prev
(
testsuite
>
TortureTests
>
Complexity
> complex3 )
next
Sample Rendering:
N/A
Your browser's rendering
:
2
∑
a
b
x
′
3
f
′
(
x
)
+
sin
cos
θ
=
1
f
(
z
)
=
∑
n
=
0
∞
a
n
z
n
,
|
z
|
<
R
(
R
≠
0
)
∫
C
(
∑
n
=
0
∞
a
n
z
n
)
ⅆ
z
=
∑
n
=
0
∞
a
n
∫
C
z
n
ⅆ
z
lim
n
→
∞
|
∫
C
[
f
(
z
)
−
∑
k
=
0
n
a
k
z
k
]
ⅆ
z
|
=
0
n
≥
N
(
ε
)
⇒
|
f
(
z
)
−
∑
k
=
0
n
a
k
z
k
|
<
ε
10
Bq
+
10
Ci
10
amol
+
10
Emol
−
10
fmol
+
10
Gmol
−
10
kmol
+
10
Mmol
10
μmol
+
10
mmol
−
10
mol
+
10
nmol
−
10
Pmol
+
10
pmol
−
10
Tmol
10
acre
+
10
hectare
−
10
ft
2
+
10
in
2
−
10
m
2
10
A
+
10
kA
−
10
μA
+
10
mA
−
10
nA
10
F
+
10
μF
−
10
mF
+
10
nF
−
10
pF
10
C
+
1.0
m/s/s
−
0.1
m
/
s
2
10
kS
+
10
μS
−
10
mS
+
10
S
10
kV
+
10
MV
−
10
μV
+
10
mV
−
10
nV
+
10
pV
−
10
V
10
GΩ
+
10
kΩ
−
10
MΩ
+
10
mΩ
−
10
Ω
10
Btu
+
10
cal
−
10
eV
+
10
erg
−
10
GeV
+
10
GJ
10
J
+
10
kcal
−
10
kJ
+
10
MeV
−
10
MJ
+
10
μJ
−
10
mJ
+
10
nJ
10
dyn
+
10
kN
−
10
MN
+
10
μN
−
10
mN
+
10
N
−
10
ozf
+
10
lbf
10
EHz
+
10
GHz
−
10
Hz
+
10
kHz
−
10
MHz
+
10
PHz
−
10
THz
10
fc
+
10
lx
−
10
phot
10
Å
+
10
am
−
10
cm
+
10
dm
−
10
fm
+
10
ft
−
10
in
10
km
+
10
m
−
10
μm
+
10
mi
−
10
mm
+
10
nm
−
10
pm
10
sb
10
lm
10
cd
10
Mx
+
10
μWb
−
10
mWb
+
10
nWb
−
10
Wb
10
G
+
10
μT
−
10
mT
+
10
nT
−
10
pT
+
10
T
10
H
+
10
μH
−
10
mH
10
u
+
10
cg
−
10
dg
+
10
g
−
10
kg
+
10
μg
−
10
mg
+
10
lb
−
10
slug
10
°
+
10
μrad
−
10
mrad
+
10
′
−
10
rad
+
10
′′
10
GW
+
10
hp
−
10
kW
+
10
MW
−
10
μW
+
10
mW
−
10
nW
+
10
W
10
atm
+
10
bar
−
10
kbar
+
10
kPa
−
10
MPa
+
10
μPa
−
10
mbar
+
10
mmHg
−
10
Pa
+
10
torr
10
sr
10
°C
+
10
°F
−
10
K
10
as
+
10
d
−
10
fs
+
10
h
−
10
μs
+
10
ms
−
10
min
+
10
ns
−
10
ps
+
10
s
−
10
y
10
ft
3
+
10
in
3
−
10
m
3
+
10
gal
−
10
l
10
ml
+
10
pint
−
10
qt
1
x
(
y
)
=
(
−
∫
e
−
1
2
y
2
sin
y
ⅆ
y
+
C
1
)
e
1
2
y
2
ⅅ
x
y
−
y
=
sin
x
(
1
2
)
(
1
2
)
(
1
2
)
[
1
2
]
(
1
2
)
{
1
2
}
〈
1
2
〉
⌊
1
2
⌋
⌈
1
2
⌉
↑
1
2
↑
↓
1
2
↓
↕
1
2
↕
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
−
(
a
−
b
)
=
b
−
a
2
5
+
3
7
=
2
⋅
7
+
3
⋅
5
35
=
29
35
|
a
|
=
{
a
if
a
≥
0
−
a
if
a
<
0
a
n
=
a
⋅
a
⋅
⋯
⋅
a
︸
n
factors
(
a
b
)
−
n
=
(
b
a
)
n
a
n
=
b
means
b
n
=
a
.
