1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Homework, Page 411 Graph the function from -2π ≤ x ≤ 2π. State whether or not the function appears to be periodic. 1. f (x) is periodic

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2 Homework, Page 411 Graph the function from -2π ≤ x ≤ 2π. State whether or not the function appears to be periodic. 5. f (x) is not periodic

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 Homework, Page 411 Verify algebraically that the function is periodic and determine its period graphically. Sketch the graph showing two periods. 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 Homework, Page 411 State the range and domain of the function and sketch a graph showing four periods. 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Homework, Page 411 State the range and domain of the function and sketch a graph showing four periods. 17.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Homework, Page 411 The graph of the function oscillates between two parallel lines. Find equations for the lines and graph the lines and the function. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Homework, Page 411 Determine whether f (x) is a sinusoid. 25.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 Homework, Page 411 Find a, b, and h so that f (x) ≈ a sin (b(x – h) 29.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Homework, Page 411 Find a, b, and h so that f (x) ≈ a sin (b(x – h) 33.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10 Homework, Page 411 The function is periodic, but not a sinusoid. Find the period graphically and sketch one period) 37.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 11 Homework, Page 411 Match the function with its graph. 41. c.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 12 Homework, Page 411 Tell whether the function exhibits damped oscillation. If so, identify the damping factor and tell whether the damping occurs as x → 0 or x → ∞. 45. There is no damping. The damping function has a constant amplitude.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 13 Homework, Page 411 Graph both f and plus or minus its damping factor in the same viewing window. Describe the behavior of f for f > 0. What is the end behavior of f? 49.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14 Homework, Page 411 Find the period and graph the function over two periods. 53.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 15 Homework, Page 411 Graph f over [-4π, 4π]. Determine whether the function is periodic and, if it is, state the period. 57.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 16 Homework, Page 411 Graph f over [-4π, 4π]. Determine whether the function is periodic and, if it is, state the period. 61.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 17 Homework, Page 411 Find the domain and range of the function. 65.

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18 Homework, Page 411 Find the domain and range of the function. 69.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19 Homework, Page 411 73. Example 3 shows that the function is periodic. Explain whether you think that is periodic. The function is not periodic because the while x changes at a uniform rate, x 3 does not

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 20 Homework, Page 411 Match the function with its graph and state the viewing window. 77. d.

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 21 Homework, Page 411 81. The function is periodic. Justify your answer. False. The function sin x is an odd function, the stated function is an even function, that is not periodic.

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 22 Homework, Page 411 85. The function is a. discontinuous b. bounded c. even d. odd e. periodic

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 23 Homework, Page 411 Predict what the graph will look like. Graph the function in one or more viewing windows, determine the main features, draw a summary sketch. Where applicable, name the period, amplitude, domain, range, asymptotes, and zeros. 89.

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 24 Homework, Page 411 Predict what the graph will look like. Graph the function in one or more viewing windows, determine the main features, draw a summary sketch. Where applicable, name the period, amplitude, domain, range, asymptotes, and zeros. 93.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.7 Inverse Trigonometric Functions

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 26 Quick Review

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 27 Quick Review Solutions

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 28 What you’ll learn about Inverse Sine Function Inverse Cosine and Tangent Functions Composing Trigonometric and Inverse Trigonometric Functions Applications of Inverse Trigonometric Functions … and why Inverse trig functions can be used to solve trigonometric equations.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Leading Questions The range for the arccosine function is [– 1, 1] The range of the arcsine function is [–π/2,π/2] sec –1 x = cos –1 x –1 sin (sin –1 x) = cos x Slide 4- 29

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 30 Inverse Sine Function

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 31 Inverse Sine Function (Arcsine Function)

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 32 Example Evaluate sin -1 x Without a Calculator

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 33 Example Evaluate sin -1 x Without a Calculator

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 34 Inverse Cosine (Arccosine Function)

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 35 Inverse Cosine (Arccosine Function)

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 36 Inverse Tangent Function (Arctangent Function)

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 37 Inverse Tangent Function (Arctangent Function)

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 38 End Behavior of the Tangent Function Recognizing that the graphs of inverse functions are reflected about the line y = x, we see that vertical asymptotes of y = tan x become the horizontal asymptotes of y = tan –1 x and the range of y = tan x becomes the domain of y = tan –1 x.

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 39 Composing Trigonometric and Inverse Trigonometric Functions

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 40 Example Composing Trig Functions with Arcsine

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 41 Example Applying Inverse Trig Functions A person is watching a balloon rise straight up from a place 500 ft from the launch point. a. Write θ as a function of s, the height of the balloon. b. Is the change in θ greater as s changes from 10 ft to 20 ft or as s changes from 200 ft to 210 ft? Explain.

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 42 Example Applying Inverse Trig Functions c. In the graph of this relationship, does the x-axis represent s height and the y-axis represent θ (in degrees) or vice- versa? Explain.

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Following Questions The angle of depression tells us how depressing an equation is relative to an equation of known depression. The angle of elevation is measured from the horizontal upwards. Simple harmonic motion is repetitive motion such as that of a pendulum. The frequency of an object in simple harmonic motion refers to the number of times it passes through a given point per unit time. Slide 4- 43

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 44 Homework Homework Assignment #32 Review Section 4.7 Page 421, Exercises: 1 – 69 (EOO) Quiz next time