Problem 14 - Calculus Early Transcendentals

Question

Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates. $$r=2 \theta ;\left(\frac{\pi}{2}, \frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

Answer: The slope of the line tangent to the polar curve $$r = 2\theta$$ at any given point is 2. 2. What is the equation of the tangent line to the polar curve $$r = 2\theta$$ at the intersection with the origin in polar coordinates? Answer: The equation of the tangent line to the polar curve $$r = 2\theta$$ at the intersection with the origin in polar coordinates is $$r = 2\theta$$.

Step 1: Find the tangent's slope in polar coordinates

To find the slope of the line tangent to the polar curve $$r = 2\theta$$ at the given points, we need to find the derivative of the curve regarding $$\theta$$, which is $$\frac{dr}{d\theta}$$. In this case, $$\frac{dr}{d\theta} = 2$$ because the function $$r$$ is a linear function of $$\theta$$.

Step 2: Find the slope at the given points

Now, we need to find the slopes at the specific points $$(\frac{\pi}{2}, \frac{\pi}{4})$$ given in the problem. The derivative is constant ($$\frac{dr}{d\theta} = 2$$), so the slope remains the same for any point on this curve: m = 2.

Step 3: Find the intersection with the origin

To find the intersection with the origin, we need to find the points on the curve $$r = 2\theta$$ where $$r = 0$$. It is important to note that this occurs only when $$\theta = 0$$, and thus the intersection point is $$(0, 0)$$.

Step 4: Find the equation of the tangent line at the intersection with the origin

In polar coordinates, the equation of a tangent line to a curve at a specific point can be found as $$r = r_0 + \left[\frac{dr}{d\theta}(\theta_0)\right](\theta-\theta_0)$$. In this case, the intersection point is $$(r_0, \theta_0) = (0, 0)$$, and we have already found $$\frac{dr}{d\theta}(\theta_0) = 2$$. Therefore, the equation of the tangent line at the origin in polar coordinates is: $$r = 2(\theta - 0) \Rightarrow r = 2\theta$$.

Key Concepts

Derivative of Polar Functions

Understanding the derivative of polar functions is vital for analyzing the behavior of curves in polar coordinates. Much like in Cartesian coordinates, where we use derivatives to find slopes of tangent lines, in polar coordinates, derivatives allow us to find the rate at which the radius r changes with respect to the angle \( \theta \).

To determine \( \frac{dr}{d\theta} \) for the function \( r = f(\theta) \), we simply differentiate r with respect to \( \theta \). In the example \( r = 2\theta \), the derivative is constant and equals to 2. This represents a steady rate of change, indicating that the curve moves away from the origin at a consistent speed as \( \theta \) increases.

How to Take the Derivative

To compute \( \frac{dr}{d\theta} \), apply the ordinary rules of differentiation to the polar equation as if it were a regular function. If the polar equation is more complex and includes trigonometric functions like sine and cosine, you'd use the respective trigonometric differentiation rules.

Slope of Tangent Line

The slope of a tangent line to a curve gives us a snapshot of the angle that tangent makes with respect to the horizontal axis. In Cartesian coordinates, this concept is straightforward, as we directly use the derivative at a point to find the slope of the tangent there.

In polar coordinates, it's a bit more nuanced. The slope of the tangent at a point \( (r, \theta) \) on the curve can still be found using derivatives, but because we're working with a radius and an angle, the process and interpretation differ. In our case, for the linear function \( r = 2\theta \), the slope at any given point is 2, which is derived from the constant rate of change in the radius as the angle increases.

Visualizing the Slope

Imagine the polar curve as a path on which an object travels. The 'slope' represents how quickly the object moves away from the origin as the angle changes. A constant slope, like in our example, means the object's distance from the origin increases uniformly with the angle.

Polar Coordinate System

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point (usually the origin, called the pole) and an angle from a reference direction (usually the positive x-axis).

In polar coordinates, the two values used to define a point are r, the radial distance from the pole, and \( \theta \), the angle measured in radians from the reference direction. This system is particularly useful when dealing with circular or spiral-shaped objects, where points are naturally oriented around a center point.

From Polar to Cartesian Coordinates

Conversion between polar and Cartesian coordinates is often necessary. For a point with polar coordinates \( (r, \theta) \), the Cartesian coordinates (x, y) can be found using x = r\cos(\theta) and y = r\sin(\theta). Understanding these conversions is essential when switching between coordinate systems to analyze curves or solve equations in a more familiar context.