parametric and polar

Slope Fields

only x - same vertically

only y - same horizontally

both - same diagonally

Euler’s Method equation

Y(n+1)=Y(n) +(change in x)(f’(x(n), y(n))

Parametric equation to Cartesian equation

solve for t and plug in

Cartesian Equation to parametric equation

let x=t and replace x with t

Slope of a parametric equation

derivative of y/derivative of x

second derivative of a parametric equation

take first derivative and do quotient rule. will end up being over (low)_³

Horizontal tangent

numerator of the derivative equal to 0

Vertical tangent

denominator of the derivative equal to 0

Length of a parametric curve

L = integration from a to b of the square root of (dx/dt)²+(dy/dt)² dt

Polar to rectangular coordinates

x=r cos(theta) and y= r sin(theta)

Rectangular to polar coordinates

tan(theta)=y/x and r²=x²+y²

Slope of a polar curve

r’sin(theta)+rcos(theta)/r’cos(theta)-rsin(theta)

area inside polar graph

A = integration from angle a to angle b (½ r²) d(theta)

area between two polar curves

A = 1/2 integration from angle a to b (R²-r²) d(theta)

length of a polar curve

L = integration from angle a to b of the (sqrt of r²+(r’)²) d(theta)

surface area of polar curves

s = (2)(pi)( r ) integration from angle a to b sin(theta) (sqrt of r²+dr/dtheta)