ALGEBRA
Logarithms
*
Exponents
and
*
P
exponent
logz
✗
work
zY=x
y
=
Word Problems
""
( diff
rates )
.
¥)
+
T
base
logarithm
common
Natural
:
logarithm
:
A
base
B-
time it takes
1-
time it takes for both A
toga
toga
✗
toga XP
=
logax
=
-
✗
to
b±
=
b
'
-
b
b
'
-
-
b-
4ac
two real
roots
conjugate
rz
rz
roots
equal
-
Addition
. .
Ex
What
.
25¢
( desired 4)
=
6×-0.5 ✗
0°
=
90°
+
(initial 4 going
+
B-
"
r r,
:
*
C-
=
,
✗
A
hands meet
)
CW
will the hands meet ?
3:00
after
time
minus ba ?
.
done
10¢
6×-0.5 ✗
the roots:
are
=
spent )
I ¢
:
A
*
job
^
→
I
1
r, +
workers )( time
clock
two real distinct roots
and
(# of
Ja
Quarters :
0
r,
job
5¢
:
Dimes
discriminant
→
4ac
Suppose
alone
& B to finish the
Wztz
Cents
i
Nickels
0
complex
4ac > 0
/
Pennies
0
-
job
coin
4ac <
=
to finish the
for B
alone
D= rt
2A
'
job
changing
accordingly
Distance
a
=
the
=
plogax
109€
bx
+
ax
finish
A to
Ji
Equations
Quadratic
for
takes
W , T,
(g)
toga
flexible by
rate)
( same
logalxy )
=
,
=
work
=
10g
*
Ye
laws :
logay
-
-
base
:
logay
t
✗
e
time it
=
is
the ratios in LHS
base 10
Naperian logarithm
commonly confused
( equation
I
=
"
16.36
=
3:16 36
.
.
Progressions
↳
sequences with
formulas
Binomial Theorem
( atb)
rth term
"
(1 )
:
,
prob stat
From
Note that
Finding
sequences
r
an
" "
-
pkqn
(1)
,
q→a
p→b
,
,
(3×2+9)?
-
K
.
k→r -1
"
.
.
=
✗
.
.
4
Checking
.
.
.
.
217
=
r
'
(31×2)
-
Cr
-
=
=
=
"
containing
×
?
( coeff
1 ))
Focus
=
Sn
=
( n 1) d
1-
12 (
-
a. + an
( 3×47
-
on
Sn
exponents
(
"
n
=
coeff b)
t
previous given
:
-
=
'
[
a"
)
.
sum
of terms
#
fails
If all else
plugging
in
an
=
#
use
,
of
terms
E
in
✗
"'
y
an
Sn
=
rn
a.
a
=
,
-
(1
.
l
"
-
r
)
1- r
i
s
-
=
¥r
solve
of
for
Sn
the limit
as
n→o
Harmonic
if either
constant
a
orb is constant
reciprocals form
an
arithmetic
"
(3+1)+-07=16384
*
specific to
general
*
General to
specific
:
:
inductive
deductive
progression
reasoning
reasoning
SGI
average
the
Geometric
'""
(F) (3×2)%5
189×4,5
"
)
average
calculator ,
(67-1)
a
a,
Recall that
constants
6
=
→
of
an
sum
Ignore
of coefficients
For the
"
'
:
6th term
=
Arithmetic
.
Find the term
4
.
progressions
-1
:
consider
sum
yr
-
)
ALGEBRA
For two numbers :
A. M
G. M
.
i
:
it
=
is
Divisor
of terms
#
=
these
are
,
from
of
R
By inspection
1st
degree
R=AxtB
,
2) ( x
No
.
R
A- ✗
=
numerator
/
22
*
R
=
✗
R= Ax
R
B
Al 2)
-
+
:
+
B
/
,
=
A- (1)
=
✗
10
Fn
term
degree
R
,
=
Ax 't Bxtc
possible
the
,
roots
and
are
Ratio
antecedent
-
a
:b
>
consequent
means
a
:b
=
c
g-
:D
extremes
third
Proportional
,
c
E. ?
=
Fourth
§
Proportional
=
c-
d
,
d
digits
of
digits
of
✗ 7
=
'd
4
=
✗
log
=
✗
1
t
✗
✗
!
:
10g ( )
In
n
a
b
-
+
"
where
=
Lucas
and
so
on
.
q
±
is the
Ip
.
constant
:
1+5
,
2
5
At B
a
sequence
=
=
"
a
+
b
2Fn+ ,
"
-
Fn
47
7
7
sequence
+ B
Ln
leading coefficient
then
,
)
,
digits
=L
-4×1-14
the
is
p
1¥
(21417
=
7
↳
"
AXTB
numerator
B
Fibonacci
:
Ln
2nd
=
.
298.1
{ -2,1 }
=
!
is
¥
.
:|
Rational Roots Theorem
If
*
→
-2A 1- B
=
R=
If
1-
=
✗
solving
,
1)
-
of
,
No of
equations using
two
base
2340.571429
=
the number of
Getting
.
It 4×2-3×+8
Get
new
Use as
manageable
.g
7
Ris
,
47
:
is
y
2
-
Factor denominator
(✗ 1-
'
,
7
linear
+ ✗
¥
.
1
.
(210198
=
.
✗
remainder
of term left out ( i.e
exponent
2980.23
Pcc )
=
3+4×2 -3×+8
'
arbitrarily
7
983
until
Repeat
✗
.
.
7
,
not
is
Get
.
146
remainder
linear
I
(2101983 g Factor
10
z
=
(2)
polynomial
a
g
=
y
Remainder Theorem :
EX
2
on
4
Get
the remainder
.
7-
210
cyclic just get
remainder
1
is
factorable
to be
=
Focus
i
-
29830
.
7
-1
=
exponent
29830 23
of reciprocals
sum
F1
=
Divisor
Rewrite
Numbers
"
Getting
*
2
.
7
AM
Imaginary
i
Ex
GMZ
=
.
remainder
9833
Fb
=
µ µ
*
9¥
=
.
the
Getting
b.
=
1-
5
2
.
DISCRETE MATH
*
Proposition
declarative sentence
-
Tautology
contradiction
Contingency
:
Negation
&
T
F
truth value )
conjunction
:
Disjunction
:
✓
( true if
any is
or :
①
( true if
Exclusive
( true if both
^
Implication :
is true)
one
to
up
✓
T
T
F
T
T
T
F
F
F
F
F
F
T
T
T
T
T
F
T
T
T
F
p
q
→
is
subset of the universal set
a
0
*
The
mill set
*
set
operations
*
Power sets
is
q→p
flipped
of
think
shoes
Converse
the
set of
power
.
p
*
q
q
→
p
→
-
→
:
contrapositive
>
a
Tp
→
q
of
converse
*
set
ordered
are
relation
T
T
T
F
F
F
T
F
symmetric
titty
F
F
T
Transitive
Tx
subset
unique
A
:
c-
proper
are
same
FX
elements
1A / =/ BI
equal
elements must be the
:
A
elements
are
of
B
(A)
subsets)
Rb
}
be B
Each ordered
.
pair (a. b)
is
.
"
is
a
"
related to b
✗
,
yRx
→
Ry
n
y Rz
→
✗
Rz
n
(B)
.
