Quadratic Formula
If
, then
Binomial Theorem
. . .
(x
2
1)
Products of Vectors
Let u be the smaller of the two angles between and .
Then
и ϭ и ϭ a x b x ϩ a y b y ϩ a z b z ϭ ab cos u
Trigonometric Identities
*See Appendix E for a more complete list.
cos a ϩ cos b ϭ 2 cos
1
2 (a ϩ b) cos
1
2 (a Ϫ b)
sin a Ϯ sin b ϭ 2 sin
1
2 (a Ϯ b) cos
1
2 (a ϯ b)
|a
: ϫ b
:
| ϭ ab sin u
ϩ (a x b y Ϫ b x a y )k ˆ
ϭ (a y b z Ϫ b y a z )i ˆ ϩ (a z b x Ϫ b z a x )j ˆ
ϭ i
ˆ
͉
a y
b y
a z
b z ͉ Ϫ j
ˆ
͉
a x
b x
a z
b z ͉ ϩ k ˆ ͉
a x
b x
a y
b y ͉
a
: ϫ b
: ϭ Ϫb
: ϫ a
: ϭ
͉
i
ˆ
a x
b x
j
ˆ
a y
b y
k ˆ
a z
b z
͉
a
:
b
:
b
:
a
:
b
:
a
:
Ͻ
(1 ϩ x)
n ϭ 1 ϩ
nx
1!
ϩ
n(n Ϫ 1)x
2
2!
ϩ
x ϭ
Ϫb Ϯ 1b
2 Ϫ 4ac
2a
ax
2 ϩ bx ϩ c ϭ 0
Derivatives and Integrals
Cramer’s Rule
Two simultaneous equations in unknowns x and y,
a 1 x ϩ b 1 y ϭ c 1 and a 2 x ϩ b 2 y ϭ c 2 ,
have the solutions
and
.
y ϭ
͉
a 1
a 2
c 1
c 2 ͉
͉
a 1
a 2
b 1
b 2 ͉
ϭ
a 1 c 2 Ϫ a 2 c 1
a 1 b 2 Ϫ a 2 b 1
x ϭ
͉
c 1
c 2
b 1
b 2 ͉
͉
a 1
a 2
b 1
b 2 ͉
ϭ
c 1 b 2 Ϫ c 2 b 1
a 1 b 2 Ϫ a 2 b 1
͵
dx
(x
2 ϩ a
2
)
3/2
ϭ
x
a
2
(x
2 ϩ a
2
)
1/2
͵
x dx
(x
2 ϩ a
2
)
3/2
ϭ Ϫ
1
(x
2 ϩ a
2
)
1/2
͵
dx
2x
2 ϩ a
2
ϭ ln(x ϩ 2x
2 ϩ a
2
)
͵e
x
dx ϭ e
x
d
dx
e
x ϭ e
x
͵cosxdx ϭ sin x
d
dx
cos x ϭ Ϫsin x
͵si n x dx ϭ Ϫcos x
d
dx
sin x ϭ cos x
SI PREFIXES*
Factor
Prefix
Symbol
Factor
Prefix
Symbol
10
24
yotta
Y
10
–1
deci
d
10
21
zetta
Z
10
–2
centi
c
10
18
exa
E
10
–3
milli
m
10
15
peta
P
10
–6
micro
m
10
12
tera
T
10
–9
nano
n
10
9
giga
G
10
–12
pico
p
10
6
mega
M
10
–15
femto
f
10
3
kilo
k
10
–18
atto
a
10
2
hecto
h
10
–21
zepto
z
10
1
deka
da
10
–24
yocto
y
*In all cases, the first syllable is accented, as in ná-no-mé-ter.
MATHEMATICAL FORMULAS*
If
, then
Binomial Theorem
. . .
(x
2
1)
Products of Vectors
Let u be the smaller of the two angles between and .
Then
и ϭ и ϭ a x b x ϩ a y b y ϩ a z b z ϭ ab cos u
Trigonometric Identities
*See Appendix E for a more complete list.
cos a ϩ cos b ϭ 2 cos
1
2 (a ϩ b) cos
1
2 (a Ϫ b)
sin a Ϯ sin b ϭ 2 sin
1
2 (a Ϯ b) cos
1
2 (a ϯ b)
|a
: ϫ b
:
| ϭ ab sin u
ϩ (a x b y Ϫ b x a y )k ˆ
ϭ (a y b z Ϫ b y a z )i ˆ ϩ (a z b x Ϫ b z a x )j ˆ
ϭ i
ˆ
͉
a y
b y
a z
b z ͉ Ϫ j
ˆ
͉
a x
b x
a z
b z ͉ ϩ k ˆ ͉
a x
b x
a y
b y ͉
a
: ϫ b
: ϭ Ϫb
: ϫ a
: ϭ
͉
i
ˆ
a x
b x
j
ˆ
a y
b y
k ˆ
a z
b z
͉
a
:
b
:
b
:
a
:
b
:
a
:
Ͻ
(1 ϩ x)
n ϭ 1 ϩ
nx
1!
ϩ
n(n Ϫ 1)x
2
2!
ϩ
x ϭ
Ϫb Ϯ 1b
2 Ϫ 4ac
2a
ax
2 ϩ bx ϩ c ϭ 0
Derivatives and Integrals
Cramer’s Rule
Two simultaneous equations in unknowns x and y,
a 1 x ϩ b 1 y ϭ c 1 and a 2 x ϩ b 2 y ϭ c 2 ,
have the solutions
and
.
y ϭ
͉
a 1
a 2
c 1
c 2 ͉
͉
a 1
a 2
b 1
b 2 ͉
ϭ
a 1 c 2 Ϫ a 2 c 1
a 1 b 2 Ϫ a 2 b 1
x ϭ
͉
c 1
c 2
b 1
b 2 ͉
͉
a 1
a 2
b 1
b 2 ͉
ϭ
c 1 b 2 Ϫ c 2 b 1
a 1 b 2 Ϫ a 2 b 1
͵
dx
(x
2 ϩ a
2
)
3/2
ϭ
x
a
2
(x
2 ϩ a
2
)
1/2
͵
x dx
(x
2 ϩ a
2
)
3/2
ϭ Ϫ
1
(x
2 ϩ a
2
)
1/2
͵
dx
2x
2 ϩ a
2
ϭ ln(x ϩ 2x
2 ϩ a
2
)
͵e
x
dx ϭ e
x
d
dx
e
x ϭ e
x
͵cosxdx ϭ sin x
d
dx
cos x ϭ Ϫsin x
͵si n x dx ϭ Ϫcos x
d
dx
sin x ϭ cos x
SI PREFIXES*
Factor
Prefix
Symbol
Factor
Prefix
Symbol
10
24
yotta
Y
10
–1
deci
d
10
21
zetta
Z
10
–2
centi
c
10
18
exa
E
10
–3
milli
m
10
15
peta
P
10
–6
micro
m
10
12
tera
T
10
–9
nano
n
10
9
giga
G
10
–12
pico
p
10
6
mega
M
10
–15
femto
f
10
3
kilo
k
10
–18
atto
a
10
2
hecto
h
10
–21
zepto
z
10
1
deka
da
10
–24
yocto
y
*In all cases, the first syllable is accented, as in ná-no-mé-ter.
MATHEMATICAL FORMULAS*