16
81
4
=
16
4
81
4
=
2
3
{
x
∣
x
≠
0
,
x
≠
1
}
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
a
3
−
b
3
=
(
a
−
b
)
(
a
2
+
a
b
+
b
2
)
(
x
+
y
)
2
H
=
{
(
a
b
c
d
)
∈
G
∣
a
d
−
b
c
=
1
}
|
x
|
+
||
y
||
+
{
z
}
−
[
a
c
]
+
(
b
)
=
[
a
,
b
]
x
=
1
x
=
1
x
=
1
x
=
1
[
−
10
3
,
−
7
3
)
∪
(
−
7
3
,
−
4
3
]
A
∂
u
∂
x
+
B
∂
u
∂
y
+
C
u
=
E
∑
x
∑
1
<
i
<
10
1
<
j
<
10
2
i
+
j
Γ
1
2
3
4
5
6
7
1
5
7
6
2
4
3
y
(
x
)
=
x
e
x
−
e
x
+
2
e
x
=
x
−
1
+
2
e
x
ⅅ
x
x
y
−
y
=
0
y
(
0
)
=
1
y
′
(
0
)
=
0
y
(
x
)
=
1
3
e
−
(
−
1
)
3
x
+
2
3
e
1
2
(
−
1
)
3
x
cos
1
2
3
(
−
1
)
3
x
y
(
t
)
=
2
tan
(
2
t
−
1
4
π
)
ℱ
(
e
2
π
i
x
2
π
Dirac
(
x
−
2
π
)
,
x
,
s
)
=
(
2
π
Dirac
(
s
−
2
π
)
2
π
e
−
2
i
π
s
)
x
=
1
x
+
3
=
123
t
x
y
z
0
1.0000
1.0000
1.0000
.1
1.1158
1.0938
.8842
.2
1.2668
1.1695
.7332
.3
1.4582
1.2173
.5418
.4
1.6953
1.2253
.3047
.5
1.9830
1.1791
.0170
.6
2.3256
1.0619
−
.3256
.7
2.7265
.8542
−
.7265
.8
3.1873
.5344
−
1.1873
.9
3.7077
.0777
−
1.7077
1.0
4.2842
−
.5424
−
2.2842
K
v
(
z
)
=
BesselK
v
(
z
)
z
2
ⅆ
2
w
ⅆ
z
2
+
z
ⅆ
w
ⅆ
z
−
(
z
2
+
v
2
)
w
=
0
∂
2
u
(
x
,
y
)
∂
x
2
−
∂
2
u
(
x
,
y
)
∂
y
2
=
0
y
(
t
,
x
)
=
F
1
(
−
x
−
a
t
)
+
F
2
(
x
−
a
t
)
1
2
3
4
5
6
2
x
+
1
=
5
1
=
3
9
=
7
a
b
c
d
e
f
x
+
2
y
−
3
=
5
4
x
−
y
−
5
=
98
x
=
z
1
=
3
A
1
=
N
0
(
λ
;
Ω
′
)
−
φ
(
λ
;
Ω
′
)
,
A
2
=
φ
(
λ
;
Ω
′
)
−
φ
(
λ
;
Ω
)
,
A
3
=
N
(
λ
;
ω
)
.
sin
θ
cos
γ
x
=
{
x
if
x
<
0
−
x
if
x
≥
0
L
M
R
M
L
M
R
M
M
A
T
H
M
A
T
H
⋮
∇×
F
=
0
∇·
F
∇·∇
F
=
∇
2
F
+
7
=
A
∇×
(
x
y
,
y
z
,
z
x
)
=
[
−
y
−
z
−
x
]
∇×
(
y
,
z
,
x
)
=
(
−
1
,
−
1
,
−
1
)
≠
0
x
+
y
+
α
=
102
a
+
b
=
c
x
+
1
x
+
f
(
x
)
−
1
=
123
T
h
e
q
u
i
c
k
b
r
o
w
n
f
o
x
j
u
m
p
s
o
v
e
r
t
h
e
l
a
z
y
d
o
g
.
