Defining and Differentiating Parametric Equations

#9.1 Defining and Differentiating Parametric Equations

🪐 Unit 9 of AP Calculus BC deals with three major topics:

1. Parametric equations
2. Polar coordinates - a two-dimensional coordinate system dealing with a line’s distance from the origin ($r$) and the angle said line makes with the positive x-axis ($θ$).
3. Vector-valued functions - functions that returns a vector after taking one or more variables.

We’ll dive deeper into the second and third topics in future sections; for now, we’ll focus on parametric functions as they actually tell us a lot of information about real-world phenomena like projectile and circular motion.

#💭 What is a Parametric Function?

Parametric functions are a set of related functions where x and y are independent from each other, but they are connected using the dummy variable t, which represents time. When we use the Cartesian graph, we assume that we are moving along the x-axis in only one direction at a constant rate. However, parametric equations give us more freedom to manipulate horizontal motion. 🗺️

A parametric equation would look something like this:

$$x(t)=t^2-1, y(t)=3t$$

In this equation, your x-coordinate would be determined by $t² - 1$ and your y-coordinate would be determined by 3t. So, when t = 1, you would plot the point (0, 3). In a parametric equation, t isn’t actually on the graph; we just use t as our constant so that our points are independent from one another.

There are several methods for calculating derivatives of real-valued functions, such as the limit definition, the power rule, the product rule, and the quotient rule. These methods can be extended to parametric functions, which are func...