DERIVATIVES

1. 3 DERIVATIVES

2. DERIVATIVES 3.4Derivatives of Trigonometric Functions • In this section, we will learn about: • Derivatives of trigonometric functions • and their applications.

3. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS • Let’s sketch the graph of the function f(x) = sin x, it looks as if the graph of f’ may be the same as the cosine curve. Figure 3.4.1, p. 149

4. Equation 1 DERIVS. OF TRIG. FUNCTIONS • From the definition of a derivative, we have:

5. DERIVS. OF TRIG. FUNCTIONS • Two of these four limits are easy to evaluate.

6. DERIVS. OF TRIG. FUNCTIONS • Since we regard x as a constant when computing a limit as h → 0,we have:

7. Equation 2 DERIVS. OF TRIG. FUNCTIONS • The limit of (sin h)/h is not so obvious. • In Example 3 in Section 2.2, we made the guess—on the basis of numerical and graphical evidence—that:

8. Proof of Eq.2 DERIVS. OF TRIG. FUNCTIONS • Assume that θlies between 0 and π/2, the figure shows a sector of a circle with center O, central angle θ, and radius 1. BC is drawn perpendicular to OA. • By the definition of radian measure, we have arc AB =θ. • Also, |BC| = |OB| sin θ = sin θ. Figure 3.4.2a, p. 150

9. Proof of Eq.2 DERIVS. OF TRIG. FUNCTIONS • We see that |BC| < |AB| < arc AB • Thus, Figure 3.4.2a, p. 150

10. Proof of Eq.2 DERIVS. OF TRIG. FUNCTIONS • Let the tangent lines at A and B intersect at E.Thus, • θ = arc AB < |AE| + |EB| • < |AE| + |ED| • = |AD| = |OA| tan θ • = tan θ Figure 3.4.2a, p. 150

11. Proof of Eq.2 DERIVS. OF TRIG. FUNCTIONS • Therefore, we have: • So, • We know that • So, by the Squeeze Theorem, we have:

12. Proof of Eq.2 DERIVS. OF TRIG. FUNCTIONS • However, the function (sin θ)/θ is an even function. • So, its right and left limits must be equal. • Hence, we have:

13. DERIVS. OF TRIG. FUNCTIONS • We can deduce the value of the remaining limit in Equation 1 as follows.

14. Equation 3 DERIVS. OF TRIG. FUNCTIONS

15. Formula 4 DERIVS. OF TRIG. FUNCTIONS • If we put the limits (2) and (3) in (1), we get: • So, we have proved the formula for sine,

16. Example 1 DERIVS. OF TRIG. FUNCTIONS • Differentiate y = x2 sin x. • Using the Product Rule and Formula 4, we have: Figure 3.4.3, p. 151

17. Formula 5 DERIV. OF COSINE FUNCTION • Using the same methods as in the proof of Formula 4, we can prove:

18. Formula 6 DERIV. OF TANGENT FUNCTION

19. DERIVS. OF TRIG. FUNCTIONS • We have collected all the differentiation formulas for trigonometric functions here. • Remember, they are valid only when x is measured in radians.

20. Example 2 DERIVS. OF TRIG. FUNCTIONS • Differentiate • For what values of x does the graph of f have a horizontal tangent?

21. Example 2 Solution: • The Quotient Rule gives: tan2 x + 1 = sec2 x

22. Example 2 DERIVS. OF TRIG. FUNCTIONS • Since sec x is never 0, we see that f’(x)=0 when tan x = 1. • This occurs when x = nπ +π/4, where n is an integer. Figure 3.4.4, p. 152

23. Example 3 APPLICATIONS • An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0. • In the figure, note that the downward direction is positive. • Its position at time t is s = f(t) = 4 cos t • Find the velocity and accelerationat time t and use them to analyze the motion of the object. Figure 3.4.5, p. 152

24. Example 3 Solution: • The velocity and acceleration are:

25. Example 3 Solution: • The object oscillates from the lowest point (s = 4 cm) to the highest point (s = -4 cm). • The period of the oscillation is 2π, the period of cos t. Figure 3.4.5, p. 152

26. Example 3 Solution: • The speed is |v| = 4|sin t|, which is greatest when |sin t| = 1, that is, when cos t = 0. • So, the object moves fastest as it passes through its equilibrium position (s = 0). • Its speed is 0 when sin t = 0, that is, at the high and low points. Figure 3.4.6, p. 153

27. Example 3 Solution: • The acceleration a = -4 cos t = 0 when s = 0. • It has greatest magnitude at the high and low points. Figure 3.4.6, p. 153

28. Example 4 DERIVS. OF TRIG. FUNCTIONS • Find the 27th derivative of cos x. • The first few derivatives of f(x) = cos xare as follows:

29. Example 4 Solution: • We see that the successive derivatives occur in a cycle of length 4 and, in particular, f (n)(x) = cos x whenever n is a multiple of 4. • Therefore, f (24)(x) = cos x • Differentiating three more times, we have:f (27)(x) = sin x

30. Example 5 DERIVS. OF TRIG. FUNCTIONS • Find • In order to apply Equation 2, we first rewrite the function by multiplying and dividing by 7:

31. Example 5 Solution: • If we let θ = 7x, then θ → 0 as x → 0.So, by Equation 2, we have:

32. Example 6 DERIVS. OF TRIG. FUNCTIONS • Calculate . • We divide the numerator and denominator by x: by the continuity of cosine and Eqn. 2