If
AEB
,
n
(A)
In (B)
) ¢ R
,
R should have
( a. b)
any pair
pairs ( a. b)
(b a)
not in A
,
A relation is
reflexive
<
x
( a. a) pairs
no
without
then the whole relation
,
if all
element
I
,
.
is
its
pair
antisymmetric
.
Asymmetric
A CB
A
n
proper
↳ set itself
EA and
a
pairs
Ry
if there is
*
A CB ,
✗
(×
,
lb , a)
B contains at least
B
all
Antisymmetric
same
B
All elements of
subset
the
exactly
cardinali ties
/
if
lrreflexive
B
have
-1
Rx
✗
,
titty VZ
A and B
/
a
,
,
theory
=
set
Reflexive
q
T
A
"
Properties
Relation
q
:
/
{ ( a. b)
=
p
Equality
If
B
inverse
*
-
(2
2
↳
Bi conditional
p
✗
these
q
empty
and the
=
the set of all its subsets
is
,
Cartesian Product
A
p
PCA )
A,
"
I P (A) I
↳
.
p⑦q= poi +59
including itself
-
any set
:
converse :
Inverse :
of
subset
a
.
npvq
q
T
consider
l
-
q
q
→
Every set
"
true)
p
p
2
=
Subsets
Proper
of Possible
,
)
true
only
equivalent
→
F
No
*
*
( flips
~
or
-
false
P
of
mix
-
or
false
all
-
true
true
all
-
that is either
.
,
,
an
do
not have their
then the whole relation
equivalence
symmetric
,
and
is
relation if it is
transitive
.
corresponding
asymmetric
.
DISCRETE MATH
Consider :
A
1,2 3,4 }
{
=
,
We define
R
R,
{
=
,
,
,
,
13,31 and (4,4 )
ALL of
II. 1)
R,
then
R,
Consider
Rz
symmetric
R2 is
Consider
Rs
Looking
Rs
( 4,4)
and
,
( 1,2)
does not
( 1,21 , ( 2. 1) , ( 2. 2) }
,
13 1)
,
,
13,2) ( 4,4 ) }
,
,
,
is transitive
,
and
( 3 1)
,
,
.
Functions
*
f
A
:
domain
codomain
one
to
-
possible outputs
input
if
must have
=/
a
unique output
a
b , then f- (a) =/ f- (b)
Surjective
set of codomain
-
outputs
Injective
/
↳
Onto /
inputs
all
set of all actual
one
-
set of
-
-
every
-
set of all
-
range
B
→
set
=
of range
Bijective
both
-
*
one
to
-
-
and onto
one
Functions
Inverse
f-
must be one to
-
also be
Alternative
a
Consider
2
.
>
✗
✗
about it !
thinking
f- (x )
:
for f-
one
function
of
way
-
:
✗
2+1
+ 1
2
7
✗
2+1
Do reverse :
✗
:
*
-
F
I
✗
f- (x )
'
.
-
<
:
o
g)
1
<
I
if
Composition of
(f
-
(x)
a
=
Function
f- Cgcx))
were
( e. g. R
.
( 3. 2) (2. 1)
at
11,17 , 12,2 )
of
.
{ 12,1)
=
:
( 4. 4) }
,
all
irreflexive
since
12,31
{ ( 1. 1)
=
contains
( 3,3 )
,
would be
with
same
,
it
( 2,2)
,
symmetric
not
is
st
,
.
,
If
A
in
( 1,17 11,2) 12,2 ) (2. 3) ( 3,31
reflexive because
is
R
relation
some
✗
'
to
=
,
not
in
R,
,
{ 11,21 12,3 ) } )
,
have
12,1 )
.
TRIGONOMETRY
*
Triangles
A
point
. .
the intersection of
is
.
.
Part
..
b
a
perpendicular bisectors
circumcenter
Sls a) ( s b) (s c)
=
-
A
altitudes
Ortho center
in center
angle
to
the
get
bisectors
(base )h
0A
CPB
IAB : I
*
Inscribed
beautiful
am
h
-
=
-
-
-
base
↳
Triangle
set
If base
:c :
b.
or
depending
c
figure
If base
If base
b:
=
a :
=
i
circumcircle
b
a
a
b
a
b
+
h
h
h
<
•
a.
as
on
✓
-
-
Sls a) ( s b) (s c)
2
CM
inside
>
tzbh
Sls a) ( s b) (s c)
2
medians
=
height :
=
Centroid
(semi perimeter )
tzabsinct
:
C
-
-
a+bz
s=
7
C
(
of
Median
*
circum radius
*
↳
General
a
Triangle
'
b
a
triangle
circumscribed
↳
%
yz
za 't
z
:
m
m
outside
C
C
circumcenter
length of
2b
'
-
c
'
↳
median
being
side
bisected
c
→ in circle
•
-
incenter
Angle
*
a
General
triangle
>
1
0¥
a
inradius
*
of
Bisector
b
t
c
CAH
-
I
Abisa
TOA
-
being
bisected
-
t
Right triangles
SOH
is
( (¥12 )
ab
=
4C
-
Cabcab
02--92+62
*
Oblique triangles
*
Inscribed circumscribed
,
an
,
described
triangles
Law of sines
asin
s¥B
=
A
since
=
of Cosines
c
'
a 't
=
b
circle
abc
=
,
hinati
ang big
saapatna regions
412
R
•
Law
A
b
a
c
'
-
2abcosC
r
2A
=
o
atbtc
r
r
A
S
7
=
As
C
T
( at
A,
°
C
Ars
b
a
n
=
btc )
r
(
=
also holds true
2A,
for
quadrilaterals
at btc
2
rs
V
In
QI
In QI
In QII
In QII
All
,
are
positive
sine is
,
,
,
tangent
cosine is
positive
is
Ao=r( s a)
↳
touching
b
Arsa
-
.
a
r
'
side
.
positive
positive
If
equilateral
:
.
r
ants
=
.
6
R
If
is
expected
R
R=a¥
to be
larger
r
so
denominator should be smaller
:
right triangle
r=
ab
atbtc
R=
hypotenuse
2
r
r
.
TRIGONOMETRY
*
Triangle
Spherical
Great circle
small
-
circle
-
largest possible
any
other circle
propositions of spherical triangle
0° <
180°C
Law
a
=
sin ✗
Law
bt
✗ +
pt 8 < 540°
of sines
sin
sin
b
since
=
sin
sing
8
of cosines
cos
=
a
cosbcosctsinbsinc cos
✗
Area
'
A
IT
=
R E
180°
E
↳
=
✗
tpt 8
spherical
terrestrial
180°
excess in
degrees
sphere
15° per hour
places
-
more
( 360° per
easterly
are
24 hours
360°
at
)
ahead of time
C
circle
L
PLANE AND SOLID GEOMETRY
One Revolution
Unit
Degree
360
Radian ( SI )
21T
/ Gradian
Gon
180°
-
Explementany
b,
B
e
th
Polygon
angles
many
"
bz
A-
bh
=
=
A
absino
-2lb
=
,
tbz )h
,
d
d
°
a
a
,
di
A-
)
di __2(a- + by
,
"
"
+
ith
b
'
°
-
d.
da
'
#
Angles
①
A
I
di
1
a
Coteminal
7
c
Rhombus
Square
Rectangle
Area
360°
-
Angles
B
Trapezoid
trapezoid
90°
-
supplementary
A
Isosceles
Parallelogram
6400
complementary
r
Trapezium
400
Mil
vertical
Quadrilaterals
salient
Reentrant
)
angle
=L did
,
"
"
# ( a' to
=
angle
smaller
angle
b
Sino
s
.
