Polar, Parametric and Vector Flashcards
x(t) =
When initial velocity is given
V(0)cos(theta)t + x(0)
y(t) =
When initial velocity is given
0.5gt^2+ v(0)sin(theta)t + y(0)
g = -9.8 m/s^2 or -32ft/s^2
dy/dx =
Given x(t) and y(t)
(dy/dt)/(dx/dt)
Same thing with theta instead of t
d^2y/dx^2
Given x(t) and y(t)
((dy/dx)’)/(dx/dt)
Arc length from t1 to t2
Given x(t) and y(t)
integral from t1 to t2 of sqrt((dx/dt)^2 + (dy/dt)^2)
What does this derivative tell you?
dx/dt
The gives the horizontal velocity of the — at time t
What does this derivative tell you?
dy/dt
This gives the vertical velocity of the — at time t
What does this derivative tell you?
dy/dx
This gives the slope of the curve/path the — takes/travels on
Speed
When you have a vector
abs(sqrt((x’(t))^2 + (y(t))^2)
This is just the length of the velocity vector
Area under a parametric equation
From t1 to t2
integral from t1 to t2 of (y(t) * (dx/dt))
Period of a circle
pi
Period of a limaçon
2pi
Period of a rose curve
2pi/n for odd petals
pi/n for even petals
n = number of petals
Convert Cartesian to Polar
x=
rcos(theta)
Convert Cartesian to Polar
y=
rsin(theta)
Convert Polar to Cartesian
r=
sqrt(x^2 + y^2)
Convert Polar to Cartesian
theta =
arctan(y/x)
What is dy/dx equal to at the pole?
tan(theta)
How to find tangent lines at the pole?
- find when r=0
- Check that dr/dtheta does not equal zero
- Find dy/dx
- Give answer as equation
Polar
What does x’ equal?
-rsin(theta) + r’cos(theta)
Polar
What does y’ equal?
rcos(theta) + r’sin(theta)
Polar
What does dy/dx equal?
(rcos(theta) + r’sin(theta))/(-rsin(theta) + r’cos(theta))
Steps to find polar graph’s tangent lines (or points furthest from the origin)
- dy/dx equals either 0 or undifined
- Not a shart point (dr/dtheta exists)
- Theta must be in domain
Area under a polar curve
From a radians to b radians
0.5 * integral from a to b of (r(theta))^2 dtheta
You only do this for one petal or section at a time
Polar
Arc length from a to b
integral from a to b of sqrt((r^2) + (dr/dtheta)^2)dtheta
