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Summer Review Packet for Students Entering Calculus Complex Fractions: Simplify each of the following. 25 a a 1. 5 a 4 x2 2. 10 5 x2 12 2x 3 3. 15 5 2x 3 x 1 4. x 1 x x 1 x 1 x 2x 3x 4 5. 32 x 3x 4 1 6 1 2 x x 6. 4 3 1 2 x x 2 4 1 3 10 x x2 7. 11 18 1 2 x x 1 1 Summation Notation: Find the sum of each of the following series. 5 8. 7 (2n 3) 9. i 1 i i 3 11. (i 2) n 1 i 1 (1)n1 n 3 n 2 6 10. 5 Write each series in expanded form. 5 12. (2 x n ) n 1 13. xi i 1 i 1 14. xi i 2 i 4 5 2 Operations on Rational Expressions Simplify each of the following by factoring: 15. x 2 x 6 x 2 x 20 12 x x 2 x 2 4 x 4 x3 y 3 2 x 2 5xy 3 y 2 17. 2 x 2 xy 3 y 2 x 2 4 xy y 2 19. 6 x 2 23x 4 4 x 2 20 x 25 6 x 2 17 x 3 2 x 2 11x 15 6 x2 x 2 2 x2 9 x 4 6 x2 7 x 2 4 7 x 2 x2 16. 2 x 2 13x 20 6 x 2 13x 5 18. 2 8 10 x 3x 2 9 x 3x 2 3x 2 10 x 8 2 x 2 9 x 10 6 x 2 13x 6 4 x 2 4 x 15 20. Simplify each of the following operations of addition or subtraction. 21. x 2 6 2 4x 1 2 x 1 8x 2 x 1 23. x 1 x 3 10 x 2 7 x 9 1 2 x 4 x 3 8 x 2 10 x 3 x 3x 2 7 x 2 24 x 28 3x 4 x 5 3x 2 11x 20 22. 3 Fractional and Integral Exponents Simplify each of the following. Leave all answers with POSITIVE exponents. 24. 3 x y 3 2 3 2 2 1 3 x y 2 2 9ab 2 3a 2b 26. 2 2 2 8a b 2a b 28. 27m n m 3 6 1 3 x 2 y 4 25. 2 x y 3 y 2 3 y 5 6 27. 19 y 9 29. y 2 3 y1 3 y 2 3 1 3 5 6 6 n 2 30. a1 6 a 5 6 a 7 6 Functions Let f ( x) 2 x 1 and g ( x) 2 x 2 1 . Find each. 31. f (2) ____________ 32. g ( 3) _____________ 33. f (t 1) __________ 34. f g (2) __________ 35. g f (m 2) ___________ 36. 4 f ( x h) f ( x ) ____ h Functions – Continued Let f ( x) x 2 , g ( x) 2 x 5, and h( x) x 2 1. Find each. 37. h f (2) _______ Find 38. f g ( x 1) _______ 39. g h( x3 ) _______ f ( x h) f ( x ) for the given function f. h 40. f ( x) 9 x 3 41. f ( x) 5 2 x Derivatives and Critical Points Find the derivative for each of the following functions. 42. f ( x) 5x 4 3x 2 2 43. f ( x) 6 x3 5x 2 4 x 2 45. f ( x) 7 46. f ( x) x5 2 x3 5x 4 5 44. f ( x) 2 x 4 Find the slope of the line tangent to each of the following functions at the given point. 47. f ( x) x 4 2 x3 5x 2 8 at the point (-2, 44) 48. f ( x) x 2 x 2 at the point (0.5, 1.25) Find the equation of the line, in slope intercept form, tangent to each of the following functions at the given point. 49. f ( x) 2 x2 3x 10 at the point (1, 11) 50. f ( x) 3x3 2 x 2 4 x 2 at the point (-2, -42) Find the critical point(s) for each of the following functions. 51. f ( x) 3x3 9 x 5 52. f ( x) x 2 2 x 15 6 Proving Trigonometric Identities Prove each of the following identities. 53. sin x sin 3 x cos 2 x sin x 54. sin x cos x 1 cos x 1 cos x 55. 1 cos x sin x 56. 2 sec x csc x csc x sec x 1 1 2 csc2 2 x 1 cos 2 x 1 cos 2 x 57. sin 2x 2sin x cos x 58. cos 4 x cos 2 2 x sin 2 2 x 59. sin( x y ) cos y cos( x y ) sin y sin x 60. sin x cos x 2 7 Trigonometric Equations: Solve each of the following equations for 0 < x < 2. 1 2 61. 2cos 2 x 3 62. cos 2 x 63. sin x sin 2x 64. 2 cos 2 x 1 cos 2 x 65. cos 2 x 1 cos x 0 66. sin 2 x cos 2 x cos x 0 67. sin x cos x 0 68. 4 cos 2 x 3 0 69. cos x tan x sin 2 x 0 70. tan 2 x 1 0 8 Inverse Trigonometric Functions: For each of the following, express the value for “y” in radians. 71. y arcsin 3 2 72. y arccos 1 73. y arctan(1) For each of the following give the value without a calculator. 2 74. tan arccos 3 12 75. sec sin 1 13 7 77. sin 2sin 1 8 1 78. sin sin 1 tan 1 3 2 9 12 76. sin 2 arctan 5 Logarithmic Functions: Evaluate each of the following logarithms. 79. log4 16 ______ 80. log2 32 ______ 81. log1000 ______ 82. log 6 216 ______ 83. log5 125 ______ 84. log3 7 ______ 85. log 6 28 ______ 86. log5 12 ______ 87. log12 9 ______ 88. log 4 x 3 89. log8 x 3log8 2 90. log 4 x log4 2 log 4 3 1 91. log x log 27 3 92. log9 x 5log9 2 log9 8 93. log3 ( x 1) 2 Solve each of the following for “x”. 94. log( x 3) log(2 x 4) log 3 95. log( x2 3) log( x 1) log 5 10 Formula Sheet Reciprocal Identities: csc x 1 sin x sec x 1 cos x Quotient Identities: tan x sin x cos x cot x cos x sin x Pythagorean Identities: sin 2 x cos 2 x 1 Sum Identities: sin( x y ) sin x cos y cos x sin y tan( x y ) Difference Identities: cot x tan 2 x 1 sec 2 x 1 cot 2 x csc 2 x cos( x y ) cos x cos y sin x sin y tan x tan y 1 tan x tan y sin( x y ) sin x cos y cos x sin y tan( x y ) 1 tan x cos( x y ) cos x cos y sin x sin y tan x tan y 1 tan x tan y cos 2 x cos 2 x sin 2 x Double Angle Identities: 1 2 sin 2 x sin 2x 2 sin x cos x 2 cos 2 x 1 tan 2 x 2 tan x 1 tan 2 x x 1 cos x 2 2 Half-Angle Identities: sin Logarithms: y log a x Product property: log b mn log b m log b n m log b log b m log b n n log b m p p log b m Quotient property: Power property: Property of equality: Change of base formula: Derivative of a Function: cos x 1 cos x 2 2 is equivalent to tan x 1 cos x 2 1 cos x x ay If log b m log b n , then m = n log b n log a n log b a Slope of a tangent line to a curve or the derivative: lim Slope-intercept form: y mx b Standard form: Ax + By + C = 0 h Point-slope form: y y1 m( x x1 ) 11 f ( x h) f ( x ) h