-1
-
b2 d2 ) tana
-
a2tcZ > bZtd
.
"
°
d
Convex
D
concave
c
A"
No
,
of
-
4s
d
d
180° ( n 2)
=
dz
-
Exterior 4s
=
,
,
"
360°
a
a
d,
b
A
=
P
=
Ina
-
cot
b
(¥ )
-
-
Bramagupta 's
¥n°
Quadrilaterals
-
-
R >
r
circumscribing
Polygon circumscribing
A
=
tuner
inscribed
nR2
A-
=
2-
circle
sin
tuner
a
Ars
from
A,
,
Circles
R>
r
so
mnemonics
with Ris
longer
=
=
K
b
c.
AA
( 211¥ ) )
ngkasalananngtagapagtanggol
b.
a
:
2nRsin( ¥ )
'
A
)
naritoangkalahatingkasalanan
ngdalawangtagapagtanggol
P=
abcd
Figures
b
ngtangongtagapagtanggol
a
=
I
( %)
nrtdtan
2hr tan
in
rs
circle :
naritoang tango ngtagapagtanggol
P=
=
Theorem
circle
a
Recall
.
similar
a
Ptolemy 's
Formula
A
Note
actbd-d.dz
( s a) ( s b) (s c) (s d)
A=
na
÷, nanakotangtagapagtanggol
Polygon
:
Cyclic Quadrilaterals
Regular Polygons
interior
-
2
A
3)
-
-
atbtctdo.AZ#=BtzDb
S
Igfn
-
-
B
Diagonals
=
( s a) ( s b) ( s c) (s d) abcdcosÉ
=
,
↳
=
⑦
scale
2
=
K2
factor
PLANE AND SOLID GEOMETRY
Area of
*
A
a
=
segment
tzi (o since )
-
base
9
tr Sino
v
of
area
d
of
If
a
o
20
c
oblique :
of
~
✓
°
✓
=
L
rectangular
"
wedge
"
:
parallelepiped
cuboid
"
4
t2=a( atb)
clad )
( total ),= Outside ( Total )
cone
Pyramid
,
-
l
>
"
:
prism
n , th , th , th ,
he
Iab
a
Abases
Prisms
-
21T
LSA 1-
=
triangular
Bhave
h3
Ellipse
P=
: TSA
Special
h'
b
outside
,
differs
K
hy
b
=
of
Note
heioht
Pe
=
Truncated Prisms
a
a
A
LSA
the
p ,
perimeter
b
V. Ke
K
,
d
-
area
area
slant
→
t
c
alatb )
base
slant
/
y
✓
d
ab=cd
-
formula
same
perimeter
Circle Theories
°
}
Bh
triangle
sector
*
:
hSA.ph
d
1.
area
of
area
-
tzro
=
*
cylinders
prisms
-
h
tbh
2
V
'
:} Bh
LSA
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Indeterminate Forms
Newton
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4b
weight
a
t
weight
n
-
1-
d
>
b
semi circle
hitting
work done in
h
3
length
stretched
,
b
n
2- bh
h
un
*
should be relative to
integration
limits of
J
f- (c)
-
b
l
a
>
1
=
fflxldx
b- a
b
±
Rolle 's theorem
^
Hemisphere
3
0
{ (4-31142)
R
g-
<
!
if :
,
^
Cone
1
0
↳ Bh
4
h
>
B
parabola
The
3
is
drawn
as
2
5
2
8
2
g- b
g-
arc
.
measures
the resistance of
Remember
=/
I×ory
strip
Use
r
only
DA
dxory
an
appropriate
,
-
axis
rotational axis
11 to
Im
=
from
→
L,
m
:
Pa
↳
.
.
I
Z
=
mr
A
'
pa
assume
=
if not
1
given
Pressure
Hydrostatic
↳ pressure exerted
on
a
submerged body by
p=pgh
/pure
use
d distance of
centroid from axis
2
'
rotate
gyration
of
Radius
object to
an
=
Pseawater
↳
F
use
=
strip
not
fpgh DA
11 to
fluid at rest
centroid to
kg / m3
1030
a
8h
kg / m3
1000
=
=
surface
level
cont
2.
f-
is
differentiable
-
.
f- (a)
=
.
on
surface
,
height of object
on
( a. b)
f- (b)
Then there exists
3
Moment of Inertia
↳
is
3
3- bh
h
f
To
<
[ a. b ]
1.
c
in
( a. b) st
.
f' (c)
=
0
Rope
DIFFERENTIAL EQUATIONS
d¥£
(
3
-
Family of
of
D. E.
( d¥)
7-
t
.
2nd order
degree
1st
"
3rd
i
zyy
>
zyy
>
u
=
Zxyy
,
Try reducing
iii.
Get
Variable
Linear DE
y
✗
o=
constants
ZYY
✗
Zxyy
of family
D. E.
'
of
→
y
=
constants
equation
in
y
curves
'
-5=0
-
,
Solve
dx
.
y
't Pcx
12
+
y
'
ftcxdxtfglyldy
dxtgly )dy=O
,iy)=IF(×,y)
N(x,y)dy=o
)y=QGDy
yes
;
"
; y
'
-
games
=
-
"
" Pd×=
i duplicates only
y×
( tn )
1m,
once
)d×
""
"
/ one
+
-
"
+
g) dy
4y
-
( for eqns
Isolate
(2y ¥ ¥
( ✗ + Ddx
=
-
C
+
=
-
=
where
×
✗
¥
for
values
9 DI
✗
+
+
c)
2
not isolated
,
isolate C instead )
implicit
interchange
(y
'
+
solve
✗
/
-1.1
andy ( x
-
choices
✗
%
=
=
¥
=
2.
.
¥
2)
Write
=
×
✗
=
=
2.2
1. I
Store to C.
→
make
sure
interchange !
Ordering in calcu
will be
flipped
✗
-
(1-4×2-2)
↳
'
✗
y=ux
=c
Caltech
use
for
implicit
y
0
+
-
3¥
2g
'
-
+
=
✗
-
I
3¥
=
C
C
.
-
✗
f( Ix Zxy ) dy
¥y ya
t
xy
C
=
integrating
Ndy
,,
)=°¥y -3¥
factor
:
0
=
plx
to
but
uz
fNdy=C
y
once
by
or
3¥
°÷
-
N
M
whichever is
choose
easier to
integrate
espcxldx
C
?
=
dY_d× /
1
( for eqns
=
y
where
and
y
=
0
exact !
→
1. I
2.2
for
,
Ne
+
Linear DE
✗
g-
Assume values
fpcx )dx
pyefpcxldx
1
given
×
-
Inc
+
:
DE
Mdx
Check :
Substitute
¥y
+
B
=
t
'
¥ ¥y
store to
CALC
✗
=
)d×
'
-
duplicates
Multiply
/
y y
+ ✗
If not exact
2.2
ON
f Mdx
t
store to
A
=
'"
-
,
=
day
x
y
for
y
.
CALC
✗ =×
,
=
-
-2g
✗
=
for
f (x2
y
2g -4×7
It
for
using
dy
,
¥É
for y
C-
=
C-
=
In (1-42)
N
}yM_
is
y
'
solve
-
l
(1-2×2 2xy)dy
+
M
0
( ✗ tuft 2×-49 )
Go back ,
=
DE
C
solve
Solve
)
uz
C
-
iii.
