Parametric Differentiation Classroom
55.1 Parametric Equations Intro
In calculus, we often describe curves using a single equation relating $x$ and $y$ (Cartesian equation). However, some curves are more naturally described by expressing both $x$ and $y$ as separate functions of a third independent variable, called a parameter. This approach uses parametric equations.
For example, a curve might be defined by $x = f(t)$ and $y = g(t)$, where $t$ is the parameter. The parameter often represents time, but it can be any variable. Parametric equations are particularly useful for describing motion along a curve, or curves that are not functions (e.g., circles, ellipses).
55.2 Common Parametric Equations
Many common geometric shapes and paths can be represented using parametric equations. Some examples include:
- Line: $x = x_0 + at$, $y = y_0 + bt$
- Circle: $x = r \cos(\theta)$, $y = r \sin(\theta)$ (parameter is $\theta$)
- Ellipse: $x = a \cos(\theta)$, $y = b \sin(\theta)$ (parameter is $\theta$)
- Parabola: $x = at^2$, $y = 2at$ (a standard form)
To convert a set of parametric equations into a Cartesian equation, the goal is to eliminate the parameter. This often involves substitution or using trigonometric identities.
55.3 Differentiating Parameters
To find the derivative $\frac{dy}{dx}$ when $x$ and $y$ are given in terms of a parameter $t$ (or $\theta$), we use the chain rule:
$$ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} $$provided that $\frac{dx}{dt} \neq 0$. This formula allows us to find the gradient of the curve at any point without first converting it to a Cartesian equation, which can be difficult or impossible for complex parametric forms.
55.4 Advanced Parametric Problems
Parametric differentiation can extend to finding second derivatives, gradients at specific points, or even tangents and normals to parametric curves.
The second derivative $\frac{d^2y}{dx^2}$ is found by differentiating $\frac{dy}{dx}$ with respect to $x$. However, since $\frac{dy}{dx}$ is usually in terms of the parameter, we again use the chain rule:
$$ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{dy}{dx}\right) \cdot \frac{dt}{dx} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}} $$These techniques allow for a comprehensive analysis of the properties of parametric curves.