T
h
e
e
n
d
.
(
∂
f
∂
x
1
(
c
1
,
c
2
,
…
,
c
n
)
,
∂
f
∂
x
2
(
c
1
,
c
2
,
…
,
c
n
)
,
…
,
∂
f
∂
x
n
1
(
c
1
,
c
2
,
…
,
c
n
)
)
∇
(
c
u
v
+
v
2
w
)
=
(
u
v
,
c
v
,
c
u
+
2
v
w
,
v
2
)
D
u
f
(
a
,
b
,
c
)
=
∇
f
(
a
,
b
,
c
)
⋅
u
=
∂
f
∂
x
(
a
,
b
,
c
)
u
1
+
∂
f
∂
y
(
a
,
b
,
c
)
u
2
+
∂
f
∂
z
(
a
,
b
,
c
)
u
3
θ
∈
{
π
+
2
X
3
π
−
(
arccos
1
7
14
)
|
X
3
∈
ℤ
}
,
θ
∈
{
2
X
4
π
−
π
+
(
arccos
1
7
14
)
|
X
4
∈
ℤ
}
P
=
A
(
A
T
A
)
−
1
A
T
det
(
x
y
1
a
b
1
a
d
1
)
=
x
b
−
x
d
+
a
d
−
a
b
=
0
A
(
θ
)
A
(
−
θ
)
=
[
cos
θ
−
sin
θ
sin
θ
cos
θ
]
[
cos
θ
sin
θ
−
sin
θ
cos
θ
]
J
(
A
)
=
[
J
n
1
(
λ
1
)
0
⋯
0
0
J
n
2
(
λ
2
)
⋯
0
⋮
⋮
⋱
⋮
0
0
⋯
J
n
k
(
λ
k
)
]
det
(
−
4
+
X
−
1
0
0
−
4
+
X
0
0
0
−
4
+
X
)
=
(
X
−
4
)
3
{
(
−
1
2
−
1
6
33
1
)
}
↔
5
2
−
1
2
33
∥
A
∥
=
max
x
≠
0
∥
A
x
∥
∥
x
∥
(
a
1
1
a
1
2
a
2
1
a
2
2
)
+
(
b
1
1
b
1
2
b
2
1
b
2
2
)
=
(
a
1
1
+
b
1
1
a
1
2
+
b
1
2
a
2
1
+
b
2
1
a
2
2
+
b
2
2
)
f
(
[
1
2
4
3
]
)
=
[
1
2
4
3
]
2
−
5
[
1
2
4
3
]
−
2
=
[
2
−
2
−
4
0
]
x
=
lim
x
=
1
∑
1
2
a
∫
a
b
f
(
x
)
ⅆ
x
=
lim
∥
P
∥
→
0
∑
i
=
1
n
f
(
x
¯
i
)
Δ
x
i
∫
a
b
f
(
x
)
ⅆ
x
=
lim
n
→
∞
b
−
a
n
∑
i
=
1
n
f
(
a
+
i
b
−
a
n
)
∫
0
2
x
5
x
3
+
1
ⅆ
x
=
∫
1
3
2
3
u
(
u
2
)
(
u
2
−
1
)
2
3
(
u
2
(
u
2
−
1
)
2
3
−
(
u
2
−
1
)
2
3
)
ⅆ
u
∫
f
(
g
(
x
)
)
g
′
(
x
)
ⅆ
x
=
∫
f
(
u
)
ⅆ
u
x
=
2
∑
n
=
1
100
n
(
n
−
1
)
lim
x
→
0
sin
(
1
x
)
=
−
1
..
