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Homework, Page 539

Homework, Page 539. The polar coordinates of a point are given. Find its rectangular coordinates. 1. Homework, Page 539. (a) Complete the table for the polar equation and (b) plot the corresponding points. 5. Homework, Page 539. Plot the point with the given polar coordinates. 9.

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Homework, Page 539

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  1. Homework, Page 539 The polar coordinates of a point are given. Find its rectangular coordinates. 1.

  2. Homework, Page 539 (a) Complete the table for the polar equation and (b) plot the corresponding points. 5.

  3. Homework, Page 539 Plot the point with the given polar coordinates. 9.

  4. Homework, Page 539 Plot the point with the given polar coordinates. 13.

  5. Homework, Page 539 Find the rectangular coordinates of the point with the given polar coordinates. 17.

  6. Homework, Page 539 Find the rectangular coordinates of the point with the given polar coordinates. 21.

  7. Homework, Page 539 Polar coordinates of point P are given. Find all of its polar coordinates. 25.

  8. Homework, Page 539 Rectangular coordinates of point P are given. Find all polar coordinates of P that satisfy: (a) 0 ≤θ≤2π (b) –π≤θ≤π (c) 0 ≤ θ≤ 4π 29.

  9. Homework, Page 539 Match the polar equation with its graph. 33. c.

  10. Homework, Page 539 Convert the polar equation to rectangular form and identify the graph. 37.

  11. Homework, Page 539 Convert the polar equation to rectangular form and identify the graph. 41.

  12. Homework, Page 539 Convert the rectangular equation to polar form and graph the polar equation. 45.

  13. Homework, Page 539 Convert the rectangular equation to polar form and graph the polar equation. 49.

  14. Homework, Page 539 53. A square with sides of length a and center at the origin has two sides parallel to the x-axis. Find polar coordinates of the vertices.

  15. Homework, Page 539 57. If r≠ 0, which of the following polar coordinate pairs represents the same point as the point with polar coordinates a. b. c. d. e.

  16. Homework, Page 539 61.

  17. Homework, Page 539

  18. Homework, Page 539

  19. Homework, Page 539 Use the results of # 61 to find the distance between the points with the given polar coordinates. 65.

  20. 6.5 Graphs of Polar Equations

  21. Quick Review

  22. Quick Review Solutions

  23. What you’ll learn about • Polar Curves and Parametric Curves • Symmetry • Analyzing Polar Curves • Rose Curves • Limaçon Curves • Other Polar Curves … and why Graphs that have circular or cylindrical symmetry often have simple polar equations, which is very useful in calculus.

  24. Polar Curves and Parametric Curves Polar curves are, in reality, a special type of parametric curves, where , for all values of θ in some parameter interval that suffices to produce a complete graph (in many cases, 0 ≤ θ≤ 2π).

  25. Symmetry The three types of symmetry figures to be considered are: • The x-axis (polar axis) as a line of symmetry. • The y-axis (the line θ = π/2) as a line of symmetry. • The origin (the pole) as a point of symmetry.

  26. Symmetry Tests for Polar Graphs The graph of a polar equation has the indicated symmetry if either replacement produces an equivalent polar equation. To Test for Symmetry ReplaceBy • about the x-axis (r,θ) (r,-θ) or (-r, π-θ) • about the y-axis (r,θ) (-r,-θ) or (r, π-θ) • about the origin (r,θ) (-r,θ) or (r, π+θ)

  27. Example Testing for Symmetry

  28. Rose Curves

  29. Limaçon Curves

  30. Example Analyzing Polar Graphs

  31. Example Analyzing Polar Graphs

  32. Example Analyzing a Polar Graph Analyze the polar graph of Domain: Range: Continuity: Symmetry: Boundedness: Maximum r-value: Asymptotes:

  33. Homework • Homework Assignment #6 • Read Section 6.6 • Page 548, Exercises: 1 – 69 (EOO) • Quiz next time

  34. 6.6 De Moivre’s Theorem and nth Roots

  35. Quick Review

  36. Quick Review Solutions

  37. What you’ll learn about • The Complex Plane • Trigonometric Form of Complex Numbers • Multiplication and Division of Complex Numbers • Powers of Complex Numbers • Roots of Complex Numbers … and why The material extends your equation-solving technique to include equations of the form zn = c, n is an integer and c is a complex number.

  38. Complex Plane

  39. Addition of Complex Numbers

  40. Absolute Value (Modulus) of a Complex Number

  41. Graph of z = a + bi

  42. Trigonometric Form of a Complex Number

  43. Example Finding Trigonometric Form

  44. Product and Quotient of Complex Numbers

  45. Example Multiplying Complex Numbers

  46. Example Dividing Complex Numbers

  47. A Geometric Interpretation of z2

  48. De Moivre’s Theorem

  49. Example Using De Moivre’s Theorem

  50. nth Root of a Complex Number

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