:
,
2uxd.li
Zuxdu
=
lnx
C
-
ZUXDU
+
✗
'
)dx
1×2 + ✗y
y
Assign appropriate
v.
dx
=
×
ii.
iv.
(
In
=
c
""
2u2d×
IF
fMd×tfNdy=C
21
¥1
'=d¥=u+×%×
/ux ) (udxtxdu)
Zuidx
-
y
degree 2
2u2xTd×t 2u¥dw
=
=
uh
-
2x
=
OR
w/
.
Rudy
:
✗
;
e
dx
-
""
"
"
-0
-
Xy
→
1- I
+ 2x
Caltech :
i.
idx
+
)dx=2×'y"dy )
d×
homogeneous
→
separation
idx
+
=
; y
'
y
G-
udxtxdu
.
Ñd×tu2xfd×
.
constants
arbitrary
dy
;
variable
using
y
+
( Zxydy )
72
=
y=u×
+
equal ( *
is
'
217×1 Ay )dy
=
( it u2x2)dx
zx
Separable DE
x2
=
iii.
iii.
Exact
¥,=×¥y
ii.
substrate
2x
y4dx= Zxydy ]
+
ii.
Y
'
Examples :
i.
YZ
2
→
Order DE
-
Variable
[
dx
Plx)y= QQ )
+
Bernoulli 's DE
Caltech
-
.
+
✗
'
72
,
'
'
2×y
:
( it y )d×
(I
Mlxiyldx
'
arbitrary
flx>
F( xx
DE
DE
Alternatively
✗
Zxydy
=
Equation
Separable DE
Homogeneous
Higher
constants
arbitrary
↳
derivative and substitute
-
Exact
arbitrary
of
#
Differential
solution to
4a=É
'
)dx
homogeneous
if
of exponents per term
sum
(4×121-44) )d×
¥
'
→
'
ya
=
>
Isolate
City
.
n
i. Test
y'=4ax
<
ii.
Ex
'
degree
'
DE
Homogeneous
72
>
i.
→
2nd order
zsinx
=
"
highest n in an
highest exponent of
→
curves
^
Ex :
d
2=0
highest
>
(dd¥ )
day /
2 1-
,
y
is
and C
✗
isolated )
explicit
(✗
=
1.1
,
C
=
any
value )
't
and
'
to
given
.
efPl×)d×
=
y
y
y
QCX)
'
.
'
,
=
Integrating factor :
4×2
+
✗
2g
'
y
PCx7y
espcxldx
'
yx
=
'
g-
¥y
+
✗
=
4x
f¥d×
,
e
21m
,
e
f4×.×2d×
¥,
+
,
eµ×2=
a
✗
yx 2=41×3
dx
yx
'
:
✗
4
+
c
DIFFERENTIAL EQUATIONS
Bernoulli 's DE
Ex
d¥ 2¥
+
PCH
yl-ne.SU
n
-
Xy
=
=
f- ¥
ty
✗
=
Case 3
+ C)
=
Csx
'
y
t
D-
t
8y +16g
8D
(c.
'"
y -5g
Ily
+
'
Non
-
{
,
"
t
e
C.
yp
yp
m
mut
,
yp
2
.
"
✗
e
yccx)
( MODE
&
5-
For RCX )
4)
)
Calculate
iii.
,
+
Ex :
.
.
.
+
Co
2+2×-1
3- 4
AX
t
BX
>
+
2é×
Ae
coslbx)
or
2e×ws2x
-
.
t
Z
+
D
✗
Acos3×tBsin3x
he"cosbx
+
3e×sin2×
Ae×cos2×
+
+
Ccos2✗tDsin2x
Be"sinbx
Bessin 2x
If math
as
v.
Acosbxt Bsinbx
Ceaisinlbx)
+
.
"
µe3×
40053×-5sin 2x
.
Bx 't Cx
g- e
Csinlbx)
t
Ax 't Bxtc
Ae
or
iv.
" "
"
Ax
"
Coos (bx)
1- X
-
l
,,
ws2×
=
To
sin 2x
denominator
m2 -4m
→
Ae
=
→
sinbxt Be
0 + Ii
:
i
✗ 2-
→
"
4X
cosbx
numerator
→
with cosine here
is
At Bi
:
.
:
✗=
CALC
✗
atbi
→
=ot2i
①
"
from
→
-
e
.
b from coslbx)
or
sincbx)
to toi
-
-
lysin2x
-
Locos 20
for
imaginary
yptx )
t
-2×2
=
Real for sin ,
.
✗
Ex :
Ex :
cos 2x
-
:
ce
Ce
Yplt)
.
-40
=
L
""
"
-12-0
=
B.
,
Cn ,X
✗
Ex :
:
3
-
solution
""
"
l
8A -413=0
Yao
-
"
¥×
t
solutions
-
At Bi
RCX )
Cnx
X
3-
Undetermined Coefficients
particular
=
.
=
'
Note
2x
C
"
4138in 2x
-
cosy ×
3
↳ sin
I
=
2.Boos 2x
auxiliary equation
particular
solution
Method of
B
-
:
yplx )
complementary
For
-
-
.
I
-2¢
=
2A C =-3
.
t
auxiliary eq
+
-13=-1
Bsin 2x
A
General solution :
=
etc
,
t
)
ii.
+
x
-4A -813=1
homogeneous (121×7--10)
ylx )
"
y
,
-4A cos 2x
=
"
cos 2x
2
=
-4A cos 2×-413sin 2×-18 Asin 2×-8 Boos 2x
0
Ca
A
-
2x
y -4g
I
-
2×2
=
-2A sin 2x
=
bi
case
3
.
'
Aws 2x
"
}
(
cos
=
"
"✗
=
2×2
LHS ± RHS
'
=
Caltech
D -5132+1113-15=0
D=
4g
-
i. Find
It 2i
c)
-
Yp
}
3
,
-
→
=
choices
Check if
"
and ma
,
X
:
y
.
}
15g
-
(
+
2A
0
=
) e-
czx
+
"
-4
,
Bx
'
bit Czsinbx
cos
,
m
,
16=0
+
D= { -4
) em
a ±
,
(C
e
distinct
Ex
terms
y
"
repeated
are
complex
:
hand side w/o
Gem
+
:(at
y
"
"
right
at
expression
→
-
,
real and
are
roots
)
o
=
'
ii. Substitute
y y
y
-
A
( A×2+B× + c)
-
2×2
=
y
B. ✗ 1- C
Differentiate
i.
-
2A
=
Ax
survival
c)
+
cosx
Rcx)
"
y
-
iii.
C. em
=
y
=
2A
✗
+ C
xaccosx
(
case 2 :
*
"
yp
y
217×+13
=
-2
"
of
A×2+
=
'
n)Pd×
I
roots
y
=
=
Order DE
-
case 1 :
y
=
(
'
=
-
eY×?
21h ✗
cosx
Homogeneous
*
n
fxfsinx.IT
-
=
Y
Higher
e-
=
.
I
ypcx )
Find
.
I
Jinx
nyfo.eu
( 1.
