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Homework, Page 682

Homework, Page 682. Find a polar equation for the conic with a focus at the pole and the given eccentricity and directrix. Identify the conic and graph it. 1. Homework, Page 682.

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Homework, Page 682

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  1. Homework, Page 682 Find a polar equation for the conic with a focus at the pole and the given eccentricity and directrix. Identify the conic and graph it. 1.

  2. Homework, Page 682 Find a polar equation for the conic with a focus at the pole and the given eccentricity and directrix. Identify the conic and graph it. 5.

  3. Homework, Page 682 Determine the eccentricity, type of conic and directrix. 9.

  4. Homework, Page 682 Determine the eccentricity, type of conic and directrix. 13.

  5. Homework, Page 682

  6. Homework, Page 682 Find a polar equation for an ellipse with a focus at the pole and the given polar coordinates as the endpoints of its major axis.. 21.

  7. Homework, Page 682 Find a polar equation for the hyperbola with a focus at the pole and the given polar coordinates as the endpoints of its transverse axis.. 25.

  8. Homework, Page 682 Find a polar equation for the conic with a focus at the pole. 29.

  9. What you’ll learn about • Analyzing Polar Equations of Conics • Orbits Revisited … and why You will learn the approach to conics used by astronomers.

  10. Example Analyzing Polar Equations of Conics Graph the conic and find the values of e, a, b, and c. 34.

  11. Example Analyzing Polar Equations of Conics Determine a Cartesian equivalent for the given polar equation. 38.

  12. Example, Writing Cartesian Equations From Polar Equations Use the fact that k = 2p is twice the focal length and half the focal width to determine the Cartesian equation of the parabola whose polar equation is given. 40.

  13. Semimajor Axes and Eccentricities of the Planets

  14. Ellipse with Eccentricity e and Semimajor Axis a

  15. Example Analyzing Orbits of Planets 42. The orbit of the planet Uranus has a semimajor axis of 19.18 AU and an orbital eccentricity of 0.0461. Compute its perihelion and aphelion distances.

  16. Homework • Homework Assignment #24 • Read Section 8.6 • Page 682, Exercises: 33 – 49(Odd), skip 43

  17. 8.6 Three-Dimensional Cartesian Coordinate System

  18. Quick Review

  19. Quick Review Solutions

  20. What you’ll learn about • Three-Dimensional Cartesian Coordinates • Distances and Midpoint Formula • Equation of a Sphere • Planes and Other Surfaces • Vectors in Space • Lines in Space … and why This is the analytic geometry of our physical world.

  21. The Point P(x,y,z) in Cartesian Space

  22. Features of the Three-Dimensional Cartesian Coordinate System • The axes are labeled x, y, and z, and these three coordinate axes form a right-handed coordinate frame. • The Cartesian coordinates of a point P in space are an ordered triple, (x, y, z). • Pairs of axes determine the coordinate planes. • The coordinate planes are the xy-plane, the xz-plane, and the yz-plane and they have equations z = 0, y = 0, and x = 0, respectively. • The coordinate planes meet at the origin (0, 0, 0). • The coordinate planes divide space into eight regions called octants. The first octant contains all points in space with three positive coordinates.

  23. The Coordinate Planes Divide Space into Eight Octants

  24. Distance Formula (Cartesian Space)

  25. Midpoint Formula (Cartesian Space)

  26. Example Calculating a Distance and Finding a Midpoint

  27. Standard Equation of a Sphere

  28. Drawing Lesson

  29. Drawing Lesson (cont’d)

  30. Example Finding the Standard Equation of a Sphere

  31. Equation for a Plane in Cartesian Space

  32. The Vector v = <v1,v2,v3>

  33. Vector Relationships in Space

  34. Equations for a Line in Space

  35. Example Finding Equations for a Line

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