Parametric and Polar

parametric slope

dy/dx= dy/dt

dx/dt

parametric second derivative

d2y/dx2= (dy/dx)’

dx/dt

length of a parametric curve

L = ∫ t2√((dx/dt)^2 + (dy/dt)^2)dt

t1

magnitude formula

√a^2 + b^2

speed formula

√(dx/dt)^2 + (dy/dt)^2

polar conversion to x

x = rcosΘ

polar conversion to y

y =rsinΘ

rate r changes in a given direction

dr/dΘ

slope of a polar curve

dy/dx = dy/dΘ

dx/dΘ

polar area

1/2 ∫Θ2 (f(Θ))^2 dΘ

Θ1

half angle formula

1+cos(2Θ) = cos^2Θ (negative for sin)

2

distance between polar curves

g(Θ)-f(Θ) (outer - inner)

area outside f(Θ) and inside g(Θ)

1/2 ∫Θ2 (g(Θ))^2 - (f(Θ))^2 dΘ

Θ1

length of polar curves

L = ∫ Θ2√((r)^2 + (dr/dΘ)^2)dΘ

Θ1

circles with range 0 < Θ < π

r=asinΘ, r=acosΘ

circles with range 0 < Θ < 2π

r=a

limacons where |a/b| < 1

inner loop

limacons where |a/b| = 1

cardioid

limacons where 1 < |a/b| < 2

dimple

limacons where |a/b| > 2

convex

range of limacons

0 < Θ < 2π

limacons

r=a+/-bcosΘ or r=a+/-bsinΘ

rose curves

r=asin(nΘ) or r=acos(nΘ)

odd rose curves range

0 < Θ < π and has n petals

even rose curves range

0 < Θ <2π and has 2n petals

lemniscates

r^2=a^2sin(2Θ) or r^2=a^2cos(2Θ)

lemniscates range

0 < Θ < 2π

types of graphs with 0 < Θ < 2π

regular circles, all limacons, even rose curves, lemniscates, and cardiods

types of graphs with 0 < Θ < π

trig circles and odd rose curves