=
-1×-2
y
2
-
yp
✗
=
=
-
yp
) Pdx
eS¥d×
n
sinx
QQ)
¥
=
-
error
usual
Multiply
taking
,
derive
auxiliary equation leg
.
cos
2m
-
6)
.
answer
derivative
by
(
"
×
or
#
where
of
n
is
#
of
times of math
times
errors
)
.
Proceed
"
DIFFERENTIAL EQUATIONS
:
Ex
complementary
the
Getting
y
"
-
dy
'
l2y=
-
C. e.
2-
:
If
Math
4×-12
not
=
f-
=
↳
ylx)
1
=
6
1
surroundings
→
✗
=
Ts
object
Tct )= Tst
↳
( To
Caltech
Ig
-
Ts )
✗
=
t
( ex )
3-5
=T
y
-
Ts
1- (f)
( To
Tst
=
↳ still
:
e-
Ts )
-
y Ts
or
:
(e.g
Ts
add
have to
"
→
yp
e6×
e-
'
Yo
C, e
=
t
Czé
"
a-
=
Ri
=
"
and
yp
.
input
concentration
input
volume
c.
+
f-
"
=
dy
+
✗
"
Q
=
①
=
%¥
mt
orthogonal trajectories
Isolate constant
¥
of the
family
of
curves
I
family
.
lines
C
=
Differentiate
y=c×
straight
through origin
of
Limited
ddtf
xy#
2
P)
p
KIM
-
check
'
y
-
M P
d¥¥
Get
↳
DE of
for
answer
family
of
d¥
=
ydy
( }
y
=
✗
)
-
integrate
d¥
,
if
2
Zty
-
( lbs )
co
Ro
R
Q
.
Vo
1-
-
Qo to Qf
-
memorize !
→
( Ri Ro)t
.
Kt
and
max
value
present
=
after
Use this
kdf
+
<
c) 2
value
+ C
C
=
0
,
→
circles centered
at
origin
curves
for K
solving
kP( M
-
P)
M
p
7
learning
for
Growth
rate is
v
Z
substance
a
a
✗ DX
✗
of
:
=
It
%
amount
difference between
y
-
-
=
and
±
=
2
Logistic
curves
negative reciprocal
y
(t ) y
b
-
↳
-
ce
-
between
equality
to dP_
0
=
M
:
to
proportional
choices
iii.
t
survival :
0
=
✗
y
"
Ts
Roco
-
limits
over
Growth
=
Rici
=
Exponential
rate is
.
dQ
Integrate
Rici
=
It
✗
:
D
e6×
do
ii.
T
.
( 9911min )
rate
flow
m
-
dx
i.
)
(t)
(1%91)
,
Find the
( choose
T
Problems
Mixing
because
✗
conflict with yo
¥
.
-
"t
orthogonal trajectories
Ex
objects
6
Note :
by getting
C,
e-
colder
1-
"t
MODE
:
:
=
hotter
objects
-
"
multiply by
must
of
solution
=
→
2×-4 /
e
( ex )
3-5
"
Axe
Proceed
✗
Caltech
using
yp
!
61-0 i
=
×
yp
.
error
Be"cosbX
+
It Oi
=
bi
at
✗
d¥
sinbx
At Bi
→
MODE
:
ft Ts )
✗
↳
↳ Ae
I
Caltech
T
e
=
↳
of Cooling
p
"
12
-
Poekt
Plt)=
:
"
( using Caltech )
4m
-
Czé
+
:
particular
m2
population problems /
decay problems
Newton 's Law
•✗
=
y
a
For
6 , -2
m=
Yc
d¥
e
4m -12=0
-
Growth
Natural
( homogeneous )
complementary :
m2
solution
"
proportional
to
present
( e-
much easier to
Mkt
value
dp
d-,
of
P
and
use
integrate
and allowable additional
of P
Survival :
check
equality
choices
between
fop(dP_ ) I f
M P
-
↳
after
:
or
conveniently :
"dP_
PIM P )
±
more
'h°"
b
kdt
-
desired
kfdt
a
solving
g Pa
for K
a
1
ADVANCED MATHEMATICS
( i=j)
Numbers
complex
i
IT
=
it
=
is
i
it
l
-
trigonometric
i
-
}
I
=
Cyclic only
,
i
,
i
'
,
and i
¥8
i "=
Rectangular
polar :
2-
:
pairs
:
(× y )
=
2-
;
z=rLO-
=
yi
✗ +
=
12-1
;
,
✗
'
+
Ex :
Make denominator
=
1
o-
tan
=
'
a
(¥ )
•
2-
:
rcjsct
=
ejoy
#
cost
+
e
=
cosho
et
=
( cos-0-tjs.int)
sina.ee#
r
=
te
sinha
et
=
Exponential
Operations
* ( rho )
Ex
.
re
=
t "Ln0-
=
↳
✗
i=
e
12
i
/
Sum
2
(
mxn
:
Difference
rows
Cols )
✗
must be same order
fax b)
Multiplication :
Inm
Division
:
>
¥
:
AB
=
( bxc )
✗
(
=
a
c)
✗
"
( press
Re
In
=
:
I
-1
×
( red )
number to
+
In 2-
-
"
=
use
In C)
↳
only
✗
i
store to B
L,
5
6
8
9
start with
just
t
=
?
?
?
?
?
+
t
-
,
A-
A
!
=
I
,
+
of
rows
AT
A
:
to
=
-
AT
for matrices with
complex numbers
*
complex conjugate
µ
columns
and
Matrix
symmetric
transpose
Adjugate
↳
of A
-
i
2+3 i
]
For
hermitian matrix
:
A*
A=
-
=
In
3t4i
/
det
A*
i×arg(3t4i )
✗
A
(
×
"
+
in
and
4)
Determinant
1.418 i
-
1.418 i
If
is
e
corresponding
or
rows
"
unchanged (
columns
" ↳
swapped
are
↳ ↳
around ,
determinant
c'
=
'
9.902
=
z
:
e
1.418C
@
19970 @
-
" 418 "
( z= rest
=
rLo
Ct
"
(A)
SHIFT
)
→
to
caku
+
adjoint
:
operations
( 3t4i )
( /
matrix
A- → C
square matrices
A=A*
:
only
of cofactor
adjunct
or
[i
for real numbers
*
Ai a.k.a classical
2-3 "
=
A-
,
transpose
of
A→AT→A*
i
Hermitian matrix
/n
?
then alternate
-
Matrix
skew
t
M
→
t
-
symmetric
A*
Zi )
?
convention :
,
columns
radians ! )
"
9.902
3
to
rows
Adjoint
.
"
?
6
ga
Transpose
skew
( in
2
minor
t
Ki )
-
3
2
"
9. 902 t
2-
signed
-
1h11AM targ (A)
A. Get
column
determinant
I
root
→
jth
and
row
cofactor
radians
jo
i×ar9( A)
( 5- 2i )
=
number
1.094 i
-
(5
=
+
the
=
IBI tixarg (B)
:(3t4i )
:
complex
from
stone to A
→
/ At
(31-41)
lnz
Inr
:
In
0.227
z
0
↳
e.
formed by removing ith
is
5
✗
principal
Tsi
-
'°
1-21-4 i )→
Evaluate
getting
and
t
→
Mij
the minor
,n -1
log (4+51)
(5-121)
In
.
3
In
=
gets
,
1- 360
-
;
In
=
.
A
2
SHIFT
arg ( 8)
109.2+4 ( 5
Evaluate
.
