Question

How do you find the slope of the line tangent to the polar graph of \(r=f(\theta)\) at a point?

Short answer

Question: Find the slope of the tangent line to the polar graph of \(r = f(\theta)\) at a given point. Answer: The slope of the tangent line to the polar graph of \(r = f(\theta)\) at a given point can be found by dividing the derivatives \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\), which is given by the expression: \(\frac{dy}{dx} = \frac{\frac{df(\theta)}{d\theta}\sin(\theta) + f(\theta)\cos(\theta)}{\frac{df(\theta)}{d\theta}\cos(\theta) - f(\theta)\sin(\theta)}\). To find the slope for a specific value of \(\theta\), substitute that value into the expression.

Step-by-step solution

Convert polar equation to parametric equation

We are given a polar equation \(r = f(\theta)\). To convert this polar equation to parametric equations for \(x\) and \(y\), we use the identities \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). So, we can write the parametric equations as: \(x(\theta) = f(\theta)\cos(\theta)\) \(y(\theta) = f(\theta)\sin(\theta)\)

Differentiate the parametric equations w.r.t. \(\theta\)

In the next step, we will find the derivatives of the parametric equations with respect to \(\theta\). This will help us determine the change in \(x\) and \(y\) as a function of \(\theta\). By applying the product rule, we get the derivatives as follows: \(\frac{dx}{d\theta} = \frac{df(\theta)}{d\theta}\cos(\theta) - f(\theta)\sin(\theta)\) \(\frac{dy}{d\theta} = \frac{df(\theta)}{d\theta}\sin(\theta) + f(\theta)\cos(\theta)\)

Finding the slope of the tangent line

Finally, we can find the slope of the tangent line by dividing the derivatives \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\): \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{\frac{df(\theta)}{d\theta}\sin(\theta) + f(\theta)\cos(\theta)}{\frac{df(\theta)}{d\theta}\cos(\theta) - f(\theta)\sin(\theta)}\) This expression represents the slope of the tangent line to the polar graph of \(r = f(\theta)\) at a given point. To find the slope for a specific value of \(\theta\), substitute that value into the expression.

Key concepts

Polar Coordinates

Polar coordinates are a way of expressing a point in the plane using a distance from the origin and an angle from a reference direction. Instead of using the familiar \(x, y\) coordinates of the Cartesian system, polar coordinates use \(r, \theta\). Here, \(r\) represents the radius \(r\) or the distance from the origin to the point, and \(\theta\) is the angle formed with the positive x-axis. This system is especially useful when dealing with circular or spiral patterns, where the radial distance and angle naturally describe the position within the plane.
To convert from polar to Cartesian coordinates, we use trigonometric identities:

\(x = r \cos(\theta)\)

\(y = r \sin(\theta)\)

These transformations allow us to switch back and forth between coordinate systems depending on the problem at hand, enabling easier computation of slope as part of calculus problems.

Parametric Equations

In many math problems, especially those involving curves, we often utilize parametric equations. These equations express the coordinates \(x\) and \(y\) not as simple functions of each other, but as separate functions of a third parameter, usually \(t\) or \(\theta\).
Parametric equations are greatly beneficial because they can represent complex curves that are otherwise difficult to describe with a single Cartesian equation.
For an object moving along a path, we could have:

• \(x(t) = f(t)\)
• \(y(t) = g(t)\)

In the case of converting polar equations to parametric, we use \(x(\theta) = f(\theta) \cos(\theta)\) and \(y(\theta) = f(\theta) \sin(\theta)\).
This approach makes it easy to apply calculus techniques like differentiation to find the slope of a tangent line, as parametric equations separate movement into easily manageable components.

Product Rule

The product rule is an essential tool in calculus used to differentiate products of two functions. If you have two functions \(u(t)\) and \(v(t)\), their product \(uv\) has a derivative given by:

• \(\frac{d}{dt}[u(t)v(t)] = u'(t)v(t) + u(t)v'(t)\)

This means that when dealing with a product of functions, you differentiate the first function and multiply it by the second function, then do the reverse. This rule allows us to handle more complex differentiation scenarios such as in the derivative of parametric equations used in finding tangent lines.
Both \(x(\theta)\) and \(y(\theta)\) derived from polar coordinates use the product rule when finding their derivatives. This helps us to separate how each part of the curve changes in response to changes in \(\theta\), enabling accurate calculation of the slope.

Derivatives

In calculus, derivatives manage the rate at which one quantity changes with respect to another.
They are fundamental in finding slopes of functions and analyzing behavior of differentiable curves. In the context of parametric equations, we consider derivatives with respect to \(\theta\), the parameter.
Using the derivatives \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\), we can find the slope of the tangent line at any point along the curve by forming the fraction \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\).
This slope tells us not only how steeply a line is changing at a specific point, but also the direction it is pointing. Understanding how derivatives work is crucial for visualizing and solving problems involving tangents and slopes in different coordinate systems.