Welcome to Section 10.1 & 10.2!

Get ready to explore the fascinating world of parametric equations and their applications in calculus! These sections will introduce you to a new way of defining curves and how to perform calculus operations on them. Let's get started!

Section 10.1: Curves Defined by Parametric Equations

Often, describing a curve with a single function $y = f(x)$ isn't enough, especially if the curve fails the vertical line test. That's where parametric equations come in! Instead of defining $y$ directly as a function of $x$, we define both $x$ and $y$ as functions of a third variable, $t$, called a parameter.

We write these as:

  • $x = f(t)$
  • $y = g(t)$

As $t$ varies, the point $(x, y) = (f(t), g(t))$ traces out a curve in the coordinate plane. This curve is called a parametric curve.

Example: Consider the parametric equations $x = t^2 - 2t$ and $y = t + 1$. By eliminating the parameter $t$ ($t = y-1$), we can rewrite this as $x = (y-1)^2 - 2(y-1) = y^2 - 4y + 3$. Therefore, the curve represented by the given parametric equations is the parabola $x = y^2 - 4y + 3$.

Important Points:

  • The parameter $t$ doesn't necessarily represent time, although it often does in applications involving motion.
  • We can restrict the values of $t$ to a finite interval, resulting in a portion of the curve being traced. The initial and terminal points are very important here.
  • Graphing devices (calculators, software) are useful for visualizing parametric curves.

Section 10.2: Calculus with Parametric Curves

Now that we know how to define curves parametrically, let's apply calculus to them!

Tangents

To find the slope of the tangent line to a parametric curve, we use the following formula:

$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$, provided $\frac{dx}{dt} \neq 0$

A curve has a horizontal tangent where $\frac{dy}{dt} = 0$ (provided $\frac{dx}{dt} \neq 0$), and a vertical tangent where $\frac{dx}{dt} = 0$ (provided $\frac{dy}{dt} \neq 0$).

We can also find the second derivative to determine the concavity of the curve:

$$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$$.

Areas

The area under a parametric curve $y = g(t)$ and $x = f(t)$ from $t = \alpha$ to $t = \beta$ is given by:

$$A = \int_{\alpha}^{\beta} g(t)f'(t) dt$$

Arc Length

The arc length of a parametric curve from $t = \alpha$ to $t = \beta$ is given by:

$$L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$$

Surface Area

The surface area obtained by rotating the parametric equations about the x-axis from $t = \alpha$ to $t = \beta$ is given by:

$$S = \int_{\alpha}^{\beta} 2\pi y \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$$

Keep Practicing!

Parametric equations provide a powerful tool for describing curves and motion. Practice applying these formulas and concepts to various examples, and you'll master this topic in no time! Good luck, and happy calculating!