Problem 82 - Essential Calculus

Question

How are the slopes of tangent lines determined in polar coordinates? What are tangent lines at the pole and how are they determined?

Step-by-Step Solution

Answer

The slopes of tangent lines in polar coordinates are determined similarly to how they would be in Cartesian coordinates, by taking the derivative with respect to theta. At the pole, technically the tangent line is undefined, however, it can be determined based on the behavior of the curve passing through the pole.

Step 1: Understand the concept of the slope

In any set of coordinates, the slope of a tangent at a certain point on the curve is the derivative of the function at that point. It is given by \(\frac{dy}{dx}\). This concept is crucial to solving this problem, as we will need it to understand further steps.

Step 2: Calculate the slope in polar coordinates

To calculate the slope in polar coordinates, we will first convert to Cartesian coordinates, i.e., \(x = r cos(\theta)\) and \(y = r sin(\theta)\). The slope of the tangent at a certain point is then given by \(\frac{dy}{dx}\), which is interpreted as the rate of change of r with respect to \(\theta\). This is given by \(\frac{dr}{d\theta}\) / \(\frac{-r}{sin(\theta)}\). The negative sign indicates that as \(\theta\) increases, r decreases (we are moving in a clockwise direction).

Step 3: Find tangent lines at the pole

In polar coordinates, the pole is the equivalent of the origin in Cartesian coordinates. Because the pole is a single point (0,0), the tangent line at the pole is undefined, as there is no unique line that can be drawn tangent to a single point. However, depending on the curve and its behavior near the pole, we can imagine many potential tangent lines with different slopes passing through the pole.