1
h
(
i
,
j
)
=
(
2
−
j
)
g
(
i
)
+
(
j
−
1
)
f
(
g
(
i
)
)
△
:
[
0
,
1
]
→
[
0
,
1
]
0
▽
x
=
x
x
△
y
=
h
−
1
(
h
(
x
)
h
(
y
)
)
x
△
y
=
f
−
1
(
max
{
f
(
x
)
+
f
(
y
)
−
1
,
0
}
)
x
▽
y
=
η
(
η
(
x
)
△
η
(
y
)
)
x
△
0
y
=
{
x
∧
y
if
x
∨
y
=
1
0
if
x
∨
y
<
1
lim
a
→
1
+
log
a
[
1
+
(
a
x
−
1
)
(
a
y
−
1
)
a
−
1
]
=
lim
a
→
1
−
log
a
[
1
+
(
a
x
−
1
)
(
a
y
−
1
)
a
−
1
]
=
x
y
g
(
x
)
=
exp
(
−
1
−
(
1
−
x
)
a
(
2
a
−
1
)
(
1
−
x
)
a
)
Aut
(
I
)
=
{
f
:
[
0
,
1
]
→
[
0
,
1
]
|
f
is one-to-one and onto, and
x
≤
y
implies
f
(
x
)
≤
f
(
y
)
}
x
2
+
y
2
=
r
2
,
tan
θ
=
y
x
2
1
−
t
2
[
(
2
+
sin
t
)
10
cos
t
,
(
2
+
cos
t
)
10
sin
t
,
3
sin
3
t
]
{
t
=
0
,
s
=
0
}
,
{
t
=
π
,
s
=
π
}
1
2
3
4
5
6
7
8
9
10
2
4
6
8
10
1
3
5
7
9
3
6
9
1
4
7
10
2
5
8
4
8
1
5
9
2
6
10
3
7
5
10
4
9
3
8
2
7
1
6
6
1
7
2
8
3
9
4
10
5
7
3
10
6
2
9
5
1
8
4
8
5
2
10
7
4
1
9
6
3
9
7
5
3
1
10
8
6
4
2
10
9
8
7
6
5
4
3
2
1
testing
x
2
end.
x
x
x
x
ⅆ
f
ⅆ
x
(
x
1
)
=
5
∫
x
ⅆ
x
=
∬
x
y
ⅆ
x
ⅆ
y
=
∭
x
y
z
ⅆ
x
ⅆ
y
ⅆ
z
=
⨌
x
y
z
t
ⅆ
x
ⅆ
y
ⅆ
z
ⅆ
t
mod
a
5
mod
3
=
2
f
(
0
)
mod
3
=
1
5
x
+
4
≡
8
(
mod
13
)
a
=
(
5
−
3
)
/
5
mod
7
=
6
(
2
x
2
+
x
+
2
)
+
(
2
x
+
1
)
mod
3
=
2
x
2
+
0
1
000
000
111
1
1
0
4. 974 9
ⅆ
ⅆ
x
F
(
x
)
[
86.333
,
146.33
,
129.33
]
BinomialDist
(
x
;
n
,
p
)
=
∑
k
=
0
x
(
n
k
)
p
k
q
n
−
k
Pr
(
X
≤
54
)
=
BinomialDist
(
54
;
100
,
.55
)
=
.45846
k
=
max
{
|
∂
f
∂
y
(
x
,
y
)
|
:
(
x
,
y
)
∈
D
}
.
m
=
lim
x
→
a
f
(
x
)
−
f
(
a
)
x
−
a
|
A
|
=
|
a
1
1
a
1
2
⋅
⋅
⋅
a
1
n
a
2
1
a
2
2
⋅
⋅
⋅
a
2
n
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
a
n
1
a
n
2
⋅
⋅
⋅
a
n
n
|
=
a
1
1
A
1
1
+
a
1
2
A
1
2
+
⋯
+
a
1
n
A
1
n
x
=
1
(
hl text
x
end.