-2
complex
store
,
-8
2- =3
Bi
It
1<=0 :
=
10.94) sinh (0.43)
j 0.57
-
of matrix :
1<=0 , 1,2
1<=2 :
.
°
360° )
0-+1
of
(
<
=
1<=1
Ex
0.56
Order
( :( )
Irl "n<
=
1- 81
(z )
jus
+
mode
v20
to
1<=0,1
,
>
* In
10.94 ) sinh (0.43)
" d" "
Minor
Find the roots
At
join
Re
not allowed in caku !
→
change
"
nrl -0
*
2-
:
10.94/ cosh 10.43)
jsinlx) sinh ( y )
-
in
2
#
-
}
x
Matrix
coso-tjsino-e-jo-coso-jsi.no
=
10.94 )cosh( 0.437
cos
Note :
e-
-
2
et
( 0.94 tj 0.437
sin
>
v
2J
°
-
e
g- 0.43 )
sin
=
-
+
=
10
<
cos
10.94 tj 0.43)
3+4 "
r
Trigonometric
G)
:
) sinh ( y )
jcos(
t
cosh ( )
y
cos
Im
-
sink ) cosh ( y )
=
( 0.94
cot
d
Laws of
Exponents
'
y
i2
-1=-1
=
.
✓
( ✗ + jy )
coscxtjy)
:
negative exponents
For
Notations
Ordered
sin
allowed
}
in calcu
)
.
If all elements
If two
rows
or
in
a
row or
columns
are
column
identical
are
zero ,
then det
/ scalar multiples det
,
is
zero
is zero
(e.g.
(e. g.
{8 )
12
24
)
ADVANCED MATHEMATICS
Laplace Transform
Rank
↳
largest
Ex
square
=
1A
/
•
=/ f- (f) e- stdt
[ flt ) ]
0
9
7
8
I
f- ( t)
I
2
3
1
function
3
6
7-
g
=/
-18
det is
A
of
Nullity
=
the determinant is
A
.
.
column
a
6
9
7
3
I
2
2
4
K
multiplied by
is
coskt
,
B
=
[A
It
-
AV
A
Eigenvector
→
v→
]u=T
"
✗
→
✗2
:
-
3
6171
9
7
317 )
I
2
2171
4
sz
eat
=
XV
Ex
multiplied
is
1
Eigenvector
Eigenvalue
? Y
to
f- (t )
replaces
,
and
t
7=6
s
Getting
0
1-6
(
the determinant for 4×4
-
0
A
,
AV
÷
] /
=
(s a) 2
-
2
1
3
O
l
-2
4
6
2
0
10
"t'
.
K
eat
choices 't
-
]
Properties
Some
Lletatflt) ]
L
L
It"fCt) ]
/ jiff
L
Caltech
Fls _+a )
→
( 1)
→
nd
/
→
]
( nth derivative )
Fcs)
dsn
Fcs)ds
s
)dt
"
-
as
[ ttflt) ]
=
as c :
!
(s a)2+1<2
3
2
I
-
n
XV
-
-
store whole
choices
a.
-
B ( column)
-
y
[? Y ][
-
a
( s a)
bigger ) matrices
or
A
:
consider
Alternatively
O
?
s
1-
eat
.
.
=
with
1-
eat
-
sinkt
l
→
s
KZ
-
t.e.at
-
choices
→
.
transformation matrix
compare :
15
$
→
s
cosh kt
-
eigenvector given
3- 6
cost
K2
-
n
Find the
.
sink → Titanic
K
sinhkt
Caltech :
Ex
→
s
sa
0
=
→
.
If
IX
-
sinkt
SZ + KZ
/ A/
7
=
K
A
remember the numerator :
to
K2
+
coskt
Eigenvalue
A
k
-
row
or
multiplied by
3
1131
'
gn.tl
s
in
!
n
rank (A)
-
columns
=
sz
sinkt
of
#
element of
If each
÷
±
th
0
rank is 2
:
FCS)
t
→
thing
I
kernel
6
O
=
Take
Fls) =L
zero
3
.
A
is not
determinant
where
matrix
} valuable
§ Fcs)
→
→
<
integral
:
N
C
Crow)
-
pivot element
-
Identify sign
Calculate
iv.
IAI
:(
of
element
i.
matrices :
highlighted
ii. store
iii.
pivot
non zero
a
Recall
pivot
↳
Choose
i.
O
t
pivot
-
+
-
ii.
-
t
t
-
+
-
t
-
8
/
A
BC
-
I
don't
-
f
-
t
-
s
( e.g
.
s
=3
)
flt)
e
st
dt
store
→
to A
0
Check :
)
choices
/
s=
?
ouosens
1
=
A
forget
by
pivot element !
of
value
calculate :
-
iii.
det
appropriate
Assume
t
:
I Pivot )
element
=/ f- (f) e- stdt
f- (s )
:
division
of
Inverse
a
Matrix
iv.
Some
modifications
:
t
A-
'
=
A*
.
IAI
.
A*
'
=
A- pal
↳ useful
using
in
→
same
formula
getting adjugate
Mata
"
✗
in
adjugate !
in calcu
det (A)
•
f. f- (E) de
Given
is
Given
includes
Given
includes
t=a
to
→
multiply f-
ult a)
-
8ft a)
-
f- (f) e-
St
,
.
or
no
Ua
factor to
,
lower
need to
integrand
limit becomes
integrate simply
,
a
substitute
Jack
ADVANCED MATHEMATICS
IT
Fourier series
ffx)
few)
9oz
=
?(
+
an 005
n
basin
+
L
L
-
period
,
z
,
+
f
DC level
-
IT
cos
ao
f f
an
bn
(
(
f- G)
sin
/ dx
"
-
2
bn
If
=
(
f- G) sin
0
Half
an
f- [ flx)
=
=
✗
2
0 <
,
21T
foflx)
=
21T
[
=
-
¥1
=
Zoos
✗
1
=
f(× )
bn
t
Period 't
sin
from
0 to
of the periodic
n=2
( 211¥ )
Given
.
L
Z
waveform
-
:# f.
bz
-29T
:
✗
sin
f.
I
26.32
=
'
↳
( n)
✗
.
assumes
that
Non
-
at
for
-
range
an
an
or
or
bn
bn
look only
,
.
Ex
Solve
setting
n
-
{
In )z
non
periodic function
-
a
is
1. 2. 5,7
0
,
¥ ¥ ¥
+
periodic function
✗ (7)
.
=
✗ (2)
.
E- ¥
+
=
with
z
attains
=
21T
/
-
1
f- (f) e-
Jwtdt
value
plane except
2=0
plane except
2=0
-
£-2
't
ROC : entire 2-
→
✗ ( Z)
81h1
1
EX
✗
.
n
-
-
-
-
az
-
i
12-1
> a
i
lzl
<
1
1)
Caltech
function
, .az
-
a
( 0.5) ucnl
=
:
i.
and
added restrictions when
still similar but uses
✗(z)
Recall
✗
=
(n )
has
"
Z
choosing
value
a
of
n=o
211-8 (
Slt)
cos
f- [
cos
13T ) ]
-
Else
ñ8( w a)
-
jitscw
-
-
etat
.
ul
2*8 ( w a)
( at )
+
a)
+
a
IT <
t< It
with
f
-
=
"
tdt
value
jw
0.5
for
Solve :
IT
f- (w)
:
f
w
.
cos
Cst )
5-
value
to
decomposition
dt
d
=
re
=
-
n
.
of
-
3)
→
-
n
-
3=0
→
z:
2- >
f) to ( t )
→
z=l
f) to f)
→
OL
( Assume
1
→
0.5^155
2-
=
5)
(2--0.3)
2- < I
"
→
store
to A
Compare :
choices
z
chosen 2-
=
?
