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11 Pages

### CalculusFormulas2005

Course: MATH 215, Fall 2005
School: Cal Poly Pomona
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Word Count: 1591

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Calculus A.P. BC Formulas 2005-2006 Hanford High School, Richland, Washington revised 3/8/06 1. 2. floor function (def) x Greatest integer that is less than or equal to x. (graph) 3. ceiling function (def) Least integer that is greater than or equal to x. 4. x (graph) 5. 6. a 3 b3 a3 b3 a a b bg c a a b bg c 2 ab b 2 2 ab h b h 2 1 of 11 Hanford High School Calculus Richland, Washington 7. f...

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Calculus A.P. BC Formulas 2005-2006 Hanford High School, Richland, Washington revised 3/8/06 1. 2. floor function (def) x Greatest integer that is less than or equal to x. (graph) 3. ceiling function (def) Least integer that is greater than or equal to x. 4. x (graph) 5. 6. a 3 b3 a3 b3 a a b bg c a a b bg c 2 ab b 2 2 ab h b h 2 1 of 11 Hanford High School Calculus Richland, Washington 7. f ( x) 1 (graph) x 3 2 1 -3 -2 -1 0 -1 -2 -3 1 2 3 x 8. 9. 10. p34 Change of base rule for logs: Circle formula: Parabola formula: loga x x h x h 2 ln x ln a y k 2 r2 2 4p y k 11. Ellipse formula: x2 a2 x2 a2 e 1 y2 b2 y2 b2 c a 1 c a 2 b2 12. Hyperbola formula: 1 c a 2 b2 13. 14. 15. 16. 17. 18. 19. eccentricity: sin 2 x cos2 x 1 tan2 x 1 cot 2 x sec2 x csc2 x sin u cos v cos u sin v cos u cos v sin u sin v b g cosb v g u sin u v tan u v b g tan u tan v 1 tan u tan v 2 of 11 Hanford High School Calculus Richland, Washington 20. 21. 22. sin(2u) cos(2u) tan(2u) 2 sin u cos u cos2 u sin 2 u 2 tan u 1 tan 2 u 1 cos 2u 2 1 cos 2u 2 1 cos 2u 1 cos 2u 1 cos u v 2 1 cos u v 2 1 sin u v 2 1 sin u v 2 a sin A u b g cosb vg u b g cosb vg 23. sin2 u 24. cos2 u 25. tan 2 u 26. sin u sin v 27. cos u cos v 28. sin u cos v u b g sinb vg u b gsinb vg b sin B c sin C 29. cos u sin v 30. 31. 32. p581 p581 p581 law of sines: law of cosines: area of triangle using trig. c2 a2 b2 2ab cos C 1 ac sin B 2 Area 33. p27 parameterization of ellipse: x2 a2 1 y2 b2 1 becomes x a cos t , y b sin t 34. p57 lim x sin x 0 x 3 of 11 Hanford High School Calculus Richland, Washington 35. 36. p67 p79 lim x sin x x 0 If a function is continuous between a and b , then it takes on every value between f (a) and f (b) . Intermediate Value Theorem 37. p95 definition of derivative f ( x) lim h 0 f ( x h) h f ( x) 38. p112 d (c) dx d x dx 0 39. bg 1 40. p113 d cu dx d n u dx bg cu 41. nu n 1u 42. p114 d u v dx b g u v 43. p115 d (uv) dx d u dx v uv vu 44. p117 FI G HJ K vu v 2 uv 45. p135 d sin u dx d cos u dx d tan u dx cosu u 46. p136 sin u u 47. p138 sec2 u u 48. p138 d cot u dx 4 of 11 csc2 u u Hanford High School Calculus Richland, Washington 49. p138 d sec u dx d csc u dx sec u tan u u 50. p138 csc u cot u u 51. p144 slope of parametrized curve: dy dx dy dt dx dt 1 x f (a) 52. p157 derivative formula for inverses df 1 dx df dx x a 53. p159 d sin 1 u dx d cos 1 u dx u 1 u2 u 1 u2 54. 55. p159 d tan 1 u dx d cot -1 u dx u 1 u2 u 1 u2 u u u2 1 u 1 56. 57. p160 d sec-1 u dx 58. d csc-1 u dx cot 1 ( x ) u u u2 1 tan 1 u 1 59. p161 2 cos x 60. p161 sec 1 ( x ) 1 61. p161 csc 1 ( x ) sin 5 of 11 1 1 FI GJ HK x 1 FI GJ HK x Hanford High School Calculus Richland, Washington 62. p164 d u e dx d ln u dx d u a dx eu u 63. p166 1 u u a u ln a u If f is continuous over a closed interval, then f has a maximum and minimum value over that interval. If f ( x) is a differentiable function over a , b , then at some point between a and b : f (b) f (a ) f (c) b a 64. 65. p178 Extreme Value Theorem 66. p186 Mean Value Theorem (for derivatives) 67. 68. p221 p223 linearization formula Newton's Method L( x) xn f (a) xn f (a) ( x a) 1 f xn f xn bg bg 69. 70. 