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ESSENTIAL CALCULUS CH09 Parametric equations and polar coordinates. 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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Presentation Transcript
slide2

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

slide3

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

slide4

if

Chapter 9, 9.2, P491

slide6

Note that

Chapter 9, 9.2, P491

slide7

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 αtoβ , then the length of C is

Chapter 9, 9.2, P494

slide8

Polar coordinates system

The point P is represented by the ordered pair (r,Θ) and r, Θare called polar coordinates of P.

Chapter 9, 9.3, P498

slide11

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

and so

1.

2.

Chapter 9, 9.3, P499

slide12

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

Chapter 9, 9.3, P500

slide13

The area A of the polar region R is

3.

Formula 3 is often written as

4.

with the understanding that r=f(Θ).

Chapter 9, 9.4, P507

slide14

The length of a curve with polar equation r=f(Θ)

, a≤Θ≤b , is

Chapter 9, 9.4, P509

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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.

Chapter 9, 9.5, P511

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An ellipse is the set of points in a plane the sum of whose distances from two fixed

points F1 and F2 is a constant. These two fixed points are called the foci (plural of focus.)

Chapter 9, 9.5, P512

slide24

The ellipse

  • has foci(± c,0), where c2=a2-b2 ,and vertices (± a,0),

Chapter 9, 9.5, P512

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A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F1 and F2 (the foci) is a constant.

Chapter 9, 9.5, P512

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2. The hyperbola

  • has foci(± c,0), where c2=a2+b2, vertices (± a,0), and asymptotes y=±(b/a)x.

Chapter 9, 9.5, P512

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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

Chapter 9, 9.5, P513

slide32

8. THEOREM A polar equation of the form

or

represents a conic section with eccentricity e. The conic is an ellipse if e<1, a parabola if e=1, or a hyperbola if e>1.

Chapter 9, 9.5, P514

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