Question

Given the polar curve \(r=e^{a \theta}, a>0,\) find the curvature \(K\) and determine the limit of \(K\) as (a) \(\theta \rightarrow \infty\) and (b) \(a \rightarrow \infty\).

Short answer

The curvature \(K\) of the polar curve is \(\frac{a^2e^{2a \theta}}{(1 + a^2)^{3/2}}\). As \(\theta \rightarrow \infty\), the limit of \(K\) is \(+ \infty\), and as \(a \rightarrow \infty\), the limit of \(K\) is \(0\).

Step-by-step solution

Finding the first and second derivatives of \(r\)

The first derivative of \(r\) with respect to \(\theta\) is given by \(r'= ae^{a \theta}\). Taking the derivative of \(r'\) with respect to \(\theta\) gives us the second derivative \(r''= a^2e^{a \theta}\).

Apply the curvature formula

The curvature \(K\) of a curve in polar coordinates is given by \(K = \frac{|r'^2 - r*r''|}{(r^2 + r'^2)^{3/2}}\). Substituting the values of \(r\), \(r'\), and \(r''\) into the curvature formula and simplifying gives \(K = \frac{|a*a^2e^{2a \theta} - a^3e^{3a \theta}|}{(e^{2a \theta} + a^2e^{2a \theta})^{3/2}} = \frac{a^2e^{2a \theta}}{(1 + a^2)^{3/2}} \)

Compute the limit of \(K\) as \(\theta \rightarrow \infty\)

The limit of \(K\) as \(\theta \rightarrow \infty\) is \(\lim_{\theta \to \infty} K = \lim_{\theta \to \infty} \frac{a^2e^{2a \theta}}{(1 + a^2)^{3/2}}. Because \(e^{2a \theta}\) grows much faster than the denominator as \(\theta\) goes to infinity, this limit is \(+ \infty\).

Compute the limit of \(K\) as \(a \rightarrow \infty\)

The limit of \(K\) as \(a \rightarrow \infty\) is \(\lim_{a \to \infty} K = \lim_{a \to \infty} \frac{a^2}{(1 + a^2)^{3/2}}. As \(a\) goes to infinity, the denominator increases much faster than the numerator, so this limit is \(0\).

Key concepts

Polar Coordinates

Polar coordinates are an alternative to the more commonly used Cartesian coordinate system. In polar coordinates, each point on a plane is determined by a distance from a reference point, called the pole (similar to the origin in Cartesian coordinates), and an angle from a reference direction, typically the positive x-axis.

The general form of a polar coordinate is \( (r, \theta) \), where \( r \) represents the radial distance from the pole, and \( \theta \) is the angle in radians, measured counter-clockwise from the reference direction.

One key aspect when working with polar coordinates is the ability to convert between polar and Cartesian coordinates. For instance, given polar coordinates \( (r, \theta) \), the corresponding Cartesian coordinates \( (x, y) \) can be found using the formulas: \[ x = r \cos(\theta) \] and \[ y = r \sin(\theta) \].

Conversely, to convert Cartesian coordinates to polar coordinates, the formulas are: \[ r = \sqrt{x^2 + y^2} \] and \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \], where \( \tan^{-1} \) represents the inverse tangent function, commonly known as arctan.

Polar coordinates are particularly useful when dealing with curves and shapes that are naturally circular or spiral, as their equation forms may be simpler and more intuitive in the polar system.

Derivatives in Calculus

Derivatives are a fundamental concept in calculus, reflecting how a function changes as its input changes. In a geometrical sense, the derivative of a function at a point is the slope of the tangent line to the function's graph at that point.

The derivative of a function \( f(x) \) with respect to \( x \) is denoted by \( f'(x) \) or \( \frac{df}{dx} \), and it's defined as the limit of the average rate of change of the function as the interval considered approaches zero: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \].

When dealing with polar coordinates, we often need to take derivatives of functions with respect to the angle \( \theta \). If a curve is given by a polar equation \( r=f(\theta) \), the derivatives \( r' \) and \( r'' \) represent the rate of change of the radial distance with respect to the angle, and the rate of change of that rate, respectively. These derivatives are crucial for calculating other properties of the curve, such as curvature.

Higher-order derivatives represent further rates of change; for instance, the second derivative \( f'' \) gives us information about the concavity of the function and helps in understanding the behavior of the curve's bending or acceleration.

Limits in Calculus

Limits are another foundational idea in calculus, describing the behavior of a function as its argument approaches a particular value or infinity. In formal terms, the limit of a function \( f(x) \) as \( x \) approaches some value \( c \) is the value that \( f(x) \) gets closer to as \( x \) gets infinitely close to \( c \).

The mathematical notation for the limit is \( \lim_{x \to c} f(x) = L \), where \( L \) is the limit value. Limits are used to define derivatives and integrals and to deal with discontinuities and asymptotic behavior.

When evaluating limits, various strategies are used, such as direct substitution, factoring and canceling, or applying limit laws. In some cases, like indeterminate forms, more advanced techniques such as L'Hôpital's rule may be necessary.

In the context of our exercise, understanding limits helps us analyze the behavior of the curve's curvature as the angle \( \theta \) becomes very large, or as the parameter \( a \) in the polar equation increases without bound. These special limits provide insight into the curve's global behavior, which can differ significantly from its local properties.