Parametric Equations: Curves, Surfaces, and Applications

All flashcards

What is a parametric equation?

An equation that expresses a quantity, such as a coordinate, as a function of one or more parameters.

What is a parametric representation?

The collection of parametric equations that together describe an object, such as a curve or surface.

Why can a parameter be something other than time?

A parameter is an independent variable used to generate points; it can be any convenient quantity.

What does implicitization do?

It eliminates the parameter from parametric equations to obtain a relation involving coordinates only.

What is the implicit equation of the circle parameterized by $x=a\cos t$, $y=a\sin t$?

$x^2+y^2=a^2$, from the identity $\cos^2 t+\sin^2 t=1$.

Quiz questions

1. How can the graph of $y=f(x)$ be parameterized using a real parameter $t$?Answer: $x=t,\ y=f(t)$
2. Which parameterization describes an ellipse centered at the origin with semi-axis lengths $a$ and $b$?Answer: $x=a\cos t,\ y=b\sin t$
3. For the helix $x=a\cos t$, $y=a\sin t$, $z=bt$, what is the vertical change after one turn when $b>0$?Answer: $2\pi b$
4. Which parametric equations describe a line through $(x_0,y_0,z_0)$ with direction vector $(a,b,c)$?Answer: $x=x_0+at,\ y=y_0+bt,\ z=z_0+ct$
5. If position is given by a differentiable vector function $\mathbf r(t)$, how is velocity obtained?Answer: Differentiate each coordinate to obtain $\mathbf r'(t)$.