Parametric Equations

Question 0

Parametric curve

Answer 0

A set of points generated by a parametric equation

Question 1

Parametric equation

Answer 1

An equation that either specifies x or y in terms of a third variable (usually ‘t’)

Question 2

Parameter

Answer 2

The third variable, usually ‘t’ upon which the independent or dependent variable depends

Question 3

Forward orientaion

Answer 3

The direction in which the curve is generated as the parameter increases

Question 4

Derivatives of Parametric curves

Answer 4

Written as [dy/dt]/[dx/dt]
h’(t)/g’(t)
h(t)- function that outputs the y-value
g(t)- function that outputs the x-value

Question 5

Parametric equations for an ellipse/circle

Answer 5

x(θ)=cos(θ)*‘r’

y(θ)=sin(θ)*‘r’

Question 6

Parametric equations for a hyperbola

Answer 6

x(θ)=sec(θ)‘r’
y(θ)=tan(θ)‘b’

b= √(c^2-a^2)
a= vertices of (0,±a) or (±a,0)
c= foci at (0,±a) or (±a,0)

Question 7

Parametric equations for a parabola

Answer 7

x(θ)=2θ*r

y(θ)=θ^2*r

Question 8

Parametric equations in colloidal dispersion/concentration

Answer 8

S

Question 9

Translations and rotations of axis

Answer 9

A

Question 10

Celestial orbit in Parametric equations

Answer 10

S

Question 11

Length of the polar curve

Answer 11

L= ∫ √(f(θ)^2+f’(θ)^2)dθ

Question 12

Arc Length with time

Answer 12

s(t)=∫|v(u)|du

from a to t

Question 13

Curvature

Answer 13

How much the tangent vector changes given time

Question 14

Curvature (given arc-length)

Answer 14

k=|d(r’/|r’|)/ds|

Question 15

Curvature (given time)

Answer 15

k=|d(r’/|r’|)/dt|/|v|
Also
=|T’(t)/r’(t)|

Question 16

Curvature (given a point)

Answer 16

k=(r”xr’)/(|v|^3)

Question 17

T-value of parametric equations

Answer 17

T=(r’/|r’|)