Math 2211 3.13.5 Review Name________________________ Just Relax and Think Positively!!!! I Know you CAN Do This!!! Differentiate: 1. f(x) = 30 3 2. F(x) = 4 8 3. f(t) = 1 6 3 4 + 2 4. h(x) = (x 2)(2x + 3) 5. B(y) = cy 6 7 6. h(x) = 7 + 2 7. y = (x - 1) 8. f(x) = 2 3+1 2 1 2 9. y = ( + 3 ) 10. y = ex + 1 + 1 11. g(x) = 12. y = 1+ +1 13. y = 3+2 14. y = (1)2 15. y = 2cscx + 5cosx 16. f(x) = sin x 17. G(x) = ex(tanx x) 18. y = 1+sin +cos 19. f(t) = 20. y = cot 1sec tan 21. y = x3 2 + 5 22. F(x) = (4x - x2)100 23. f(x) = (1 + x4)2/3 3 24. f(t) = 1 + tan 1 2 25. y = ( 3 5) (3 3 + 5 2 7) 26. y = sin((3 2 + 2 + 3)4 ) 27. y = 5sin 3 28. y = 6 (4 3 7 + 5) 29. y = e2t cos 4t 30. h(t) = (t4 1)3 (t3 + 1)4 31. y = 101 2 2 5 32. G(y) = (+1) 33. f(t) = 2 +4 34. y = esec 3x 35. y = 2x 2 + 1 36. Find an equation of the tangent line to the curve y = x4 + 2x2 x at the point (1, 2). 37. Find an equation of the normal line to the curve y = (1 + 2x)2 at the point (1, 9). 38. Find an equation of the tangent line to the curve y = x at the point (1, 0). 39. Find an equation of the tangent and normal lines to the curve y = (2 + x)ex at the point (0, 2). 3 40. Find the first and second derivatives of the function G( r) = + . 41. The equation of motion of a particle is s = t4 2t3 + t2 - t, where s is in meters and t is in seconds. a. Find the velocity and acceleration as functions of t. b. Find the acceleration after 1 second. 42. For what values of x does the graph of f(x) = x3 + 3x2 + x + 3 have a horizontal tangent? 43. Find an equation of the tangent line to the curve y = (1 + 2x)10 at the point (0, 1). 44. Find an equation of the tangent line to the curve y = 1 + 3 at the point (2, 3). 45. Find f '(x) and f ''(x): f(x) = 21 46. Find f '(x) and f ''(x): f(x) = cos(x2) 47. Find f '(x) and f ''(x): f(x) = cos2x 48. Find f '(x) and f ''(x): f(x) = Find dy/dx by implicit differentiation: 49. 2 + = 3 50. 2x3 + x2y xy3 = 2 51. y5 + x2y3 = 1 + y 2 52. Use implicit differentiation to find an equation of the tangent line to the curve x2 + 2xy - y2 + x = 2 at the point (1, 2) Math 2211 1. 0 2. 6x7 3. 3t5 12t3 +1 4. 4x 1 5. 6cy7 6. 7. 7 2 3 2 2 52 + 7 57 1 12 2 12 8. 3x2 2x3 1 2 9. 1 + 3 56 3 53 10. ex+1 1 11. 12 + ( 12 ) 2 12. 13. 14. (+1)2 2 3 3 2 3 ( 3 + 2)2 1 ( 1)3 15. 2cscxcotx 5sinx 16. cosx + 2 17. ( 2 1 + ) 18. (+)2 19. 2 + 3.1-3.5 Review SOLUTIONS 20. (1) 2 21. 4 ( 2 + 5)12 + 3 2 ( 2 + 5)12 22. 100(4x x2)99 (4 2x) 23. 24. 8 3 3 3 1+ 4 2 3 3 (1+)2 25. 70 6 + 51 4 + 12 3 5 2 26. cos(3 2 + 2 + 3)4 [4(3 2 + 2 + 3)3 (6 + 2)] 27. 53 23 28. 6(12 2 7) [tan(4 3 7 + 5)]5 2 (4 3 7 + 5) 29. 2 2 (24 + 4) 30. 12 2 ( 4 1)2 ( 3 + 1)3 (2 4 + 1) 31. 2x(ln10)101 32. 33. 2 5 9 (+2) (+1)6 4 2 2 12 ( 2 +4)32 34. 3 3 33 35. 2(2 2 +1) 2 +1 36. y = 7x - 5 37. y = 1 12 x+ 1 1 2 2 38. y = 109 12 39. Equation of the tangent line: y = x + 2; Equation of the normal line: y = x + 2. 1 1 40. G1(r) = 2 12 + 3 23; 1 G11(r) = 4 32 41. a v(t) = 4t3 6t2 +2t 1; a(t) = 12t2 12t + 2 1 42. 1 6 3 2 53 9 b. a(1) = 2m/s2 43. y = 20x + 1 44. y = 2x - 1 45. f1(x) = 2 1 ( 2 1)2 ; ; \"()= 2 3 +6 ( 2 1)3 46. y'= 2xsin(x2); y'' = 4x2cos(x2) 2 sin(x2) 47. y' = 2cosxsinx; y'' = 2cos2x + 2sin2x 48. y'= ; 49. - 50. 2 6 2 2+ 3 2 3 2 2 51. 2( 2 ) 5 4 +3 2 2 7 3 2 2 52. y = y''= + 2