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Sec 11.1 Curves defined by Parametric Equations

Sec 11.1 Curves defined by Parametric Equations. DEFINITIONS : A parametric curve is determined by a pair of parametric equations : x = f ( t ), y = g ( t ), where f and g are continuous on an interval a ≤ t ≤ b . The variable t is called a parameter .

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Sec 11.1 Curves defined by Parametric Equations

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  1. Sec 11.1 Curves defined by Parametric Equations DEFINITIONS: A parametric curve is determined by a pair of parametric equations: x = f (t), y = g (t), where f and g are continuous on an interval a ≤ t≤b. The variable t is called a parameter. The points P = (f(a), g(a)) and Q = (f(b), g(b)) are called the initial and terminal points, respectively. The direction in which the curve is traced for increasing values of the parameter is called the orientation of the curve. Note: A curve can have different parametrizations.

  2. DEFINITIONS:If the endpoints of the curve are the same, the curve is called closed. If distinct values of t yield distinct points in the plane (except possibly for t = a and t = b), we say the curve is a simple curve. A simple closed curve is a curve that is both (i) closed and (ii) simple. A curve: x = f (t), y = g (t), a ≤ t ≤ b is called smooth if f ′ and g′ exist and are continuous on [a, b], and f ′(t) and g′(t) are not simultaneously zero on (a, b).

  3. If a circle has radius r and rolls along the x-axis, and if one position of P is the origin, the parametric equations for the cycloid are A Cycloid: A cycloid is the curve traced by a point P on the circumference of a circle as the circle rolls along a straight line without slipping. Seehttp://www.ies.co.jp/math/java/calc/cycloid/cycloid.html

  4. Sec 11.2 Calculus with Parametric Curves Theorem: Let f and g be continuously differentiable with f′(t) ≠ 0 on α < t < β. Then the parametric equations: x = f (t), y = g (t) define y as a differentiable function of x and

  5. Second Derivatives: Theorem: If the equations x = f (t), y = g (t) define y as a twice-differentiable function of x, then at any point where dx/dt≠ 0,

  6. Area under a curve y = F(x) from a to b: Theorem: If the curve is traced out once by the parametric equations: x = f (t), y = g (t), α ≤ t≤ β, then the area can be calculated by

  7. Arc Length Theorem: The arc length of a smooth curve C given by x = f (t), y = g (t), α ≤ t≤ β, which does not intersect itself except possibly at the endpoint is given by

  8. Surface Area Theorem: If the curve C given by: x = f (t), y = g (t), α ≤ t≤ β, is rotated about the x-axis, where f and g have continuous first derivatives, and g(t) ≥ 0, then the area of the resulting surface is given by

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