Problem 11 - 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. \(r=4 \cos 2 \theta ;\) at the tips of the leaves

Step-by-Step Solution

Answer

#tag_title# Step 2: Find the critical points of \(r(\theta)\) # #tag_content# To find the critical points, we set \(\frac{dr}{d\theta}\) to zero and solve for \(\theta\): \(-8 \sin{2\theta} = 0\) This implies that \(\sin{2\theta} = 0\). So, for the tips of the leaves, we must find values of \(\theta\) where \(\sin{2\theta} = 0\). This occurs at \(\theta = k \frac{\pi}{4}\), where \(k\) is an integer.

Step 1: Use the polar tangent formula

For a polar curve \(r = f(\theta)\), the slope of the tangent is:
\(\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta}\)

Step 2: Evaluate at the given points

Compute \(\frac{dr}{d\theta}\) and substitute the given values of \(\theta\) to find the slope at each point. At the origin (\(r=0\)), the tangent line equation is \(\theta = \theta_0\) where \(f(\theta_0) = 0\).

Key Concepts

Polar Coordinates

Polar coordinates are an alternative to the more familiar Cartesian (or rectangular) coordinate system, and they are particularly useful in situations where the geometry of a problem exhibits some form of radial symmetry.

In polar coordinates, each point in the plane is represented by the pair \( (r, \theta) \) where \( r \) is the radial distance from the origin (the fixed point from which all distances are measured), and \( \theta \) is the angular coordinate, representing the counterclockwise angle, usually in radians, from the positive x-axis.

A polar curve is then defined as a set of points \( (r(\theta), \theta) \) where \( r(\theta) \) is a function describing the radial distance for each angle \( \theta \). These curves can often be more complex and beautiful than Cartesian graphs, and they are described by equations such as \( r = 4 \cos 2\theta \) in the given exercise.

Tangent Line Equation

The equation of a tangent line to a curve at a given point provides important information about the behavior of the curve at that locality. In the context of polar coordinates, determining the equation of the tangent line involves understanding both the geometry of the curve and the calculus associated with it.

Ideally, we're interested in the slope of the tangent line at the point of interest, as this will give us insight into the direction the line heads off from that point. In a polar system, this slope, typically denoted as \( \frac{dy}{dx} \), can be described through \( r \) and \( \theta \) by implicitly differentiating \( r(\theta) \) and then using trigonometric identities or conversion factors to relate it to Cartesian coordinates.

It's important to remember that polar equations can intersect the origin; in this case, special attention is needed as the conventional notion of slope may not apply, and the tangent line might be better described using polar terms.

Derivative of Polar Equations

The derivative in polar coordinates carries the essence of finding how rapidly the distance \( r \) from the origin changes with respect to the angle \( \theta\). This derivative can symbolize the slope of the tangent line to a polar curve if properly transformed.

In the example exercise, we have the polar equation \( r = 4 \cos 2\theta \) for which we found the derivative with respect to \( \theta \) as \( \frac{dr}{d\theta} = -8 \sin{2\theta} \). This result details how \( r \) changes as \( \theta \) varies, giving us information we need to further determine the slope of the tangent line.

It's also worth noting that when dealing with curves defined in polar coordinates, using the chain rule for differentiation is often necessary, especially when the polar equation features trigonometric functions, which is common. This process ultimately assists in finding the tangent lines to polar curves at given points.