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INTERESTING CANCELLING

INTERESTING CANCELLING. The first one is. LIMITS OF TRIGONOMETRIC FUNCTIONS. In order to understand the derivatives that the trigonometric functions will produce, we must first understand how to evaluate two important trigonometric limits. The Sandwich Theorem.

bernard-williams
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INTERESTING CANCELLING

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  1. INTERESTING CANCELLING

  2. The first one is LIMITS OF TRIGONOMETRIC FUNCTIONS In order to understand the derivatives that the trigonometric functions will produce, we must first understand how to evaluate twoimportant trigonometric limits.

  3. The Sandwich Theorem • First evaluate something that we know to be smaller • Second evaluate something that we know to be larger. • Make a conclusion about the value of the limit in between these small and large values.

  4. First we will examine the value of for values of x close to 0. Graph Y1 = 1 0 We see in the table as x 0 1

  5. Graph Y1 = Since and then

  6. We need to review some trigonometry before we can proceed to the proof that Slides 7 to 13 are included for those students interested in looking at the formal proof of this limit. We will now move on to slide 14

  7. r y x The Circle x 2 + y 2 = r 2

  8. (0, 1) (-1, 0) (1, 0) (0,-1) Unit Circle x 2 + y 2 = 1 (cos θ, sin θ)

  9. Areas of Sectors in Degrees Area of circle = p r2 If θ = 90o then the sector is or of the circle.

  10. Areas of Sectors in Radians 360o = 2p radians

  11. (0, 1) A (0, sin θ) cos θ, sin θ C O B D (cos θ, 0) (1, 0) r = 1 r = cos θ The size of ∆OAB is between the areas of sector OCB and sector OAD

  12. Area of sector OCB Area of ∆OAB Area of sector OAD < < < < < < < < < < < < divide by ½ divide byq cosq

  13. Conclusion: Inorder to evaluate our limit, we now need to look at what happens as θ→0 REMEMBER: cos0o = 1 As we approach this limit from the left and from the right, it approaches the value of 1.

  14. Example 1: Estimate the limit by graphing x 0 1 = 1

  15. If the coefficients on the x are equal the limit value will be 1

  16. Example 2: Evaluate the limit Solution: Multiply top and bottom by 2: Separate into 2 limits: Evaluate

  17. Example 3: Evaluate the limit Solution: Multiply top and bottom by 3: Separate into 2 limits: Evaluate

  18. We see in the table as x→0 →0 THE SECOND IMPORTANT TRIGONOMETRIC LIMIT 0 0

  19. Mathematical Proof for Multiply top and bottom by the conjugate cos x + 1 Pythagorean Identity sin 2x + cos 2 x = 1 cos 2x – 1 = – sin 2x

  20. Example 4: Evaluate the limit

  21. Example 5: Evaluate the limit

  22. Example 6: Evaluate the limit

  23. REMEMBER: EXAMPLE 7: Solution

  24. ASSIGNMENT QUESTIONS 1. 2

  25. 2. 9

  26. 3.

  27. 4. Use your calculator to estimate the value of the following limit. 0 2 2

  28. Algebraic Method

  29. ASSIGNMENT QUESTIONS 5. Multiply by the conjugate Remember cos2 x + sin2 x = 1 so cos2 x– 1 = –sin2x Substitute

  30. 6.

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