FORMULAS
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360040_B01.qxd 2/1/05 A12 2:31 PM APPENDIX B B B.1 Page A12 Formulas FORMULAS D I F F E R E N T I AT I O N A N D I N T E G R AT I O N F O R M U L A S Use differentiation and integration tables to supplement differentiation and integration techniques. Differentiation Formulas 1. d cu cu dx 2. d u ± v u ± v dx 3. d uv uv vu dx 4. d u vu uv dx v v2 5. d c 0 dx 6. d n u nun1u dx 7. d x 1 dx 8. u d ln u dx u 9. d u e e uu dx 10. d sin u cos uu dx 11. d cos u sin uu dx 12. d tan u sec2 uu dx 13. d cot u csc2 uu dx 14. d sec u sec u tan uu dx 15. d csc u csc u cot uu dx Integration Formulas Forms Involving u n 1. 2. un du un1 C, n 1 n1 1 du ln u C u Forms Involving a bu 3. 4. 5. 6. 7. u 1 du 2bu a ln a bu C a bu b u 1 a du 2 lna bu C a bu2 b a bu u 1 1 a du 2 C, n 1, 2 a bun b n 2a bun2 n 1a bun1 u2 1 bu du 3 2a bu a2 ln a bu a bu b 2 C C u2 1 a2 du 3 bu 2a ln a bu 2 a bu b a bu 360040_B01.qxd 2/1/05 2:31 PM Page A13 APPENDIX B 8. 9. 11. 12. 13. u2 1 1 2a du 3 a bun b n 3a bun3 n 2a bun2 10. 15. 16. 17. 18. 19. 20. a2 C, n 1a bun1 n 1, 2, 3 1 1 u du ln C ua bu a a bu 1 1 1 1 u du ln ua bu2 a a bu a a bu 1 1 1 b u du ln a bu a u a a bu u2 C C 1 1 a 2bu 2b u du 2 ln u2a bu2 a ua bu a a bu Forms Involving a bu 14. C u2 1 2a a2 du ln a bu a bu3 b3 a bu 2a bu2 un a bu du 2 una bu32 na b2n 3 C un1a bu du a bu a 1 1 du ln C, a > 0 a a bu a ua bu a bu 1 1 2n 3b du an 1 un1 2 una bu a bu u a bu un u a bu du 2a bu a du 1 du ua bu 1 a bu32 2n 5b an 1 un1 2 du 22a bu a bu C 3b2 un 2 du una bu na 2n 1b a bu 1 du , un1a bu a bu un1 un1 du a bu du , n1 n1 Forms Involving u 2 a 2, a > 0 21. 1 du u2 a2 22. u2 1 du a2 u2 ua 1 ln C 2a u a u 1 1 du 2 2n 3 2 n 2 a 2a n 1 u a2n1 u2 1 du , a2n1 n1 Formulas A13 360040_B01.qxd 2/1/05 A14 2:31 PM Page A14 Formulas APPENDIX B Integration Formulas (Continued) Forms Involving u2 ± a2, a > 0 23. 24. 25. 26. 27. 28. 29. 30. 31. u2 ± a2 du 1 uu2 ± a2 ± a2 ln u u2 ± a2 C 2 1 u2u2 ± a2 du u2u2 ± a2u2 ± a2 a4 ln u u2 ± a2 C 8 u2 a2 u u2 ± a2 du u2 1 du 2 uu2 a u ± a2 u 1 u2 ± a2 u2 33. 34. 35. du a u2 a2 C u ln u u2 ± a2 C 1 a u2 a2 ln C a u 1 uu2 ± a2 a2 ln u u2 ± a2 C 2 u2 ± a2 1 du C 2 2 a2u u u ± a 2 1 ±u du 2 2 C u2 ± a232 a u ± a2 a2 u2 u du a2 u2 a ln 1 1 du ln a ua2 u2 a a2 u2 u 1 u du 2 2 C a2 u232 a a u2 eu du eu C a a2 u2 C u 1 a2 u2 du C a2u u2a2 u2 Forms Involving e u 36. du ln u u2 ± a2 C u2 ± a2 2 du u2 a2 a ln Forms Involving a 2 u2, a > 0 32. C 360040_B01.qxd 2/1/05 2:31 PM Page A15 APPENDIX B 37. 38. 39. 40. ueu du u 1eu C uneu du uneu n un1eu du 1 du u ln1 eu C 1 eu 1 1 du u ln1 enu C 1 enu n Forms Involving In u 41. 42. 43. 44. 45. ln u du u1 ln u C u ln u du u2 1 2 ln u C 4 un ln u du un1 1 n 1 ln u C, n 12 n 1 ln u2 du u2 2 ln u ln u2 C ln un du uln un n ln un1 du Forms Involving sin u or cos u 46. 47. 48. 49. 50. 51. 52. sin u du cos u C cos u du sin u C 1 sin2 u du u sin u cos u C 2 1 cos2 u du u sin u cos u C 2 sinn u du cosn u du sinn1 u cos u n 1 n n cosn1 u sin u n 1 n n u sin u du sin u u cos u C sinn2 u du cosn2 u du Formulas A15 360040_B01.qxd A16 2/1/05 2:31 PM Page A16 Formulas APPENDIX B Integration Formulas 53. 54. 55. 56. 57. 