Parametrics, Polar Curves, Vectors

Parametrics, Polar Curves, Vectors By: Kyle Dymanus, Linda Fu, Jessica Haswell

Parametrics Parametric form: x(t) = t y(t) = t² Cartesian(rectangular) form: y = x²

Graphing parametrics Put into rectangular form or use vectors. Example: Graph x=t y=t² Rectangular: y = x² *Must indicate direction of movement.

Slope of the tangent line (dy/dx) = (dy/dt) / (dx/dt) Example: Find Tangent line at t=3 of x(t) = t² y(t) = 2t³

Solution 1. Find coordinates at t=3 : (9,54) x(3) = 9 y(3) = 54 2. Find slope: (dy/dx) = (6(3)²) / (2(3)) = 9 Answer: (y - 54) = 9(x - 9)

Vectors c(t) = ﴾ x(t) , y(t) ﴿ = < x(t) , y(t) > = xi + yj

Velocity Vector v = ( x’(t) , y’(t) ) v = ( dx/dt , dy/dt )

Acceleration Vector ā = (x”(t) , y”(t)) ā = (d²x/dt² , d²y/dt²)

Speed Speed = √[(x’(t))² + (y’(t))²]

Example Write the velocity and acceleration vector and find the speed at t=1. x = t² - 4 , y=t/2

More Examples Find the minimum speed. c(t) = ( t³ , 1/t²) , t≥.5

Coordinates of Polar Curves (3, π/4) (-3, 5π/4) (3, -7π/4) (-3, -3π/4) (r, θ)

Polar Rectangular Conversions • x2+y2=r2 • tanθ=y/x • x≠0 • Convert to polar • (1,0) • (3, √3) • (-2, 2) (1,0) (√12,π/6) (2 √2,π/4)

Graph Polar Curves • Window • θmin = 0 • θmax = 2π • θstep = π/24 • Shapes to know • sin(nx) n=1, odd, even • cos(nx) n=1, odd, even

Polar Rectangular Conversions • x=rcosθ • y=rsinθ • x2+y2=r2 • tanθ=y/x • x≠0 • Rectangular  Polar: • x=5 • xy=1 • Polar  Rectangular • r=2cscθ r=5secθ r2=1/(cosθsinθ) y=2

Slope of Tangent Line • Example: • r=4cos(3θ) Find the equation of the tangent line in rectangular coordinates at θ=π/6

Area Bounded by a Polar Curve • A= (1/2)∫ᵝᵅ r2dθ r=f(θ) • Calculator • Math9:fnInt( • fnInt(f(θ),θ,a,b)

Area Bounded by a Polar Curve • Example • Find the area of region A.

Parametrics, Polar Curves, Vectors