Hanford High School, Richland, Washington Revised 3/8/06

A.P. Calculus BC Formulas 2005-2006 Hanford High School, Richland, Washington revised 3/8/06 

1. floor function (def) Greatest integer that is less than or equal to x.

2. x (graph)

3. ceiling function (def) Least integer that is greater than or equal to x.

4. x (graph)

2 2 5. a3  b3  ba  bgca  ab  b h

2 2 6. a3 b 3  ba  bgca  ab  b h

1 of 11 Hanford High School Calculus Richland, Washington 1 7. f (x)  (graph) x

-3 -2 -1 0 1 2 3 x -1

ln x 8. p34 Change of base rule for logs: log x  a ln a

2 2 9. Circle formula:  x h  y  k  r 2

2 10. Parabola formula: ( x- h) =4 p( y - k )

x2 y2 11. Ellipse formula:   1 c  a2  b2 a2 b2

x2 y2 12. Hyperbola formula:   1 c  a2  b2 a2 b2

c 13. eccentricity: e  a

14. sin2 x  cos2 x  1

15. 1 tan2 x  sec2 x

16. 1 cot 2 x  csc2 x

17. sinbu  vg sinucosv  cosusinv

18. cosbu  vg cosucosv ∓sinusinv

tan u  tanv 19. tan u  v  b g 1∓tanu tanv

2 of 11 Hanford High School Calculus Richland, Washington 20. sin(2u)  2sinucosu

21. cos(2u)  cos2 u  sin2 u

2 tan u 22. tan(2u)  1 tan2 u

1 cos2u 23. sin2 u  2

1 cos2u 24. cos2 u  2

1 cos2u 25. tan2 u  1 cos2u

1 26. sinusin v  cos u  v  cos u  v 2 b g b g

1 27. cosucosv  cos u  v  cos u  v 2 b g b g

1 28. sinucosv  sin u  v  sin u  v 2 b g b g

1 29. cosusinv  sin u  v  sin u  v 2 b g b g

a b c 30. p581 law of sines:   sin A sin B sinC

31. p581 law of cosines: c2 a 2  b 2  2 ab cos C

1 32. p581 area of triangle using trig. Area  ac sin B 2

x2 y 2 33. p27 parameterization of ellipse:  1 becomes x  a cos t , y  b sin t a2 b 2

sin x 34. p57 lim 1 x0 x

3 of 11 Hanford High School Calculus Richland, Washington sin x 35. p67 lim 0 x x

36. p79 Intermediate Value Theorem If a function is continuous between a and b , then it takes on every value between f (a) and f (b) .

f (x  h)  f (x) 37. p95 definition of derivative f (x)  lim h0 h

d 38. p112 (c)  0 dx

d 39. x  1 dx bg

d 40. p113 cu  cu dx bg

d n 41. u  nun1u dx

d 42. p114 u  v  u  v dx b g

d 43. p115 (uv)  uv  vu dx

d u vu  uv FI  44. p117 GJ 2 dx HvK v

d 45. p135 sinu  cosuu dx

d 46. p136 cosu   sinuu dx

d 47. p138 tanu  sec2 uu dx

d 48. p138 cot u   csc2 uu dx

4 of 11 Hanford High School Calculus Richland, Washington d 49. p138 secu  secu tanuu dx

d 50. p138 cscu   cscucot uu dx

dy dy 51. p144 slope of parametrized curve:  dt dx dx dt

df 1 1  52. p157 derivative formula for inverses df dx x f (a)

dx xa

d u 53. p159 sin1 u  dx 1 u2

d u 54. cos1 u  dx 1 u2

d u 55. p159 tan1 u  dx 1 u2

d u 56. cot-1 u  dx 1 u2

d u 57. p160 sec-1 u  u  1 dx u u 2 1

d u 58. csc-1 u  u  1 dx u u 2 1

 59. p161 cot 1(x)   tan1  x 2

1 1F1I 60. p161 sec (x)  cos GJ HxK

1 1F1I 61. p161 csc (x)  sin GJ HxK 5 of 11 Hanford High School Calculus Richland, Washington d u 62. p164 e  euu dx

d 1 63. p166 ln u  u dx u

d u 64. a  au lna u dx

65. p178 Extreme Value Theorem If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.

66. p186 Mean Value Theorem If f (x) is a differentiable function over a,b , (for derivatives) then at some point between a and b : f (b)  f (a)  f (c) b  a

67. p221 linearization formula L(x)  f (a)  f (a)  (x  a)

f bxn g 68. p223 Newton’s Method xn1  xn  f bxn g

69. p269  k f u du  k f u du 70. p269 zf (u)  g(u) du  zf (u)du  zg(u)du

71. p272 Mean Value Theorem If f is continuous on a, b , then at some 1 b (for definite integrals) point c in a, b , f c  f x dx b a a

d u 72. p277 First fundamental theorem: f( t ) dt f ( u )  u dx a

h 73. p290 Trapezoidal Rule: T  y  2y  2y ...2y  y 2 b0 1 2 n1 n g

h 74. p292 Simpson’s Rule: S  y  4y  2y ...2y  4y  y 3 b0 1 2 n2 n1 n g 6 of 11 Hanford High School Calculus Richland, Washington 75. zdu  u  c