58. (Continued) u cos u du cos u u sin u C un sin u du un cos u n un cos u du un sin u n un1 cos u du un1 sin u du 1 du tan u sec u C 1 ± sin u 1 du cot u ± csc u C 1 ± cos u 1 du ln tan u C sin u cos u Forms Involving tan u, cot u, sec u, or csc u 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. tan u du ln cos u C cot u du ln sin u C sec u du ln sec u tan u C csc u du ln csc u cot u C tan2 u du u tan u C cot2 u du u cot u C sec2 u du tan u C csc2 u du cot u C tann u du tann1 u n1 cotn u du secn u du cotn1 u n1 cotn2 u du, secn2 u tan u n 2 n1 n1 cscn u du n1 tann2 u du, cscn2 u cot u n 2 n1 n1 n1 secn2 u du, n 1 cscn2 u du, n1 360040_B01.qxd 2/1/05 2:31 PM Page A17 APPENDIX B 71. 72. 73. 74. B.2 1 1 du u ± ln cos u ± sin u C 1 ± tan u 2 1 1 du u ln sin u ± cos u C 1 ± cot u 2 1 du u cot u csc u C 1 ± sec u 1 du u tan u ± sec u C 1 ± csc u FORMULAS FROM BUSINESS AND FINANCE Summary of business and finance formulas Formulas from Business Basic Terms x number of units produced (or sold) p price per unit R total revenue from selling x units C total cost of producing x units C average cost per unit P total profit from selling x units Basic Equations R xp C C x PRC Typical Graphs of Supply and Demand Curves p Demand Equilibrium p0 price Supply Equilibrium point (x0, p0 ) x x0 Equilibrium quantity Supply curves increase as price increases and demand curves decrease as price increases. The equilibrium point occurs when the supply and demand curves intersect. Formulas A17 360040_B01.qxd A18 2/1/05 2:31 PM APPENDIX B Page A18 Formulas Formulas from Business (Continued) Demand Function: p f x price required to sell x units px price elasticity of demand dpdx If < 1, the demand is inelastic. If > 1, the demand is elastic. Typical Graphs of Revenue, Cost, and Profit Functions C R Elastic demand P Inelastic demand Maximum profit Fixed cost x Break-even point x x Negative of fixed cost Revenue Function Cost Function The low prices required to sell more units eventually result in a decreasing revenue. The total cost to produce x units includes the fixed cost. Profit Function The break-even point occurs when R C. Marginals dR marginal revenue dx the extra revenue from selling one additional unit dC marginal cost dx the extra cost of producing one additional unit dP marginal profit dx the extra profit from selling one additional unit Marginal revenue 1 unit Extra revenue for one unit Revenue Function 360040_B01.qxd 2/1/05 2:31 PM Page A19 APPENDIX B Formulas from Finance Basic Terms P amount of deposit r interest rate n number of times interest is compounded per year t number of years A balance after t years Compound Interest Formulas 1. Balance when interest is compounded n times per year AP 1 r n nt 2. Balance when interest is compounded continuously A Pert Effective Rate of Interest reff 1 r n n 1 Present Value of a Future Investment A P 1 nr nt Balance of an Increasing Annuity After n Deposits of P per Year for t Years 1 nr AP nt 1 1 n r Initial Deposit for a Decreasing Annuity with n Withdrawals of W per Year for t Years nr1 1 1rn nt PW Monthly Installment M for a Loan of P Dollars over t Years at r% Interest MP r12 1 1 1 r12 12t Amount of an Annuity T erT ctert dt 0 ct is the continuous income function in dollars per year and T is the term of the annuity in years. Formulas A19
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