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
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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
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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
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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
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csc2 u u
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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
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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
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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
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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
[e-mail] 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
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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
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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
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Hanford High School Calculus Richland, Washington