M408M Learning Module Pages
Main page ## Chapter 10: Parametric Equations and Polar Coordinates## Learning module LM 10.1: Parametrized Curves:3 kinds of functions, 3 kinds of curvesThe cycloid Visualizing parametrized curves Tracing curves and ellipses Lissajous figures ## Learning module LM 10.2: Calculus with Parametrized Curves:## Learning module LM 10.3: Polar Coordinates:## Learning module LM 10.4: Areas and Lengths of Polar Curves:## Learning module LM 10.5: Conic Sections:## Learning module LM 10.6: Conic Sections in Polar Coordinates:## Chapter 12: Vectors and the Geometry of Space## Chapter 13: Vector Functions## Chapter 14: Partial Derivatives## Chapter 15: Multiple Integrals |
## 3 kinds of functions; 3 kinds of curvesFor most of calculus, the word ``function'' has meant a machine that inputs a number and outputs another number. Something like $f(x)=x^2$. But a function can have more than one input or more than one output. Each kind of function gives a different way to describe a curve in the plane. Every ordinary function with one input and one output is associated
with a Some curves don't pass the vertical line test and can't be written as
graphs. We usually describe the unit circle by an equation:
$x^2+y^2=1$. This is an example of a The third way to get a curve is to track the trajectory of a moving
point. If we specify both $x$ and $y$ as functions of a Every graph can be written as a parametrized curve. Just take $x(t)=t$, $y(t)=f(t)$. We move from left to right at constant horizontal speed, and let the graph do the rest. The unit circle is a parametrized curve with $x(t)=\cos(t)$ and $y(t)=\sin(t)$. This traces the circle counter-clockwise, starting at $(1,0)$, going around once every $2\pi$ seconds. |