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Parametrics, Polar Curves, Vectors. By: Kyle Dymanus , Linda Fu, Jessica Haswell. Parametrics. Parametric form: x(t) = t y(t) = t ² Cartesian(rectangular) form: y = x ². Graphing parametrics. Put into rectangular form or use vectors. Example: Graph x=t y=t ² - PowerPoint PPT Presentation
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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.