n1 n u 76. p315 u du   c n  1 z n 1

77. p317  sinu du   cosu  c

78. p317  cosu du  sinu  c

79. p317  sec2 u du  tanu  c

80. p317  csc2 u du  cot u  c

81. p317  secu tan u du  secu  c

82. p317  cscu cot u du   cscu  c

1 83. du  ln u  c zu 84. zeudu  eu  c

1 85. audu  au  c z lna

86.  tanu du   ln cosu  c

87.  cotu du  ln sin u  c

88.  secu du  ln secu  tanu  c

89.  cscu du   ln cscu  cot u  c

du u 90.  arcsin  c za2  u2 a

7 of 11 Hanford High School Calculus Richland, Washington du 1 u 91.  arctan  c za2  u2 a a

du 1 u 92.  arcsec  c zu u2  a2 a a

93. p323 Integration by parts: zudv  uv  zvdu 94. p323 order for choosing u in LIPET logs, inverse trig., polynomial, integration by parts: exponential, trig.

kt 95. p330 exponential change: y  y0e

ln 2 96. p332 half-life k

rt 97. continuous compound interest: A(t)  Aoe

dP K 98. p343 logistics differential equation: P M  P dt M

M 99. p343 logistics growth model P  1 Aekt

2 b dy  100. p389 surface area about x axis (Cartesian): S2 y 1    dx a dx 

2 b dy  101. p397 length of curve (Cartesian): L1    dx a dx 

102. Mr. Kelly’s e-mail address: [email protected]

ln n 103. p417 lim  0 n n

104. lim n n  1 n

1 105. lim x n  1 n

8 of 11 Hanford High School Calculus Richland, Washington n 106. lim x  0 x  1 n c h

x n F I x 107. limG1 J  e nH nK

xn 108. lim  0 n n!

n n(n 1) 109.  k  k 1 2

n n(n 1)(2n 1) 110.  k 2  k 1 6

n n(n 1) 2 111.  k 3  GF JI k 1 H 2 K

a 1 r n 112. partial sum of geometric series: S  c h n 1 r

 a 113. p459 What series? ar n1 geometric, converges to if r  1 n1 1 r

f (0) f (0) 114. p473 Maclaurin Series: P(x)  f (0)  f (0)x  x 2  x 3 ... 2! 3!

f (a) 2 115. p475 Taylor Series: P(x)  f (a)  f (a)(x  a)  x  a  2! b g f (a) 3 x  a ... 3! b g

1 1 116. p477 Maclaurin Series for  1 x  x 2  x 3 ... 1 x 1 x

1 1 117. p477 Maclaurin Series for  1 x  x 2  x 3 ... 1 x 1 x

x 2 x 3 x 4 118. p477 Maclaurin Series for e x e x  1 x    ... 2! 3! 4!

9 of 11 Hanford High School Calculus Richland, Washington x 3 x 5 x 7 119. p477 Maclaurin Series for sin x sin x  x    ... 3! 5! 7!

x 2 x 4 x 6 120. p477 Maclaurin Series for cos x cos x  1   ... 2! 4! 6!

x2 x3 x4 121. p477 Maclaurin Series for ln(1 x) ln(1 x)  x    ... 2 3 4

3 5 7 1 x x x 122. p477 Maclaurin Series for tan  x : tan1(x)  x    ... 3 5 7

n1 f c n1 123. p482 Lagrange form of remainder R x  x  a n n 1 !

M n1 124. p483 Remainder Estimation Theorem Rn  x  x  a n 1 !

 1 125. p484 What series?  reciprocal of factorials, converges to e n0 n!

b  b b1- lim bn+ 1 126. p494 What series? bn n1g telescoping series, converges to n n1

 1 p p  1 127. p497 What series?  p series, converges if n1 n

 1 128. p498 What series?  harmonic, diverges n1 n

 n1 1 129. p500 What series? b1g alternating harmonic, converges n1 n

dy d 2 y 130. p514 2nd deriv. of parametrized curve:  dt dx2 dx dt

10 of 11 Hanford High School Calculus Richland, Washington 2 2 b dx dy 131. p514 length of curve (parametric): L  GF JI  GF JI dt za Hdt K Hdt K

2 2 b dx dy 132. p517 surface area (parametric): S  2y GF JI  GF JI dt za Hdt K Hdt K

133. p532 position vector (standard form): r t  f ti  g t j  h t k

134. p533 speed from velocity vector: speed = vbtg

velocity vector vbtg 135. p533 direction from velocity vector: direction =  speed vbtg

136. p555 polar to Cartesian: x  r cos , y  r sin

x xo   v o cos  t 137. p543 trajectory equations: 1 y y  vsin  t  gt 2 o o 2

rsin  r cos 138. p560 slope of polar graph: slope at (r,)  r cos  r sin

139. slope of polar graph at origin: slope = tan

 1 140. p562 area inside polar curve: A  r 2d z 2

2  dr 141. p564 length of curve (polar): L  r 2  GF JI d z Hd K

2  dr 142. p565 surface area (polar): S  2r sin r 2  GF JI d z Hd K

11 of 11 Hanford High School Calculus Richland, Washington