A.P. Calculus Formulas 2008-2009
Hanford High School, Richland, Washington
revised 8/25/08

1.
2.
3.
4.
floor function (def)
x
(graph)
ceiling function (def)
x
Greatest integer that is less than or equal to x.
Least integer that is greater than or equal to x.
(graph)
5.
a 3  b3 
ba  bgca  ab  b h
6.
a 3  b3 
ba  bgca  ab  b h
2
2
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2
2
Hanford High School Calculus
Richland, Washington
f ( x) 
7.
1
(graph)
x
3
2
1
-3
-2
-1
0
1
x
2
3
-1
-2
-3
ln x
ln a
8.
p42
Change of base rule for logs:
log a x 
9.
p579
Circle formula:
 x  h   y  k 
10.
p580
Parabola formula:
 x  h
11.
p583
Ellipse formula:
x2 y2

1
a 2 b2
c  a 2  b2
12.
p585
Hyperbola formula:
x2 y2

1
a 2 b2
c  a 2  b2
13.
p591
eccentricity:
e
14.
sin2 x  cos2 x 
1
15.
1  tan2 x 
sec 2 x
16.
1  cot 2 x 
csc2 x
17.
sin u  v 
sin u  cos v  cos u  sin v
18.
b g
cosb
u  vg

19.
tan u  v 
b g
2
2
2
 r2
 4p y  k
c
a
cos u  cos v  sin u  sin v
tan u  tan v
1 tan u  tan v
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Hanford High School Calculus
Richland, Washington
20.
sin(2u) 
2 sin u  cos u
21.
cos(2u) 
cos2 u  sin2 u
22.
tan(2u) 
2 tan u
1  tan 2 u
23.
sin2 u 
1  cos 2u
2
24.
cos2 u 
1  cos 2u
2
25.
tan 2 u 
1  cos 2u
1  cos 2u
26.
sin u  sin v 
1
cos u  v  cos u  v
2
27.
cos u  cos v 
1
cos u  v  cos u  v
2
28.
sin u  cos v 
1
sin u  v  sin u  v
2
29.
cos u  sin v 
1
sin u  v  sin u  v
2
30.
law of sines:
a
b
c


sin A sin B sin C
31.
law of cosines:
c 2  a 2  b 2  2ab cos C
32.
area of triangle using trig.
Area 
b g b g
b g b g
b g b g
b g b g
1
ac sin B
2
33.
p32
parameterization of ellipse:
x2 y 2

 1 becomes x  a cos t , y  b sin t
a 2 b2
34.
p60
lim
sin x
x 0
x
1
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Hanford High School Calculus
Richland, Washington
sin x
x 
x
35.
p71
lim
0
36.
p83
Intermediate Value Theorem
If a function is continuous between a and b ,
then it takes on every value between f (a ) and
f (b) .
37.
p99
definition of derivative
38.
p116
d
( c) 
dx
0
bg
1
d
cu 
dx
bg
cu
d n
u 
dx
nun1u
d
x 
dx
39.
40.
p117
41.
b g
f ( x )  lim
h 0
42.
p117
d
uv 
dx
u  v 
43.
p119
d
(uv ) 
dx
uv   vu 
44.
p120
d u

dx v
F
I
G
HJ
K
vu   uv 
v2
45.
p142
d
sin u 
dx
cosu  u 
46.
p142
d
cos u 
dx
 sin u  u
47.
p145
d
tan u 
dx
sec2 u  u
48.
p145
d
cot u 
dx
 csc2 u  u
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f ( x  h)  f ( x )
h
Hanford High School Calculus
Richland, Washington
49.
p145
d
sec u 
dx
sec u  tan u  u
50.
p145
d
csc u 
dx
 csc u  cot u  u
slope of parametrized curve:
dy
dy
 dt
dx
dx
dt
derivative formula for inverses
df 1
dx
51.
52.
53.
p151
p165
p166
55.
p167
56.
57.
p168
df
dx
x a
1  u2
u
1  u2
d
tan 1 u 
dx
u
1  u2
d
cot -1 u 
dx
u
1  u2
d
sec-1 u 
dx
u
u u2 1
u
d
csc-1 u 
dx
58.
x f (a)
1
u
d
sin 1 u 
dx
d
cos 1 u 
dx
54.

