Parametrics, Polar Parametrics, Polar Curves, VectorsCurves, Vectors

By: Kyle Dymanus, Linda Fu, By: Kyle Dymanus, Linda Fu, Jessica HaswellJessica 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=ty=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

Solution1. 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)

VectorsVectors

c(t) = c(t) = ﴾﴾ x(t) , y(t) x(t) , y(t) ﴿﴿ = < x(t) , y(t) >= < x(t) , y(t) >

= xi + yj= xi + yj

Velocity VectorVelocity Vector

vv = ( x’(t) , y’(t) ) = ( x’(t) , y’(t) )

vv = ( dx/dt , dy/dt ) = ( dx/dt , dy/dt )

Acceleration VectorAcceleration Vector

āā = (x”(t) , y”(t)) = (x”(t) , y”(t))

āā = (d²x/dt² , d²y/dt²) = (d²x/dt² , d²y/dt²)

SpeedSpeed

Speed = Speed = √[(x’(t))² + (y’(t))²]√[(x’(t))² + (y’(t))²]

ExampleExample

Write the velocity and acceleration vector Write the velocity and acceleration vector and find the speed at t=1. and find the speed at t=1.

x = tx = t² - 4 , y=t/2² - 4 , y=t/2

More ExamplesMore Examples

Find the minimum speed.Find the minimum speed.

c(t) = ( tc(t) = ( t³ , 1/t²) , t≥.5³ , 1/t²) , t≥.5

Coordinates of Polar CurvesCoordinates of Polar Curves

(r, (r, θθ))

(3, π/4)(-3, 5π/4)(3, -7π/4)(-3, -3π/4)

Polar Rectangular ConversionsPolar Rectangular Conversions

xx22+y+y22=r=r22

tantanθθ=y/x=y/x x≠0x≠0

Convert to polarConvert to polar (1,0)(1,0) (3, (3, √3)√3) (-2, 2)(-2, 2)

(1,0)

(√12,π/6)

(2 √2,π/4)

Graph Polar CurvesGraph Polar Curves

WindowWindow θθmin = 0min = 0 θθmax = 2max = 2ππ θθstep = step = ππ/24/24

Shapes to knowShapes to know sin(nx) n=1, odd, evensin(nx) n=1, odd, even cos(nx) n=1, odd, evencos(nx) n=1, odd, even

Polar Rectangular ConversionsPolar Rectangular Conversions

x=rcosx=rcosθθ

y=rsiny=rsinθθ

xx22+y+y22=r=r22

tantanθθ=y/x=y/x x≠0x≠0

Rectangular Rectangular Polar: Polar: x=5x=5 xy=1xy=1

Polar Polar Rectangular Rectangular r=2cscr=2cscθθ

r=5secθ

r2=1/(cosθsinθ)

y=2

Slope of Tangent LineSlope of Tangent Line

Example:Example: r=4cos(3r=4cos(3θθ)) Find the equation of the tangent Find the equation of the tangent

line in rectangular line in rectangular coordinates at coordinates at θθ==ππ/6/6

Area Bounded by a Polar CurveArea Bounded by a Polar Curve

A= (1/2)∫A= (1/2)∫ᵝβᵅ ᵅ rr22ddθθ r=f(r=f(θθ))

CalculatorCalculator MathMath9:fnInt(9:fnInt( fnInt(f(fnInt(f(θθ),),θθ,a,b),a,b)

Area Bounded by a Polar CurveArea Bounded by a Polar Curve

ExampleExample Find the area of region A.Find the area of region A.