9.1-9.8 parametric/polar/vector Flashcards
What is the standard form of a parabola and its directrix and foci?
A vertical parabola: (x-h)2=4p(y-k) where the directrix is y=k-p and the focus is (h,k+p)
A horizontal parabola: swap (y-k) for (x-h), directrix is x=h-p, focus is (h+p,k)
How is arc length found normally and for parametrics?
integral from a to b of the sqrt of 1+[dy/dx]2
or integral from a to b of the sqrt of [dx/dt]2+[dy/dt]2 or integral from α to β or the sqrt of r2+[dr/dθ]2
What is the standard form of an ellipse and its eccentricity and foci?
Where a is bigger than b, major axis of 2a is horizontal:(x-h)2/a2+(y-k)2/b2=1
major axis is vertical: switch a and b as denominators
Eccentricity is e=c/a
The two foci are on the major axis c from the center, where c2=a2-b2
What is the standard form of a hyperbola and its vertices and foci?
Transverse axis is horizontal:
(x-h)2/a2-(y-k)2/b2=1
Transverse axis is vertical:
switch which is subtracted
Vertices are a from the center
Foci are c from the center, where c
What is the second (or higher) derivative of parametric equations?
d2y/dx2=d/dx[dy/dx]=d/dt[dy/dx]/dx/dt
What is slope in polar form?
r=f(θ)
[f(θ)cos(θ)+f’(θ)sin(θ)]/[-f(θ)sin(θ)+f’(θ)cos(θ)] provided that dx/dθ≠0
What is the equation and shape of a limacon?
a>0 and b>0
r=a+/-bcos(θ)
horizontal loop/dimple at π
r=a+/-bsin(θ)
vertical loop/dimple at 3π/2
a/b<1 inner loop
a/b=1 cardioid (no loop, touch pole)
1<a/b<2 dimpled
a/b>2 convex
What is the equation and shape of a rose curve?
r=acos(nθ)
r=asin(nθ)
n petals when n odd
2n petals when n even
n>=2
What is the equation and shape of a circle or lemniscate?
circle going from x=0 to x=a (radius of a/2): r=acosθ
circle going from y=0 to y=a (radius of a/2): r=asinθ
lemniscate (infiniti symbol) with full length not on axis: r2=a2sin(2θ)
with full length on axis:
r2=a2cos(2θ)
