ESSENTIAL CALCULUS CH09 Parametric equations and polar coordinates

1. ESSENTIAL CALCULUSCH09 Parametric equations and polar coordinates

2. 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

3. 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

4. if Chapter 9, 9.2, P491

5. Chapter 9, 9.2, P491

6. Note that Chapter 9, 9.2, P491

7. 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

8. 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

9. Chapter 9, 9.3, P498

10. Chapter 9, 9.3, P499

11. 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

12. 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

13. 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

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

15. 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

16. Chapter 9, 9.5, P511

17. Chapter 9, 9.5, P511

18. Chapter 9, 9.5, P511

19. Chapter 9, 9.5, P511

20. 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

21. Chapter 9, 9.5, P512

22. Chapter 9, 9.5, P512

23. Chapter 9, 9.5, P512

24. The ellipse • has foci(± c,0), where c2=a2-b2 ,and vertices (± a,0), Chapter 9, 9.5, P512

25. 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

26. 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

27. 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

28. Chapter 9, 9.5, P514

29. Chapter 9, 9.5, P514

30. Chapter 9, 9.5, P514

31. Chapter 9, 9.5, P514

32. 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