71. p269 p269 p272 k f u du k f u du z f (u) g(u) du z z f (u)du g (u)du Mean Value Theorem (for definite integrals) If f is continuous on a, b , then at some point c in a, b , f c 1 b a b a f x dx 72. p277 First fundamental theorem: d dx T u a f (t )dt f (u ) u 73. p290 Trapezoidal Rule: h y0 2 y1 2 y2 ... 2 yn 2 b 1 yn g 1 74. p292 Simpson's Rule: S 6 of 11 h y0 4 y1 2 y2 ... 2 yn 3 b 2 4 yn yn g Hanford High School Calculus Richland, Washington 75. 76. 77. 78. 79. 80. 82. 81. 83. p315 p317 p317 p317 p317 p317 p317 z z n du u c un 1 c n 1 u du n 1 sin u du cos u du sec2 u du csc2 u du sec u tan u du cosu c sin u c tanu c cot u c sec u c csc u cot u du cscu c ln u c 84. 85. 86. 87. 88. 89. 90. z z z 1 du u e u du eu c a u du 1 u a c ln a ln cosu ln sin u c c tan u du cot u du c sec u du csc u du ln sec u tan u ln csc u cot u c z du 2 a u 2 arcsin u c a 7 of 11 Hanford High School Calculus Richland, Washington 91. 92. z z p323 p323 du a u2 2 1 u arctan c a a du a2 u u2 u 1 arcsec a a c 93. 94. Integration by parts: order for choosing u in integration by parts: exponential change: half-life z y ln 2 k udv uv z vdu LIPET logs, inverse trig., polynomial, exponential, trig. 95. 96. p330 p332 y0e kt 97. 98. p343 continuous compound interest: logistics differential equation: A(t ) dP dt P Ao e rt K P M M P 99. p343 logistics growth model M 1 Ae b a kt 100. p389 surface area about x axis (Cartesian): S 2 y 1 dy dx 2 2 dx 101. p397 102. 103. p417 length of curve (Cartesian): Mr. Kelly's e-mail address: L b a 1 dy dx dx Greg_Kelly@rsd.edu 0 n lim ln n n 104. n lim n n 1 n 1 105. n lim x 1 8 of 11 Hanford High School Calculus Richland, Washington 106. n lim x n n 0 x c 1h 107. n F xI limG J 1 H nK xn n! n ex 108. n lim 0 109. k 1 n k n(n 1) 2 n(n 1)(2n 1) 6 110. k 1 k2 n 111. k 1 k3 112. partial sum of geometric series: n F(n 1) I G2 J H K ac r h 1 S 2 n n 1 r 113. p459 What series? n 1 ar n 1 geometric, converges to a 1 r if r 1 114. p473 Maclaurin Series: P( x ) f (0) f (0) x f (0) 2 x 2! f (0) 3 x ... 3! 115. p475 Taylor Series: P( x ) f (a ) f (a )( x a ) 3 f (a ) x a 2! b g 2 f (a ) x a 3! 116. p477 Maclaurin Series for b g ... 1 x x2 x 3 ... 1 1 x 1 1 x 1 1 x 1 1 x 117. p477 Maclaurin Series for 1 x x2 x 3 ... 118. p477 x Maclaurin Series for e e 9 of 11 x 1 x x2 2! x3 3! x4 ... 4! Hanford High School Calculus Richland, Washington 119. p477 Maclaurin Series for sin x sin x x x3 3! x2 2! x x5 5! x4 4! x2 2 x3 3 x3 3 x5 5 x7 ... 7! x6 ... 6! x4 ... 4 x7 ... 7 120. p477 Maclaurin Series for cos x cos x 1 121. p477 Maclaurin Series for ln(1 x) ln(1 x) 122. p477 Maclaurin Series for tan 1 x : tan 1 ( x) x 123. p482 Lagrange form of remainder Rn x f n 1 c n 1! x a n 1 124. p483 Remainder Estimation Theorem Rn x M n x a n 1! 1 125. p484 What series? n 1 0 n! reciprocal of factorials, converges to e 126. p494 What series? n 1 b b n bn 1 g telescoping series, converges to b1 lim bn n 1 127. p497 What series? n 1 p 1 n p series, converges if p 1 128. p498 What series? n 1 1 n 1 b1g n n 1 harmonic, diverges 129. p500 What series? n 1 alternating harmonic, converges dy dt dx dt Hanford High School Calculus Richland, Washington 130. p514 2 deriv. of parametrized curve: nd d2y dx 2 10 of 11 131. p514 length of curve (parametric): L dx dy GK GK zFdt IJ Fdt IJdt H H b 2 2 a 132. p517 surface area (parametric): S z b a 2 y dx dy F I F I dt GJ GJ HK HK dt dt 2 2 133. p532 134. p533 position vector (standard form): speed from velocity vector: r t f t i g t j h t k speed = v t b g velocity vector speed vt vt 135. p533 direction from velocity vector: direction = b g b g 136. p555 137. p543 polar to Cartesian: trajectory equations: x r cos , y r sin x y xo yo vo cos vo sin t t 1 2 gt 2 138. p560 139. 140. p562 slope of polar graph: slope of polar graph at origin: area inside polar curve: slope at (r , ) slope = tan r sin r cos r cos r sin A 141. p564 length of curve (polar): L 142. p565 surface area (polar): S z z z 1 2 r d 2 2 r dr F Id GJ HK d 2 2 r sin r 2 dr F Id GJ HK d 2 11 of 11 Hanford High School Calculus Richland, Washington
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