Please Answer Questions 1-7, 9-14, 16-24 (Please type, I can't read a lot of handwriting. Thank you!)

3.3 Exercises SECTION 3.3 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 197 1-16 Differentiate. 1. f (x) = 3x3 - 2 cos x 3. f(x) = sin x + 2 cot.x 2. f (x) = Vx sin x 29. If H(0) = 0 sin 0, find H'(0) and H"(0). 30. If f (t) = csc t, find f"( 7/ 6). 5. y = sec 0 tan 0 4. y = 2 sec x - csc x 31. (a) Use the Quotient Rule to differentiate the function 6. g(0) = e"(tan 0 - 0) 7. y = c cost + 12 sint f (x) = tan x - 1 8. f (t ) = cot sec x (b) Simplify the expression for f(x) by writing it in terms 9. y = - 2 - tan x of sin x and cos x, and then find f'(x). 10. y = sin 0 cos 0 (c) Show that your answers to parts (a) and (b) are 11. f(0) = Sec 6 equivalent. 1 + sec 0 12. y = - COS X 32. Suppose f( Tr/3) = 4 and f'( 7/3) = -2, and let - sin x g(x) = f(x) sin x and h(x) = (cos x)/f(x). Find 13. y = t sin t a) g' (T/ 3) ( b) h' ( TT / 3 ) 1+ 14. y = - sec x tan x 33-34 For what values of x does the graph of f have a horizontal 15. f(x) = xet csc x tangent? 16. y = x sin x tan x 33. f(x) = x + 2 sin x 34. f(x) = ex cos x 17. Prove that - dx a (csc x) = - csc x cot x. 35. A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is 18. Prove that - (sec x) = sec x tan x. x(t) = 8 sin t, where t is in seconds and x in centimeters. (a) Find the velocity and acceleration at time t. worte sought after (b) Find the position, velocity, and acceleration of the mass 19. Prove that - (cot x) = -csc2x. soldue today digrist at time t = 27/3. In what direction is it moving at that time? 20 Prove, using the definition of derivative, that if f (x) = cos x, equilibrium then f'(x) = - sin x. position 21-24 Find an equation of the tangent line to the curve at the given point. 21. y = sec x, ( TT / 3, 2 ) 22. y = e* cos x, (0, 1) 23. y = cos x - sin x, ( 7, - 1) 24. y = x + tan x, (TT, TT) 36. An elastic band is hung on a hook and a mass is hung on the ower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of 25. (a) Find an equation of the tangent line to the curve motion is s = 2 cost + 3 sin t, t > 0, where s is measured y = 2x sin x at the point (T/2, Tr). in centimeters and t in seconds. (Take the positive direction (b) Illustrate part (a) by graphing the curve and the tangent to be downward.) line on the same screen. (a) Find the velocity and acceleration at time t. (b) Graph the velocity and acceleration functions. 26. (a) Find an equation of the tangent line to the curve (c) When does the mass pass through the equilibrium y = 3x + 6 cos x at the point (T/3, T + 3). position for the first time? (b) Illustrate part (a) by graphing the curve and the tangent (d) How far from its equilibrium position does the mass line on the same screen. travel? 27. (a) If f(x) = sec x - x, find f'(x). (e) When is the speed the greatest? b) Check to see that your answer to part (a) is reasonable by 37. A ladder 10 ft long rests against a vertical wall. Let 0 be the graphing both f and f' for | x |