)
x
=
1
(
hl to URI
x
end
)
x
=
1
(
sex
)
x
=
1
(
jbm
)
f
(
x
)
g
[
y
]
h
{
z
}
+
⌊
a
⌋
⌈
b
⌉
〈
c
〉
123
456
A
|
∥
A
B
A
/
1
2
A
/
(
3
4
A
)
↕
5
6
A
↕
7
8
A
⇕
9
20
10
A
⇕
↑
11
12
A
↑
⇑
13
14
A
⇑
↓
15
16
A
↓
⇓
17
18
A
⇓
x
x
x
x
x
x
(
a
1
,
a
2
,
…
,
a
n
)
⋅
(
b
1
,
b
2
,
…
,
b
n
)
=
a
1
b
1
*
+
a
2
b
2
*
+
⋯
+
a
n
b
n
*
⌊
n
5
⌋
+
⌊
n
5
2
⌋
+
⌊
n
5
3
⌋
+
⌊
n
5
4
⌋
+
⋯
x
1
+
⋯
+
x
n
x
+
⋯
+
x
︸
k
times
x
1
x
2
⋯
x
n
n
n
!
=
1
×
2
×
3
×
4
×
⋯
×
n
P
:
a
=
x
0
<
x
1
<
x
2
<
⋯
<
x
n
=
b
f
(
x
)
=
30
13
cos
x
+
10
3
(
100
+
9
cos
2
x
−
60
cos
x
sin
(
x
+
29
90
π
)
)
∫
cos
(
A
x
)
sin
(
B
x
)
ⅆ
x
=
−
cos
(
B
−
A
)
x
2
(
B
−
A
)
+
−
cos
(
B
+
A
)
x
2
(
B
+
A
)
+
C
.
235.3
+
813
=
1048. 3
max
−
2
≤
x
≤
2
(
x
3
−
6
x
+
3
)
=
8.0
x
decade
=
2
century
ⅆ
5
(
x
7
−
3
x
6
)
ⅆ
x
5
ⅆ
n
sin
x
ⅆ
x
n
ⅆ
3
ⅆ
x
3
f
(
x
)
ⅆ
2
ⅆ
t
2
(
4
t
5
−
3
t
)
f
(
x
)
=
30
13
cos
x
+
10
3
(
100
+
9
cos
2
x
−
60
cos
x
sin
(
x
+
29
90
π
)
)
∫
R
3
(
|
u
1
|
2
+
|
∇
u
0
|
2
2
+
|
u
0
|
6
6
)
ⅆ
x
<
∞
(
∇×
F
)
⋅
k
=
z
+
1
M
M
M
M
M
ⅅ
x
x
2
ⅅ
x
(
x
2
)
ⅅ
x
x
(
x
2
)
ⅅ
x
2
(
x
2
)
ⅅ
x
y
(
x
2
y
3
)
ⅅ
x
s
y
t
(
x
2
y
3
)
5
24
!
x
6
x
+
a
y
−
1
12.34
2
sin
θ
1
0
1
1
0
(
0
−
i
i
0
)
[
1
0
0
−
1
]
|
a
b
c
d
|
∥
1
0
1
0
11
∥
1
2
3
4
5
testing
sin
θ
a
̂
+
b
ˇ
+
c
˜
+
d
´
+
e
`
+
f
˘
+
g
¯
+
h
+
i
˚
+
j
˙
+
k
¨
+
l
⃛
+
m
⃜
+
n
→
f
(
g
(
x
)
)
=
sin
3
x
2
+
sin
x
2
sin
(
sin
x
2
)
(
x
2
+
12
x
2
+
12
)
+
1234
x
=
1
not
here
x
2
merged
y
1
jbm
lowlife
The end.
