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§3.3 Derivatives of Trig Functions

§3.3 Derivatives of Trig Functions. The student will learn about:. Derivative formulas for trigonometric functions. Some Preliminary Work #1. Remember if a, b and c are positive 0 < a < b < c, then. If then. Reciprocals and inequalities. 2. Some Preliminary Work #2. Q.

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§3.3 Derivatives of Trig Functions

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  1. §3.3 Derivatives of Trig Functions The student will learn about: Derivative formulas for trigonometric functions.

  2. Some Preliminary Work #1 Remember if a, b and c are positive 0 < a < b < c, then If then Reciprocals and inequalities. 2

  3. Some Preliminary Work #2 Q P Consider the drawing to the right. tan x sin x x O A M (1, 0) We need to establish the following important limit. ∆OMP is inside sector OMP which is inside ∆OMQ. Area of ∆OMP ≤ area sector OMP ≤ area of ∆OMQ. 3

  4. Preliminary Work #2 Q P tan x sin x x O A M (1, 0) Area of ∆OMP ≤ area sector OMP ≤ area of ∆OMQ. and and since So by the “squeeze” theorem 4

  5. Some Preliminary Work #3 We will also need to establish that = 0 ∙ 1 ∙ ½ = 0 5

  6. Derivative of sin x Let f (x) = sin x and we will use the five step procedure to calculate the derivative. 2. f (x) = sin x 1. f (x + h) = sin x cos h + cos x sin h 3. f (x + h) – f (x) = sin x cos h + cos x sin h – sin x = sin x cos h – sin x + cos x sin h = sin x (cos h – 1) + cos x sin h = (sin x) · (0) + (cos x) · (1) = cos x

  7. Derivative Formulas for Sine and Cosine Basic Form General Form - Chain Rule For u = u (x): Derivative Formulas

  8. Examples a. y = sin 5x y’ = 5 cos 5x b. y = cos x 2 y’ = (2x)(- sin x2) = - 2x sin x 2 c. y = (cos x) 2 y’ = (cos x)(- sin x) + (cos x) (-sin x = 2 sin x cos x

  9. slope Examples Find the slope of the graph of f (x) = cos x at (π/4, 2/2), and sketch the tangent line to the graph at that point. We will use our graphing calculator to do this problem.

  10. Examples And the tangent goes through the point (π/4, 2/2), so using the point-slope form of a line Find the slope of the graph of f (x) = cos x at (π/4, 2/2), and sketch the tangent line to the graph at that point. Or we can use algebra to solve this problem. y’ = - sin x y’ = - sin x so the m = y’ = - sin (π/4) y’ = - sin x so the m = y’ = - sin (π/4) = -√2/2. 10

  11. Derivative Formulas for Sine and Cosine Basic Form General Form For u = u (x): Summary

  12. ASSIGNMENT §3.3; Page 53; 1 to 21 odd.

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