Parametric Equations and Polar Coordinates - Concise Notes

  • Curves can be described using parametric equations, giving xx and yy as functions of a parameter tt. x=f(t)x = f(t), y=g(t)y = g(t).
  • Each tt value corresponds to a point (x,y)(x, y) which traces a parametric curve.
  • Eliminating the parameter tt can reveal the Cartesian equation of the curve.
  • Parametric equations provide information about the position and direction of motion along the curve.
  • Restricting tt to a finite interval defines a curve segment with initial and terminal points.
  • Multiple parametric equations can represent the same Cartesian curve.
  • Circle with center (h,k)(h, k) and radius rr: x=h+rcos⁡(t)x = h + r\cos(t), y=k+rsin⁡(t)y = k + r\sin(t), 0≤t≤2π0 ≤ t ≤ 2π.
  • Graphing devices are essential for visualizing complicated parametric curves.
  • A cycloid is traced by a point on a rolling circle: x=r(t−sin⁡t)x = r(t - \sin t), y=r(1−cos⁡t)y = r(1 - \cos t).
  • Eliminating the parameter might lead to complicated Cartesian equations.
  • Tangent slope: dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, if dxdt≠0\frac{dx}{dt} ≠ 0
  • Horizontal tangent: dydt=0\frac{dy}{dt} = 0 and dxdt≠0\frac{dx}{dt} ≠ 0
  • Vertical tangent: dxdt=0\frac{dx}{dt} = 0 and dydt≠0\frac{dy}{dt} ≠ 0
  • Area under a parametric curve: A=∫<em>abydx=∫</em>αβg(t)f′(t)dtA = \int<em>a^b y dx = \int</em>\alpha^\beta g(t)f'(t) dt
  • Arc length: L=∫αβ(dxdt)2+(dydt)2dtL = \int_{\alpha}^{\beta} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt
  • Surface area (rotation about x-axis): S=∫αβ2πy(dxdt)2+(dydt)2dtS = \int_{\alpha}^{\beta} 2πy \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt
  • Polar coordinates (r,θ)(r, θ): rr is the distance from the pole, θθ is the angle from the polar axis.
  • Conversion: x=rcos⁡θx = r\cos θ, y=rsin⁡θy = r\sin θ, r2=x2+y2r^2 = x^2 + y^2, tan⁡θ=yx\tan θ = \frac{y}{x}.
  • Polar equation r=f(θ)r = f(θ) represents a curve.
  • Symmetry tests: Replace θθ with −θ-θ (polar axis), θθ with π−θπ - θ (pole), rr with −r-r (pole), θθ with π/2−θπ/2 - θ (vertical line θ=π/2θ = π/2)
  • Tangent slope in polar coordinates: \frac{dy}{dx} = \frac{\frac{dr}{dθ} \sin θ + r \cos θ}{\frac{dr}{dθ} \cos θ - r \sin θ}}
  • Area in polar coordinates: A=∫<em>ab12r2dθ=∫</em>ab12[f(θ)]2dθA = \int<em>a^b \frac{1}{2}r^2 dθ = \int</em>a^b \frac{1}{2}[f(θ)]^2 dθ
  • Arc length in polar coordinates: L=∫abr2+(drdθ)2dθL = \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} dθ
  • Conic Sections: parabolas, ellipses, and hyperbolas.
  • Parabola: Set of points equidistant from a focus and a directrix.
  • Ellipse: Set of points where the sum of distances from two foci is constant.
  • Hyperbola: Set of points where the difference of distances from two foci is constant.
  • Ellipse with foci (±c,0)(\pm c, 0) and vertices (±a,0)(\pm a, 0): x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, where b2=a2−c2b^2 = a^2 - c^2
  • Hyperbola with foci (±c,0)(\pm c, 0) and vertices (±a,0)(\pm a, 0): x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, where b2=c2−a2b^2 = c^2 - a^2
  • Eccentricity: e=cae = \frac{c}{a} (e<1e < 1 for ellipse, e>1e > 1 for hyperbola, e=1e = 1 for parabola).
  • Polar equation of a conic with focus at the origin:
    • r=ed1±ecos⁡θr = \frac{ed}{1 ± e \cos θ} or r=ed1±esin⁡θr = \frac{ed}{1 ± e \sin θ}, where ee is eccentricity and dd is the distance from focus to directrix. Kepler's Laws briefly discussed.