AP Calculus BC - Parametric and Polar Equations

Parametric and Polar Equations
83Q:
How do you eliminate the parameter?

Parametric and Polar Equations
83A:
To eliminate the parameter solve the x equation to be t = and then plug that answer into any t’s for the y equation.

Parametric and Polar Equations
84Q:
What is the formula for the derivative of a parametric equation? What is the formula for the second
derivative of a parametric equation?

Parametric and Polar Equations
84A:
For dy/dx: take the derivative of the y equation of the derivative of the x equation.
dy/dx= [dy/dt]/[dx/dt]
For d²y/dx²: take the derivative of the (dy/dx) equation with respect to t and put the x original derivative on the bottom of the equation
d²y/dx²= [d(dy/dx)/dt]/[dx/dt]

Parametric and Polar Equations
85Q:
What are the formulas for converting rectangular/Cartesian and polar equations back and forth?

Parametric and Polar Equations
85A:
x = r⋅cos(θ)
y = r⋅sin(θ)
x²+y²=r²
tan(θ) = y/x

Parametric and Polar Equations
86Q:
What is the formula for dy/dx of a polar equation?

Parametric and Polar Equations
86A:
dy/dx=[dy/dθ]/[dx/dθ]
=[r'sin(θ)+rcos(θ)]/[r'cos(θ)+rsin(θ)]
This is just the product rule of y over x

Parametric and Polar Equations
87Q:
How do you find the area of a polar region?

Parametric and Polar Equations
87A:
To find area of a polar region:
1. Sketch the polar curve.
2. Shade the region.
3. Find the a and b
4. take the integral ₐ∫ᵇ(1/2)r²dθ

Parametric and Polar Equations
88Q:
What does dr/dθ and dy/dx stand for polar equations?

Parametric and Polar Equations
88A:
dr/dθ gives how fast at which the curve moves away from the origin.
dy/dx equals the slope of the path of the particle on the curve.