u u2 1

p169
cot 1 ( x) 
60.
p169
sec 1 ( x) 
cos1
61.
p169
csc 1 ( x) 
sin 1
5 of 11
u 1
 tan 1  x 
59.
2
u 1
1I
F
G
Hx J
K
1I
F
G
Hx J
K
Hanford High School Calculus
Richland, Washington
62.
p172
d u
e 
dx
euu
63.
p174
d
ln u 
dx
1
u
u
64.
p173
d u
a 
dx
a u ln a  u
65.
p188
Extreme Value Theorem
If f is continuous over a closed interval, then
f has a maximum and minimum value over
that interval.
66.
p196
Mean Value Theorem
(for derivatives)
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
67.
p233
linearization formula
L( x )  f ( a )  f  ( a )  ( x  a )
68.
p236
Newton’s Method
x n 1  x n 
69.
p285
 k  f  u  du 
k  f  u  du
70.
p285
z
z z
71.
p288
Mean Value Theorem
If f is continuous on  a, b , then at some
(for definite integrals)
point c in  a, b , f  c  
f (u)  g (u) du 
bg
bg
f xn
f  xn
f (u)du  g (u)du
1 b
f  x  dx
b  a a
72.
p294
First fundamental theorem:
d u
f (t )dt  f (u )  u 
dx a
73.
p307
Trapezoidal Rule:
T
h
y0  2 y1  2 y2 ...2 yn 1  yn
2
74.
p309
Simpson’s Rule:
S
h
y0  4 y1  2 y2 ...2 yn  2  4 yn 1  yn
3
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b
b
g
g
Hanford High School Calculus
Richland, Washington
75.
z
z
du 
uc
u du 
u n1
c
n 1
n
76.
p332
77.
p332
 sin u du 
 cosu  c
78.
p332
 cos u du 
sin u  c
79.
p332
 sec
80.
p332
 csc
81.
p332
 sec u  tan u du 
secu  c
82.
p332
 csc u  cot u du 
 cscu  c
83.
p332
84.
p332
85.
p332
2
u du 
tan u  c
2
u du
 cot u  c
z
z
z
1
du 
u
ln u  c
eu du 
eu  c
a u du 
1 u
a c
ln a
n  1
86.
 tan u du 
 ln cosu  c
87.
 cot u du 
ln sin u  c
88.
 sec u du 
ln sec u  tan u  c
89.
 csc u du 
 ln csc u  cot u  c
90.
z
arcsin
du
a u
2
2

7 of 11
u
c
a
Hanford High School Calculus
Richland, Washington
z
z
du

a  u2
91.
1
u
arctan  c
a
a
2
92.
du
u u a
2
2
u
1
arcsec
c
a
a

93.
p341
Integration by parts (formula):
z
94.
p341
order for choosing u in
integration by parts:
LIPET  logs, inverse trig., polynomial,
exponential, trig.
95.
p351
exponential change:
y  y0 e kt
96.
p353
half-life
ln 2
k
continuous compound interest:
A(t )  Ao ert
97.
z
udv  uv  vdu
98.
p365
logistics differential equation:
dP
 kP  M  P 
dt
99.
p367
logistics growth model
P
100. p409
M
1  Ae Mk  t
2
 dy 
surface area about x axis (Cartesian): S   2 y 1    dx
a
 dx 
b
b
2
 dy 
1    dx
 dx 
101. p413
length of curve (Cartesian):
L
102.
Mr. Kelly’s e-mail address:
Greg.Kelly@rsd.edu
a
FORMULAS BELOW HERE ARE BC ONLY:
103. p450
104.
lim
ln n

n  n
0
lim n n 
1
n
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Hanford High School Calculus
Richland, Washington
1
105.
106.
lim x n 
1
lim x n 
0
n 
n
F1  x IJ 
limG
H nK
ex
xn
lim

n n !
0
cx  1h
n
107.
108.
n
n(n  1)
2
n
109.
k 
k 1
n
110.
k
2

n( n  1)(2n  1)
6

n(n  1) I
F
G
H2 J
K
k 1
2
n
111.
k
3
k 1
112.
partial sum of geometric series:
113. p475
What series?

