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Level 640

Parametrics & Polars


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also has a derivative at t.
If f and g have derivatives at t, then the parameterized curve...
(dy/dt)/(dx/dt)
Slope of parameterized curve
(dy¹/dt)/(dx/dt)
Second derivative of parameterized curve
∫√((dx/dt)²+(dy/dt)²)dt
Length of paramterized curve
Surface area of parameterized curve
∫2πy√((dx/dt)²+(dy/dt)²)dt (about x-axis) OR ∫2πx√((dx/dt)²+(dy/dt)²)dt (about y-axis)
Component form of vector
v = <v₁, v₂>
Magnitude of vector
‖v‖ = √(v₁² + v₂²)
<u₁+v₁, u₂+v₂>
Vector addition (u+v)
<u₁-v₁, u₂-v₂>
Vector subraction (u-v)
<ku₁, ku₂>
Scalar multiplication (ku)
Angle between two vectors
cos⁻¹((u₁v₁+u₂v₂)/‖u‖‖v‖ (dot product over product of magnitudes)
Distance
∫|v(t)|dt = ∫√((v₁(t))²+(v₂(t))²)dt
Speed
‖v(t)‖
direction
v(t)/‖v(t)‖ (velocity vector over speed)
rcosθ
Polar x
rsinθ
Polar y
Polar x²+y²
arctan(y/x)
Polar θ
Rose
r=acos(nθ) OR r=asin(nθ)
Lemniscate
r²=a²cos(2θ) OR r²=a²sin(2θ)
limacon
r=a+bcosθ OR r=a+bsinθ
Cartioid
r=a+acosθ OR r=a+asinθ
(r,-θ), (-r,-θ), (-r,θ)
If (r,θ) is on the graph, so are...
it has all three
If a graph has two symmetries...
(rcosθ+r¹sinθ)/(-rsinθ+r¹cosθ)
Slope of polar curve
∫¹/₂r²dθ
Area inside polar curve
¹/₂∫(R²-r²)dθ
Area between two polar curves
∫√((r¹)²+r²)dθ
Length of polar curve
limacon
r = a +/- a sinθ, r = a +/- bcosθ
Y-axis Limacon Equation
r = a +- b sinθ
X-axis Limacon Equation
r = a +- b cosθ
|a| < |b|
There is an inner loop in a limacon if
|a| > |b|
There is NO inner loop in a limacon if
|a| - |b|
Limacons: inner loop or kidney bean shape of what expression
|a| + |b|
Lengthwise distance of Limacon
r = a sin (bθ)
Y-axis symmetric Rose (where b≠1)
r = a cos (bθ)
X-axis symmetric Rose (where b≠1)
Odd
An integer n is defined to be odd if n=2k+1 for some integer k.
Even
An integer n is defined to be even if n=2k for some integer k.
r = a θ
Spirals (θ is in radians)
θ = constant
Line through pole equation
r = constant
Circle with center at pole
Y-axis shifted Circle
r = d sin θ
X-axis shifted Circle
r = d cos θ
|d|
diameter of circle
d>0
circle located on positive side of the axis
d<0
circle located on negative side of the axis
distance formula: between 2 points (r₁, θ₁) and (r₂, θ₂)
d² = r₁² + r₂² - 2 r₁ r₂ cos (θ₂ - θ₁)
circle (general formula)
a² = r₀² + r² - 2 r₀ r cos (θ - θ₀)
r = 2acos(θ-θ₀)
circle through pole
skew lines (not through pole)
d = r cos (θ - α) (or r = d sec (θ - α))
vertical line
r = h sec θ (or rcosθ = h or x = h or r = h/cosθ)
horizontal line
r = k csc θ (or rsinθ = k or y = k or r = k/sinθ)
roses
r = a cos (bθ) r = a sin (bθ)