A.P. Calculus Formulas

A.P. Calculus Formulas 2004-2005

1.
floor function (def)
x
2.
Greatest integer that is less than or equal to x.
(graph)
3.
a 3 + b3 =
4.
a 3 − b3 =
5.
f ( x) =
ba + bgca − ab + b h
ba − bgca + ab + b h
2
2
2
2
1
(graph)
x
3
2
1
-3
-2
-1
0
1
x
2
3
-1
-2
-3
ln x
ln a
Change of base rule for logs:
log a x =
7.
Circle formula:
( x − h)
8.
Ellipse formula:
x2 y2
+
=1
a 2 b2
6.
p34
1
2
+ ( y − k ) = r2
2
c = a 2 − b2
9.
Hyperbola formula:
x2 y2
−
=1
a 2 b2
10.
sin 2 x + cos2 x =
1
11.
1 + tan 2 x =
sec 2 x
12.
sin u ± v =
13.
b g
cosb
u ± vg
=
14.
sin(2u) =
2 sin u ⋅ cos u
15.
cos(2u) =
cos2 u − sin 2 u
16.
sin 2 u =
17.
cos2 u =
sin u ⋅ cos v ± cos u ⋅ sin v
cos u ⋅ cos v sin u ⋅ sin v
1 − cos 2u
2
1 + cos 2u
2
sin x
x →0
x
sin x
lim
x →∞
x
lim
18.
p57
19.
p67
20.
p79
Intermediate Value Theorem
21.
p95
definition of derivative
22.
p112
d
( c) =
dx
0
bg
1
d
cu =
dx
bg
cu′
d n
u =
dx
nu n−1u ′
24.
25.
1
0
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 )
f ′( x ) = lim
h→ 0
h
d
x =
dx
23.
p113
c = a 2 + b2
2
b g
26.
p114
d
u±v =
dx
u′ ± v ′
27.
p115
d
(uv ) =
dx
uv ′ + vu ′
28.
p117
d u
=
dx v
F
I
G
HJK
vu ′ − uv ′
v2
29.
p135
d
sin u =
dx
cosu ⋅ u ′
30.
p136
d
cos u =
dx
− sin u ⋅ u ′
31.
p138
d
tan u =
dx
sec 2 u ⋅ u ′
32.
p138
33.
p138
34.
p138
35.
36.
37.
38.
p144
p157
p159
d
cot u =
dx
d
sec u =
dx
− csc 2 u ⋅ u ′
sec u ⋅ tan u ⋅ u ′
d
csc u =
dx
− csc u ⋅ cot u ⋅ u ′
slope of parametrized curve:
dy
dy
= dt
dx
dx
dt
df −1
dx
derivative formula for inverses
=
x= f (a)
u′
d
sin −1 u =
dx
1 − u2
−u ′
d
cos −1 u =
dx
1 − u2
3
1
df
dx
x =a
39.
p159
d
tan −1 u =
dx
u′
1 + u2
40.
p164
d u
e =
dx
euu′
41.
p166
d
ln u =
dx
1
u′
u
d u
a =
dx
a u ln a ⋅ u ′
42.
43.
p178
Extreme Value Theorem
If f is continuous over a closed interval, then
f has a maximum and minimum value over
that interval.
44.
p186
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
45.
p221
linearization formula
L( x ) = f (a ) + f ′(a ) ⋅ ( x − a )
46.
p269
∫ k ⋅ f ( u)
k ∫ f ( u ) du
47.
p269
48.
p272
z
du =
z z
f ( u) ± g (u) du =
f (u)du ± g (u)du
If f is continuous on [ a, b ] , then at some
1 b
f ( x ) dx
point c in [ a, b ] , f ( c ) =
b − a ∫a
Mean Value Theorem
(for definite integrals)
49.
p277
Second fundamental theorem:
d u
f (t )dt = f (u ) ⋅ u ′
dx ∫a
50.
p290
Trapezoidal Rule:
T=
51.
z
du =
u+c
4
b
h
y0 + 2 y1 + 2 y2 +...+2 yn −1 + yn
2
g
z
u n+1
+c
n +1
p317
∫ sin u du =
− cosu + c
54.
p317
∫ cos u du =
sin u + c
55.
p317
∫ sec
2
u du =
tan u + c
56.
p317
∫ csc
2
u du
− cot u + c
57.
p317
∫ sec u ⋅ tan u du =
sec u + c
58.
p317
∫ csc u ⋅ cot u du =
− csc u + c
52.
p315
53.
59.
60.
61.
u n du =
z
z
z
1
du =
u
ln u + c
e u du =
eu + c
a u du =
1 u
a +c
ln a
n ≠ −1
62.
∫ tan u du =
− ln cosu + c
63.
∫ cot u du =
ln sin u + c
64.
∫ sec u du =
ln sec u + tan u + c
65.
∫ csc u du =
− ln csc u + cot u + c
66.
67.
z
z
du
a −u
2
2
u
arcsin + c
a
=
du
=
a + u2
1
u
arctan + c
a
a
2
5
exponential change:
y = y0e kt
69.
continuous compound interest:
A(t ) = Ao e rt
70.
F1 + x IJ =
limG
H nK
68.
p330
n
ex
n→∞
6