 ar
n 1
Sn 
c h
a 1 rn
1 r
geometric, converges to
n 1
114. p487
Maclaurin Series:
115. p489
Taylor Series:
P( x )  f (0)  f (0) x 
a
if r  1
1 r
f (0) 2 f (0) 3
x 
x ...
2!
3!
P( x )  f (a )  f (a )( x  a ) 
f (a )
3
x  a ...
3!
f (a )
xa
2!
b g
2
b g
116. p491
Maclaurin Series for
1
1 x
117. p491
Maclaurin Series for
1
1 x
1
 1  x  x 2  x 3 ...
1 x
1
 1  x  x 2  x 3 ...
1 x
Hanford High School Calculus
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Richland, Washington
ex  1 x 
x
x2 x3 x4


...
2! 3! 4!
118. p491
Maclaurin Series for e
119. p491
Maclaurin Series for sin x
sin x  x 
x3 x5 x7


...
3! 5! 7!
120. p491
Maclaurin Series for cos x
cos x  1 
x2 x4 x6


...
2! 4! 6!
121. p491
Maclaurin Series for ln(1 x )
ln(1  x )  x 
1
 x :
x 3 x5 x 7
tan ( x)  x    ...
3 5 7
1
122. p491
Maclaurin Series for tan
123. p496
Lagrange form of remainder
Rn  x  
124.
Taylor’s Inequality
Rn  x  

125. p498
What series?
1
 n!
x2 x3 x4
  ...
2
3 4
 c  x  a n1


 n  1!
f
n 1
M
n 1
xa
 n  1!
reciprocal of factorials, converges to e
n0

126. p510
What series?
b b
b
n
n 1
n 1

127. p514
What series?
1
n
g
telescoping series, converges to b1  lim bn 1
n 
p series, converges if p  1
p
n1

128. p514
What series?
1
n
harmonic, diverges
n1

129. p517
What series?
1g
b
n 1
n 1
1
n
alternating harmonic, converges
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Hanford High School Calculus
Richland, Washington
2nd deriv. of parametrized curve:
dy 
d y
 dt
dx
dx 2
dt
131. p533
length of curve (parametric):
Fdx I Fdy I
L  z G J  G Jdt
Hdt K Hdt K
132.
surface area (parametric):
S
133.
position vector (standard form):
r t   f t  i  g t  j  h t  k
134. p542
speed from velocity vector:
speed = v t
135. p542
direction from velocity vector:
direction =
136. p551
polar to Cartesian:
x  r cos , y  r sin 
137.
trajectory equations:
x  xo   vo cos   t
2
130. p532
2
b
2
a
dx I F
dy I
F
 G Jdt
G
J
Hdt K Hdt K
z
b
2
2y
a
2
bg
bg
bg
velocity vector v t

speed
vt
1
y  yo   vo sin   t  gt 2
2
138. p552
slope of polar graph:
slope at (r , ) 
139.
slope of polar graph at origin:
slope = tan
140. p553
area inside polar curve:
A
141.
142.
length of curve (polar):
surface area (polar):
L
S
11 of 11
z
z
r  sin   r cos
r  cos  r sin 

1 2
r d
 2


z


r
Fdr I
 G Jd
Hd K
2
2
2r sin  r
Fdr I
 G Jd
Hd K
2
2
Hanford High School Calculus
Richland, Washington