1. Homework, Page 530 1. b. The window appears to be
Match the parametric equations with their graph. Identify the viewing window that seems to have been used. 1.
b. The window appears to be

2. Homework, Page 530 5. t –2 –1 1 2 x 3 4 y –0.5 Und. 2.5
(a) Complete the table for the parametric equations and (b) plot the corresponding points 5. t –2 –1 1 2 x 3 4 y
–0.5
Und.
2.5

3. Homework, Page 530
Graph the parametric equations x = 3 – t2, y = 2t in the specified parameter interval. Use the standard viewing window. 9.

4. Homework, Page 530
Eliminate the parameter and identify the graph of the parametric curve.
13.

5. Homework, Page 530
Eliminate the parameter and identify the graph of the parametric curve.
17.

6. Homework, Page 530
Eliminate the parameter and identify the graph of the parametric curve.
21.

7. Homework, Page 530
Eliminate the parameter and identify the graph of the parametric curve.
25.

8. Homework, Page 530
Find a parameterization for the curve.
29. The line segment with endpoints (3, 4) and (6, –3).

9. Homework, Page 530 33. Quadrant I
Refer to the graph of the parametric equations
given below. Find the values of the parameter t that produce the graph in the indicated quadrant.
33. Quadrant I

10. Homework, Page 530
37. Ben can sprint at the rate of 24 fps and Jerry sprints at the rate of 20 fps. Ben gives Jerry a 10-ft head start. The race can be modeled by the parametric equations:
a. Find a viewing window to simulate a 100-yard dash. Graph simultaneously with t starting at t = 0 and t-steps of 0.05.
b. Who is ahead after three seconds and by how much.
Ben is ahead by 2 ft.

11. Homework, Page 530
41. The graph of the parametric equations x = 2 sin t and y = 2 cos t is a circle of radius 2 centered at the origin. Find an interval of values for t so that the graph is the given portion of the circle.
a. The portion in the first quadrant.
b. The portion above the x-axis.
c. The portion to the left of the y-axis.

12. Homework, Page 530
45. Kirby hits a ball 4ft above the ground with an initial velocity of 120 fps at 30° above the horizontal toward a 30-ft high fence 350 ft away. Suppose that the moment Kirby hits the ball, there is a 5 fps wind gust blowing in the direction Kirby hit the ball.
a. Does the ball clear the fence?

13. Homework, Page 530
45. Kirby hits a ball 4ft above the ground with an initial velocity of 120 fps at 30° above the horizontal toward a 30-ft high fence 350 ft away. Suppose that the moment Kirby hits the ball, there is a 5 fps wind gust blowing in the direction Kirby hit the ball.
b. If so, by how much does it clear? If not, could the ball be caught?
The ball clears the fence by about 1.5 ft.

14. Homework, Page 530
49. Orlando hits a ball 4ft above the ground with an initial velocity of 160 fps at 20° above the horizontal toward a 30-ft high fence 440 ft away. How strong in fps must a wind gust be toward or away from the fence to cause the ball to hit within a few inches of the top?

15. Homework, Page 530
53. The graph of the parametric equations x = t – sin t and
y = 1 – cos t is a cycloid.
a. What is the maximum value of y = 1 – cos t? How is that value related to the graph
b. What is the distance between neighboring x-intercepts?

16. Homework, Page 530
A particle moves along a horizontal line so that its position at any time t is given by s (t). Write a description of the motion.
57.

17. Homework, Page 530
Which of the following points corresponds to t = –1 in the parameterization a. b. c. d. e.

18. Homework, Page 530 65. Consider the parametric equations , and on .
a. Graph the parametric equations for a = 1, 2, 3, 4 in the same window.
b. Eliminate the parameters in the parametric equations to verify that they are circles. What is the radius?

19. Homework, Page 530 65. Now consider the parametric equations and
c. Graph the equations for a = 1 using the following pairs of values for h and k. h 2 –2 –4 3 k –3

20. Homework, Page 530
65. d. Eliminate the parameter t in the parametric equations and identify the graphs.
e. Write a parameterization for the circle with center (–1 , 4) and radius 3.

21. 6.4
Polar Coordinates

22. Quick Review

23. Quick Review
Use the Law of Cosines to find the measure of the third side of the given triangle. 4.
40º 8 10 5.
35º 6 11

24. Quick Review Solutions

25. Quick Review Solutions
Use the Law of Cosines to find the measure of the third side of the given triangle. 4.
40º 8 10 5.
35º 6 11
6.4 7

26. What you’ll learn about
Polar Coordinate System
Coordinate Conversion
Equation Conversion
Finding Distance Using Polar Coordinates
… and why
Use of polar coordinates sometimes simplifies
complicated rectangular equations and they are useful in
calculus.

27. The Polar Coordinate System

28. Example Plotting Points in the Polar Coordinate System

29. Finding all Polar Coordinates of a Point

30. Coordinate Conversion Equations

31. Example Converting from Polar to Rectangular Coordinates

32. Example Converting from Rectangular to Polar Coordinates

33. Example Converting from Polar Form to Rectangular Form

34. Example Converting from Polar Form to Rectangular Form

35. Example
Find the distance between two aircraft if one is located at (5, 65º) and the second is at (9, 135º).

36. Homework
Homework Assignment #6
Read Section 6.5
Page 539, Exercises: 1 – 65 (EOO), 51

37. Graphs of Polar Equations
6.5
Graphs of Polar Equations

38. Quick Review

39. Quick Review Solutions

40. 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.

41. 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).

42. 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.

43. 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 Replace By
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)

44. Example Testing for Symmetry

45. Rose Curves

46. Limaçon Curves

47. Example Analyzing Polar Graphs

48. Example Analyzing Polar Graphs

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