Problem 24 - Calculus

Question

When is there a dimple in \(r=1+b \cos \theta\) ? From \(x=\) \((1+b \cos \theta) \cos \theta\) find \(d x / d \theta\) and \(d^{2} x / d \theta^{2}\) at \(\theta=\pi .\) When that second derivative is negative the limacon has a dimple.

Step-by-Step Solution

Answer

A dimple occurs in the limacon when \( 0 < b < 1 \).

Step 1: Understand the problem

The given function is a polar equation of the form \( r = 1 + b \cos\theta \), representing a limacon. We want to determine when this limacon has a dimple. A dimple occurs if the second derivative \( \frac{d^2 x}{d\theta^2} \) at \( \theta = \pi \) is negative.

Step 2: Substitute and simplify

Start by substituting \( \theta = \pi \) into the equation for \( x \). The equation is given as \( x = (1 + b\cos\theta)\cos\theta \). Calculate \( x \) at \( \theta = \pi \):\[\begin{align*}\cos\pi &= -1, \1 + b\cos\pi &= 1 - b, \x &= (1 - b)(-1) = b - 1.\end{align*}\]

Step 3: First derivative

Find the first derivative of \( x = (1+b\cos \theta) \cos \theta \) with respect to \( \theta \). Use the product rule:\[ \begin{align*}x &= (1 + b\cos\theta)\cos\theta \\frac{dx}{d\theta} &= \frac{d}{d\theta}[(1 + b\cos\theta)\cos\theta] \&= (1+b\cos\theta)(-\sin\theta) + (-b\sin\theta)(\cos\theta).\end{align*} \] Simplify to:\[\frac{dx}{d\theta} = -(1+b\cos\theta)\sin\theta - b\sin\theta\cos\theta.\]

Step 4: Second derivative

Now, find the second derivative by differentiating \( \frac{d x}{d \theta} \) with respect to \( \theta \):\[ \frac{d^2 x}{d\theta^2} = -\frac{d}{d\theta}[(1 + b\cos\theta)\sin\theta + b\sin\theta\cos\theta].\] Use the product rule on each term and simplify.

Step 5: Evaluate second derivative

Evaluate \( \frac{d^2 x}{d\theta^2} \) at \( \theta = \pi \). Given that \( \sin\pi = 0 \) and \( \cos\pi = -1 \), simplify to check when the second derivative is negative, indicating a dimple.Substitute \( \theta = \pi \) and simplify:\[ \frac{d^2 x}{d\theta^2} = -b(1 + b\cos\pi)(1) = -b(1 - b)\] \[= -b + b^2.\] Evaluate for negativity:\[ b-b^2 > 0 \Rightarrow b(1-b) > 0. \]

Step 6: Determine condition for dimple

Given \( b(1-b) > 0 \), the dimple exists for values of \( b \) such that \( 0 < b < 1 \). This is determined from conditions when a negative product occurs.

Key Concepts

Limacon

The limacon is a fascinating type of polar curve, which is derived from the French word for "snail." Its general form is given by the equation \( r = 1 + b \cos \theta \). This curve exhibits different shapes depending on the value of \( b \).

• When \( b = 0 \), the limacon becomes a circle.
• If \( 0 < b < 1 \), it exhibits a dimple, a subtle inward bend.
• With \( b = 1 \), it becomes a cardioid, which resembles the shape of a heart.
• For \( b > 1 \), the limacon has an inner loop.

It's the distinctive ability of the limacon to morph between these forms that makes it an interesting subject of study in polar coordinates.

Second Derivative

The second derivative is a vital concept in calculus, representing the rate of change of the rate of change. It's the derivative of the derivative of a function.
This concept helps to understand the concavity and convexity of curves.
For a curve described by \( x \), the second derivative \( \frac{d^2 x}{d\theta^2} \) is used to determine the nature of the curve at a given point. In the context of limacons, when the second derivative is negative, it indicates a concave section known as a dimple.
This tells us about changes in the curve's behavior, making it an invaluable tool for analyzing the geometry of polar curves like the limacon.

Dimple Condition

The dimple condition in a limacon occurs when there is a slight inward dent on the curve. For a limacon described by \( r = 1 + b \cos \theta \), the presence of a dimple depends on the parameter \( b \). Specifically, a dimple is observed when \( 0 < b < 1 \).
In this range, the second derivative \( \frac{d^2 x}{d\theta^2} \) at \( \theta = \pi \) becomes negative, confirming the presence of a dimple.
This condition comes from the analysis of the derivative:

• The first negative result for the second derivative ensures that the concavity at \( \theta = \pi \) is inward, forming a dimple.

The dimple lends the limacon its unique aesthetics, which is just one of the many interesting features this versatile curve can have.

Cosine Function

The cosine function is a fundamental trigonometric function that varies between -1 and 1. Crucial to polar coordinates, it describes the horizontal component of a point in the polar plane.
When used in the equation \( r = 1 + b \cos \theta \), the cosine function modifies the radial distance depending on the angle \( \theta \).

• At \( \theta = 0 \), cosine is maximum, adding its full potential to the radius.
• At \( \theta = \pi \), cosine reaches its lowest, reducing the radius.

It's the periodic nature and amplitude of cosine that primarily influence the shape and characteristics of polar curves such as the limacon. Therefore, understanding how cosine interacts with \( b \) in different scenarios is key to analyzing the behavior of these curves.