Derivative of Trigonometric Functions

Derivative of Trigonometric Functions
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2.5

1. Most people draw sine curves that are very steep and rounded. Given the results of this section, discuss the actual shape of the sine curve. Starting at (0, 0) , how steep should the graph be drawn? What is the steepest the graph should be drawn anywhere? In which regions is the graph almost straight and where does it curve a lot?

2. In many common physics and engineering applications, the term sin x makes calculations difficult. A common simplification is to replace sin x with x, accompanied by the justification “ sin x approximately equals x for small angles. ” Discuss this approximation in terms of the tangent line to y = sin x at x = 0. How small is the “small angle” for which the approximation is good? The tangent line to y = cos x at x = 0 is simply y = 1, but the simplification “ cos x approximately equals 1 for small angles” is almost never used. Why would this approximation be less useful than sin x x ?

3. Use a graphical analysis as in the text to argue that the derivative of cos x is - sin x.

4. In the proof of where do we use the assumption that x is in radians? If you have access to a CAS, find out what the derivative of sin x° is; explain where the /180 came from.

In exercises 5-24, find the derivative of each function.

5. f (x) = 4sin x-x 6. f (x) = x²+2cos x

7. f (x) = tan x-csc x 8. f (x) = 4sec x-3cot x

9. f (x) = xcos x 10. f (x) = 4x²-3tan x

11. f (x) = 4-2sin x 12. f (x) = 3sec xtan x

13. 14.

15. f (t ) = sin tsec t 16.

17. 18.

19. f (x) = 2sin xcos x 20. f (x) = cos ²x-sin ²x

21. f (x) = 4x²tan x 22.

23. f (x) = 4sin ²x+4cos²x 24. f (x) = 2tan ²x

In exercises 25-28, use your CAS or graphing calculator.

25. Repeat exercise 19 with your CAS. If its answer is not in the same form as ours in the back of the book, explain how the CAS computed its answer.

26. Repeat exercise 23 with your CAS. If its answer is not in the same form as ours in the back of the book, explain how the CAS computed its answer.

27. Find the derivative of f (x) = 2sin²x+cos 2x on your CAS. Compare its answer to 0. Explain how to get this answer and your CAS' answer, if it differs.

28. Find the derivative of on your CAS. Compare its answer to sec x tan x. Explain how to get this answer and your CAS' answer, if it differs.

In exercises 29-32, find an equation of the tangent line to y = f (x) at x = a.

29. 30. f (x) = tan x, a = 0

31. 32.

In exercises 33-36, use the position function to find the velocity at time t = a.

33. s(t ) = t ²-sin t, a = 0 34. s(t ) = tcos t, a = 0

35. 36. s(t ) = 4+3sin t, a =

37. Suppose an object is traveling in a circular path counterclockwise at constant speed. If the circle has radius 1 and it takes 2 seconds to make one revolution, explain why the x - coordinate of the object can be modeled by x(t ) = cos t . What is the y coordinate? What changes if the radius is 3? What changes if it takes seconds for a revolution? What if it takes 2 seconds? What if the object travels clockwise? What changes if at time t = 0 the object is at the top of the circle?

38. Based on your answers to exercise 37, explain the meanings of the parameters a , b and c in the general model x(t ) = acos (bt+c) , y(t ) = asin (bt+c).

39. A spring hanging from the ceiling vibrates up and down. Its vertical position is given by f (t ) = 4sin t . Find the velocity of the spring at time t .

40. In exercise 39, for what time values is the velocity 0? What is the location of the spring when its velocity is 0?

In exercises 41 and 42, refer to Example 5.4.

41. If the total charge in an electrical circuit is given by Q(t ) = 3sin t+t+4 coulombs, compare the current at times t = 0 and t = 1.

42. If the total charge in an electrical circuit is given by Q(t ) = 4cos t-3t+1 coulombs, compare the current at times t = 0 and t = 1.

43. Using the identities sin 2x = 2sin xcos x and cos 2x = cos ²x- sin ²x, prove that if f (x) = sin 2x, then f ' (x) = 2cos 2x.

44. Using the identities sin 2x = 2sin xcos x and cos 2x = cos ²x-sin ²x , prove that if f (x) = cos 2x , then f ' (x) = -2sin 2x.

45. For f (x) = sin x , find f ⁽⁷⁵⁾(x) and f ⁽¹⁵⁰⁾(x).

46. For f (x) = cos x , find f ⁽⁷⁷⁾(x) and f ⁽¹²⁰⁾(x).

47. For Lemma 5.1, show that

48. Use Lemma 5.1 and the identity cos²+sin² = 1 to prove Lemma 5.2.

49. Use the identity cos(x+h) = cos xcos h-sin xsin h to prove Theorem 5.2.

50. Use the quotient rule to derive formulas for the derivatives of cot x, sec x and csc x.

51. Use the basic limits and to find the following limits:

(a) (b)

(c) (d)

52. Use the basic limits and to find the following limits:

(a) (b)

(c) (d)

53. The sex of newborn Mississippi alligators is determined by the temperature of the developing egg. If the temperature is not between 26° C and 36° C, the egg does not develop at all. Between 26° C and 30° C, the eggs always develop into females. Between 34° C and 36° C, the eggs always develop into males. The probability of an egg developing into a female decreases from 1 at 30° C to 0 at 34° C. Thus, the probability of getting a female as a function of temperature is

Find a function of the form g(t ) = acos (bt+c) to make f (t ) differentiable at t = 30 and t = 34. At this point, there is not enough information to compute the probability of a Mississippi alligator egg developing into a female. What information is missing?

54. We have seen that the approximation sin x x for small x derives from the tangent line to y = sin x at x = 0. We can also think of it as arising from the result Explain why the limit implies sin x x for small x. How can we get a better approximation? Instead of using the tangent line, we can try a quadratic function. To see what to multiply x² by, numerically compute a = . Then

sin x-x a x² or sin x x+ax² for small x. How about a cubic approximation? Compute . Then sin x-(x+ax²) bx³ or sin x x+ax²+bx³ for

small x. Starting with , find a cubic approximation of the cosine function. If you have a CAS, take the approximations for sine and cosine out to a seventh-order polynomial and identify the pattern. (Hint: Write the coefficients a , b etc.in the form 1/n! for some integer n. ) We will take a longer look at these approximations, called Taylor polynomials, in Chapter 8.