x
2
+
y
2
=
z
2
−
1
x
2
+
y
2
=
z
2
−
1
x
+
y
3
=
z
3
x
2
+
y
2
=
z
2
−
1
x
+
y
3
=
z
3
x
2
+
y
2
=
1
x
=
1
−
y
2
(
a
+
b
)
2
=
a
2
+
2
a
b
+
b
2
(
a
+
b
)
⋅
(
a
−
b
)
=
a
2
−
b
2
First line of equation
Middle line of equation
Other middle line of equation
Last line of equation
L
1
=
R
1
L
2
=
R
2
L
3
=
R
3
L
4
=
R
4
(
a
+
b
)
4
=
(
a
+
b
)
2
(
a
+
b
)
2
=
(
a
2
+
2
a
b
+
b
2
)
(
a
2
+
2
a
b
+
b
2
)
=
a
4
+
4
a
3
b
+
6
a
2
b
2
+
4
a
b
3
+
b
4
x
2
+
y
2
=
1
x
=
1
−
y
2
(
a
+
b
)
2
=
a
2
+
2
a
b
+
b
2
(
a
+
b
)
⋅
(
a
−
b
)
=
a
2
−
b
2
Vertex
V
(
0
,
0
)
Focus
F
(
0
,
p
)
Directrix
y
=
−
p
ⅆ
ⅆ
x
(
csc
−
1
x
)
=
−
1
|
x
|
x
2
−
1
tanh
−
1
x
=
1
2
ln
(
1
+
x
1
−
x
)
−
1
<
x
<
1
∠
α
+
∠
A
B
C
+
∠
1
=
▵
a
b
c
y
=
e
−
∫
P
ⅆ
x
[
∫
e
∫
P
ⅆ
x
Q
ⅆ
x
+
c
]
x
=
1
+
y
3
x
=
1
+
y
$
1.00
+
25
¢
−
3
£
+
2.45
¤
−
0.7
¥
−
a
₠
+
20
₣
+
30
₤
−
4.56
₧
2
x
+
y
=
3
3
x
−
4
y
=
5
a
+
b
=
c
+
12345
Unrestricted
Symmetric
Antisymmetric
Triangular
a
≠
b
≠
x
c
≮
d
≮
y
e
≯
f
≯
11
g
∉
h
∉
Z
k
≁
l
≁
3
A
⊄
B
⊂
C
A
⊈
B
⊈
C
10
≢
11
≡
12
x
≰⃥
y
≰⃥
z
lim
¯
x
lim
̲
x
lim
→
x
lim
←
x
x
=
y
+
z
=
k
+
m
College Algebra
Second Edition
James Stewart
McMaster Universitiy
Lothar Redlin
Pennsylvania State University
Saleem Watson
California State University, Long Beach
Copyright 1996, ISBN 0 534-33983-2
Brooks/Cole Publishing Company
An International Thomson Publishing Company
{
1
2
1
2
↑
∑
1
2
}
〈
1
2
1
2
|
∑
1
2
〉
⌈
1
2
1
2
|
∑
1
2
⌉
⇓
1
2
1
2
↕
∑
1
2
⇓
[
1
2
1
2
]
(
1
2
1
2
)
{
1
2
1
2
}
〈
1
2
1
2
〉
⌊
1
2
1
2
⌋
⌈
1
2
1
2
⌉
↑
1
2
1
2
↑
↓
1
2
1
2
↓
↕
1
2
1
2
↕
⇑
1
2
1
2
⇑
⇓
1
2
1
2
⇓
⇕
1
2
1
2
⇕
1
2
1
2
\arrowvert
1
2
1
2
\arrowvert
\Arrowvert
1
2
1
2
\Arrowvert
\bracevert
1
2
1
2
\bracevert
|
1
2
1
2
|
|
1
2
1
2
|
|
1
2
1
2
|
∥
1
2
1
2
∥
∥
1
2
1
2
∥
/
1
2
1
2
/
\
1
2
1
2
\
⎱
1
2
1
2
⎰
\lgroup
1
2
1
2
\rgroup
⌞
1
2
1
2
⌟
⌜
1
2
1
2
⌝
A
←
n
+
μ
−
1
B
→
T
n
±
i
−
1
C
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
…
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
…
(
sin
θ
M
⌋
(
sin
θ
M
⌋
(
sin
θ
M
⌋
(
sin
θ
M
⌋
(
sin
θ
M
⌋
(
sin
θ
M
⌋
(
sin
θ
M
⌋
(
sin
θ
M
⌋
(
sin
θ
M
⌋
sin
θ
M
sin
θ
M
sin
θ
M
sin
θ
M
sin
θ
M
sin
θ
M
sin
θ
M
sin
θ
M
sin
θ
M