=
1
A
to
cosotjsino
Quick recap
:
Fcs)
Laplace :
f
=
-
2-
(f)
to 0
Oleg ul
to
0
/ jw
"*
IT
but
-25
0.5
=
↳ avoids
iii.
laplace
jw
0
as
:
25
v.
a
appropriate
Assume
w
f- A) e-
solve
iv.
as
set limits
(t)
is
n
to 25
set limits
appropriate
_+jw
,
Flw)=
:
jñ8( wta)
similar to
Recall
If
:
)
I
go
ii.
a)
n
-
equate argument
:
Set
iii.
+8 ( wt
Caltech :
i.
uln )
If
ii.
-
sin ( at )
Ex
w)
1
ejat
→
"
(n)
•
f- ( )
w
1
z
Roc
1
,
and
✗
kernel
f- ( t)
finite
a
ROC : entire 2-
→
( )
ul
an
-
N
1
Ftw)
Is
+
+
which
z
-7
Unitary
! f- (f) e- Jwtdt
✗ (z )
plane representation
}
I
,
anulnl
unitary
-
"
✗ n
a
complex
→
a
It
:
(n )
✗
-
then compare with choices
N
=
✗
↳ set of values for
,
a
Region of convergence
infinite period
an
=
=
Transform
Fourier
signal
for discrete-time
.
✗( z )
,,
dx
✗
limit becomes
lower
ya
transform
/ 2x )d×
For half
21T
1
or
-
21T
I fflxldx
1-
go
ult a)
includes
n=
bz
=
±
0.5
=
a
dx
store to C
→
modifications :
some
L
✗( z )
'
JB
A +
=
or
anas
Limits
w
) dx
"
=
Ftw)
%
:
n =/
,,
a.
↳
dx
f- (w )
C
ntx
EX
,
92
formula
to
IT
=
21T
a,
(
cos
B
choices
< 21T
✗
d
A
:
Compare
only appears
O
2
-1T
I
.
in the
once
Determine the Fourier coefficients at
flx)
an
↳
The 2
I
( NY ) dx
cos
0
Example:
wsfo.s-tldttjfcoststlsinfo.SI) at
(A)
cos
If
.
L to L
2
)
"
cosine series
range
-
-
iv.
=
2
h
use
,
dt
IT
calculate :
period
sine series
range
limits
no
)d×
"
↳
Half
limits
specified
Use
f- Chaos
If
=
cos
IT
f- G) DX
If
=
(
=
-
coefficients
=
rlwsotjsino)
fast ) tjsinc-0.5T) )
re
=
( 3T )
µ
f
Fourier
dt
-
sines
cosines
Cst )
cos
it
L is the
hat :¥
t
t
t
)
" "✗
"" ✗
"*
f
:
rlcosotjsino)
Fourier
2-
:
:
f- ( w )
✗ (z )
=
=
f- ( t )
e- stat
→
ff (f) e-Jwtdt
✗
(n) z
→
0 to
-8 to
8
8
"
→
0
to 25
or
-25 to
0
n
=-3
)
z
•
PROBABILITY AND STATISTICS
*
of
Rule
sum :
Rule of
*
if
after the other
one
*
son !
" "
"
this
ways
(n
arranging
start 4 end
n
in
people
row
a
,
*
are
the
same chair
EIX ]
*
!
Odds
Odds
!
p! q
r
(0.05×0.2571-10.04) (0.357+(0.4/10.02)
defective
=
working
!
For
#
Expectation
E.
=
Pnxn
I
E [✗
=
P
:
=
"
2
]
E[ ✗ ]
-
(A)
'
MA )
against
Remember
p( Ac ,
:
p( A)
probabilities involving
the bell
curve :
✗
=
e
with Partitions
Permutation
µ
n.
hi
*
na
!
ns !
(A)
=
partitions
successful outcomes
total
P( A)
1
=
A
possible
outcomes
'
PCA )
-
B
Mutually
A
Exclusive
P( AUB )
Non
"
-
B
mutually
PCAU B)
=P (A) + PCB)
↳ union
,
=
Exclusive
PCA )tP( B)
-
intersection
Independent Events
↳
event has
one
Plan B)
*
Binomial
P
=
no
effect
Poisson 's
P
=
the other
Distribution
(1) pkqn
-
↳ 1
*
on
P (A) PCB )
=
k
-
p
Distribution
zk
-
t
e
k!
Alam Ko , ba 't
PCAAB)
t
"
or
"
*
Pt )
Correlation Coefficient
Probability
P
21T
Use STAT :
of
size
→
!
0
!
n
=
40.58%
defective
&(
foft
p.q
identical elements
1- r
fcx ) .
*
(" ° " )( " 35 )
=
working
-
→
.
m
Elements
Identical
n
100
P( defective / B)
defective
2%
Mathematical
var [ ✗ ]
=
"%
c
n
Permutation of
N
as
1) !
-
B
40%
same
but the
,
?
-
the
as
working
s%
( n 1) !
=
n
tree :
following
A
%
g. round table
of
think
go
consider the
of ways
and
up
Binomial
PIA)
Permutation
#
=
approximation of
an
( BA A)
P
=
arrangement
,
e.
is
Probability
conditional
grouping arrangement regardless of order
Cyclic
✗
where
P( BI A)
Combination
↳
*
ways
mxn
Poisson 's distribution
distribution
Permutation
↳
*
the
ways if either
mtn
Product :
↳ ordered
*
of counting
Fundamental Principle
nangengeelam
ka ! ?
and
"
,
QC )
and RC )
.
.
,
,
P( A)
+
'
PCA )
=/
STATISTICS
Statistics
- methods of collecting, processing, analyzing and summarizing data.
Descriptive Stats - describe what is there in our data.
Inferential Stats
- make inferences from our data to more general
conditions
- data taken from a sample is used to estimate a
population parameter
Data
- it is a collection of facts from experiments, observations, sample
surveys and censuses, administrative reporting systems.
Universe
- collection or set of units or entities from whom we got the data.
Variable
- it is characteristic that is observable or measurable in every unit of
the universe.
Normal Distribution
- a continuous probability distribution that is symmetrical around its
mean with most values near the central peak
Population
- set of all possible values of a variable.
Inferential Stats
Sample
- a subgroup of a universe or of a population is a sample.
Slovin’s Formula (How many samples do we need?)
Point Estimation - point estimator is a statistic that provides an estimate of a
population parameter. The value of the statistic from a
sample is called a point estimate.
Confidence Interval
MEASURE OF CENTRAL TENDENCY
- a range of values so constructed that there is a
specified probability of including the true value of a
parameter within in.
- finding the center of the data
Mean
Median
Mode
MEASURE OF POSITION
- finding the rank
Percentiles
Quartiles
Deciles
MEASURE OF VARIATION - determining the dispersion of the data
Range
Inter-Quartile Range
Variance
Standard Deviation
Coefficient of Variation
How to set the confidence interval?
