occurs at \(x=1\) and \(x=-1\).

Find the Limit as \(x \rightarrow \infty\)

To find the limit of curvature as \(x \rightarrow \infty\), substitute \(\infty\) into the curvature. As \(x\) approaches infinity, the second term in the denominator of the curvature equation goes to zero and the curvature becomes zero. Thus, as \(x \rightarrow \infty\), \(K \rightarrow 0\).

Key concepts

Curvature Maximum

Understanding when a curve reaches its maximum curvature is essential in fields like engineering and physics. In the case of the function \(y=\frac{1}{x}\), we are looking for the point or points where the curvature \(K\) is at its highest.

The curvature of a curve at a point is a measure of how quickly the curve deviates from a straight line at that point. To find the maximum curvature, one would generally take the derivative of \(K\), the curvature function, and find the critical points by setting this derivative equal to zero. However, analyzing the behavior of \(K\) can sometimes offer a quicker solution.

For example, since the curvature of \(y=\frac{1}{x}\) minimizes as \(x\) increases or decreases from zero and the function is not defined at \(x=0\), we can assert that the curvature maximum occurs at the points closest to zero — namely \(x=1\) and \(x=-1\). From this observation, we deduce that the maximum curvature for this function occurs at these points without solving the derivative of \(K\).

First Derivative

The first derivative of a function is a fundamental concept in calculus, representing the rate at which the function's value is changing at any given point. To find the first derivative, also known as \(y'\), we apply rules of differentiation such as the power rule, product rule, or quotient rule depending on the form of the function.

In our example where \(y = \frac{1}{x}\), we use the power rule to find that \(y' = -\frac{1}{x^2}\). This derivation informs us about the slope of the tangent line to the curve at any point \(x\). A positive slope indicates an increasing function, while a negative slope indicates a decreasing function. At any point where \(y'=0\), the curve has a horizontal tangent, known as a critical point, which could be either a local maximum, a minimum, or a point of inflection.

Second Derivative

The second derivative, denoted as \(y''\), provides information about the concavity of a function. In other words, it tells us whether the function is curving upwards or downwards. It also plays a crucial role in determining points of inflection, where the concavity of the function changes.

For the given function \(y = \frac{1}{x}\), we find the second derivative by differentiating the first derivative. The result \(y'' = \frac{2}{x^3}\) helps us understand how the curvature of the curve changes. If the second derivative is positive, the graph of the function is concave up, resembling the shape of a cup, and if it is negative, it is concave down, like an arch. These insights are pivotal in analyzing the geometry of curves and optimizing their properties.

Limit of a Function

The limit of a function is another cornerstone of calculus, describing the behavior of the function as the input approaches a certain value, which could be a real number or even infinity. It helps us understand the function's end behavior or predict the value that the function approaches near a specific point.

When evaluating the limit of the curvature \(K\) as \(x \rightarrow \infty\) for our function \(y = \frac{1}{x}\), we examine how the value of \(K\) changes. As \(x\) becomes very large, the terms containing \(x\) in the denominator of the curvature formula grow, effectively shrinking \(K\). Thus, the limit of \(K\) as \(x\) approaches infinity is zero. This tells us that as we move along the curve away from the origin, the curve becomes flatter, which is consistent with the idea of the curvature decreasing.