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ESSENTIAL CALCULUS CH09 Parametric equations and polar coordinates

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In this Chapter: 9.1 Parametric Curves 9.2 Calculus with Parametric Curves 9.3 Polar Coordinates 9.4 Areas and Lengths in Polar Coordinates 9.5 Conic Sections in Polar Coordinates Review

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Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x=f (t) y=g (t) (called parametric equations). Each value of t determines a point (x.y), which we can plot in a coordinate plane. As t varies, the point (x,y)=(f(t). g(t)), varies and traces out a curve C, which we call a parametric curve. Chapter 9, 9.1, P484

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Chapter 9, 9.2, P491 if

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Chapter 9, 9.2, P491

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Note that

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Chapter 9, 9.2, P494 5. THEOREM If a curve C is described by the parametric equations x=f(t), y=g(t), α ≤ t ≤ β, where f ’ and g ’ are continuous on [ α,β] and C is traversed exactly once as t increases from α to β, then the length of C is

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Chapter 9, 9.3, P498 The point P is represented by the ordered pair (r, Θ) and r, Θ are called polar coordinates of P. Polar coordinates system

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Chapter 9, 9.3, P498

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Chapter 9, 9.3, P499

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If the point P has Cartesian coordinates (x,y) and polar coordinates (r,Θ), then, from the figure, we have and so 1. 2.

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Chapter 9, 9.3, P500 The graph of a polar equation r=f( Θ), or more generally F (r, Θ)=0, consists of all points P that have at least one polar representation (r,Θ) whose coordinates satisfy the equation

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Chapter 9, 9.4, P507 The area A of the polar region R is 3. Formula 3 is often written as 4. with the understanding that r=f( Θ).

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Chapter 9, 9.4, P509 The length of a curve with polar equation r=f( Θ), a≤ Θ ≤b, is

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Chapter 9, 9.5, P511 A parabola is the set of points in a plane that are equidistant from a fixed point F (called the focus) and a fixed line (called the directrix). This definition is illustrated by Figure 1. Notice that the point halfway between the focus and the directrix lies on the parabola; it is called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola.

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Chapter 9, 9.5, P511

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Chapter 9, 9.5, P512 An ellipse is the set of points in a plane the sum of whose distances from two fixed points F 1 and F 2 is a constant. These two fixed points are called the foci (plural of focus.)

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Chapter 9, 9.5, P512

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1.The ellipse has foci(± c,0), where c 2 =a 2 -b 2,and vertices (± a,0),

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Chapter 9, 9.5, P512 A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F 1 and F 2 (the foci) is a constant.

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Chapter 9, 9.5, P512 2. The hyperbola has foci(± c,0), where c 2 =a 2+ b 2, vertices (± a,0), and asymptotes y=±(b/a)x.

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Chapter 9, 9.5, P513 3.THEOREM Let F be a fixed point (called the focus) and I be a fixed line (called the directrix) in a plane. Let e be a fixed positive number (called the eccentricity). The set of all points P in the plane such that (that is, the ratio of the distance from F to the distance from I is the constant e) is a conic section. The conic is (a) an ellipse if e<1 (b) a parabola if e=1 (C) a hyperbola if e>1

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Chapter 9, 9.5, P514

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8. THEOREM A polar equation of the form or represents a conic section with eccentricity e. The conic is an ellipse if e 1.

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