- Sample Size, Variability of Population and Width of Interval
Standard Error
STATISTICS
Sample Size
Variability of Population
Width of Interval
HYPHOTESIS
- a statement of expectation or prediction that will be tested by research
REFRESHER
iii.
It
0-1
6m
0-1
h
IF
Ol
1
>
The
become
0s
because the
M
N
'
You
.
can
:
You
are
a
S :
You
are
a
major
freshman
>
tano
d-
=
=
zpg
253h thx
d
+
6th
comsci
Nuns
campus
tano
.
bD= grztd
.
d
l
253
.
6th
h
internet from
access
sun
equal
is far
away
=
2¥
=
Tano
2%
=
.
0
M
6 ✗ thx
60°
=
8
1300
a
240m
Police
Horror
§
35
60°
)
Floor :
h
,
f
2
2402
+
2402
D=
✗
35
✗
d
=
2
2
+ ✗
25
20
n
tan 60°
15
15
10
tan
g
=
300=1
d
h
=
2402
I
80
2s
go
75
✗
tan 60°
=
h
146.97
=
1- ✗
2
in
20
60
10
Romance
150
If
exponent
is
1:41
4
B
(seca
+
tan
tana
A)
Recall
=p
seca
seen
.
-
-
tana
tana
Alternatively
:
seix
-
tan
>
✗
-
l
-
1
:
.
.
.
.
.
=p
with 1
.
.
.
.
.
digit
tens
digit
-
tana
sec
A
see
At
tana
2tanA=
-
tan A
=
f
:
units
digit
.
is
units
digit of
digit
units
Numbers
w/
3,9
Apply
laws
(33 )
=
1.1 ,
=
9
36
=
61
"
✗
✗
-
-
with
,
-
0.7601
5
is 1
tp p
£ ( p tp )
>
I
01
4
=p
557
61
exponent
and
=
2789
21
1
-
41
g-
as
unit
tana
see A-
.
,
For #s
=P
.
.
5 :
At
.
81
. .
.
3:
sec
.
=
2 :
A
6:
,
and 7
digit
of exponents
"
=
( 1185921 )
↳
0.7601
i.
last
as
41
"
2×2
=
4
product
of
base tens digit
REFRESHER
Special
#
even
s
:
choose 2 #s
integer
any
76
>
last 2
digits
is
always
76
>
last 2
digits
is
always
24
(
odd #
24
"" #
24
56283
>
digits
is
16,96
,
76
,
.
,
.
}
283=56
s
2,4
,
28
,
if
62586
:
:(565/56.563
.
(
=
.
.
-76)( 56 )
=
.
=
43C 2) ( 4C 2) ( 4C3 )
G1
16
555kt
) ( 210158.26
=
.
.
84
P( D)
%
=
P( KID )
:
p( RAD )=
(F) (0.521%0.48)
=
Red
:
King
=
215
'
0.183958
2
=
2
:
out of
I
=
15
males
0.4
0.7
13 out of 52
PCA / B)
a.)
0.4
52
"
0.3
P( red king
red
U
heart )
king
nor
¥2
:
heart )
+
=
¥2 £2
-
14
I -5J
19
=
52
P(A)
-26
PCB)
t
t
PLA)
PCA / B)
5-21-211--10 -9
=
52 51.50.49
-
.gg
33
16660
OR
( 401 )( 1305 )
52C 5
33
=
16660
=
0.4
0.3
:
=
0.3
:
-
-
PIAAB )
0.3
=
=
37
0.7
P( A)
=
p( B)
PCBIA )
b.)
0.3
=
14
:
PCAAB)
=
I
1 out of 52
Both :
p( neither
PCD )
PCRAD )
females
PCRAD )
PCRAD )
43
=
?
I
PCRID )
Heart
0.00144
=
52C 5
@ 1.21586
( 1024758.26
81.76.26
.
5
-
+ trio
81
=
=
G
w
pair
.
}
.
KK 555
3,586 2586
=
( 550731776156.563
:(
0.04754
=
KKK 55
or
.
=
remaining
1)
36,32
,
6 , and 8 :
5628° 563
=
used since they are
choose any
,
from
→
2x
1
in
( paired
pairs
52C 5
NIA
ending
2
( 13C 2) (4Cz)(4C2)( 44C
16
Numbers
8 cards cannot be
choose the
suits of the
:
62,44
.
(
pairs
76
always
62586
:
56,36
i.
last 2
as
PCAUB)
0.7
0.7
0.3
0.3
=/
PCB / A)
=
0.3
0.7
"
¥
£4
REFRESHER
I
tired
2
¥ ¥
p=
green
% ¥
+
red
=
green
4g
2
heads
b-
already
i
5 tails
means
P
6
7
7
8
6
78
9
7
8
9
10
78
9
10
11
9
10
11
12
2
3
2
3
45
5
4
56
5
6
8
7
6
6
5
45
I
34
4
3
6
1st die is 6
/
1
=
is 7
sum
6
( ¥ ) ( 0.57=10.575--0.2461
5 white
7- black
1-
3
white
12
black
10
(100×110.5)×10.5110
✗
-
✗
2
=¥÷
=6
É
ya
b.
"
Fn
a
=
"
b
-
where
a
:
926 626
:
1-
white
+
black
Fb
37
Fn
=
b
+
-
,
+
Fnt ,
flx )
( x 2)
:
'
f- (x )
2
f- (xn )
l
-
-
✗
bn
-
=
5
no
,
✗
Ln it Ln
=
Ln
-
:
Ah
2
=
fyxn )
(x
-
d
d-✗
-
2)
3.25020
=
4
I
-
(1×-272-1)
4.000539
:
3
2
=
3.02503
=
3.0003
✗
=
9.33
✗
=
Ans
✗ =×
,
.
.
.
2
2
F. =/
,
1,3 , 4,7
✗
"
B
1-
-
CALC
✗
Ln
✗n
5.7332
=
Xz
46367
Fo
=
✗
✗
"
nti
21×-2 )
=
:O
a
12
=
<
"
(2) ( %)
(E) (E) (E)(8)
=
2
F , 1-01=22
Fo
#
tails
Ptails / whitebait
"
"
a
=
Ln
↳
black
121393
=
Ln
:
b
heads
"
-
:
FAlso
and
2
5
1=26
11-55
""
g-
✗ 6=3.000000047
135--20633239
✗
14
An
"
t
B
Uses
"
=
:
5
Fcs)
→
2178
-6
.
Ptwo girls
Pat
least I
=
0.25
girl
=
0.51-0.25=0.75
"
f- (t )
Ptwo girls / atleast
tis
a
girl
=
Ptwo
girls
Pat
1st
2nd
0.2667=4-5
=
BB G G
BG BG
:
hat least I
least I
44
314
Recall :
girl
L{
L {
girl
=
Yg
f- (f)
t
f'Ctl
}
f- (t ) }
=
"
=
=
cospt
sfcs )
s
'
51=6)tsF(s )
s
_f#
Fcs)
-
s
=
°
sff'µ°
⑤ 1- 1) Fcs )
Fcs )
=
l
52+1
-
+
p2
s
=
sztpz
s
sZtp2
REFRESHER
s
-
f- ( s)
=
e-
+
e
2s
S
•
L { ult ) }
Recall :
I
=
-
✗ (z)
=
✗
s
(n ) z
n :O
Assume
z
0.5187
3
n
.
✗
to
-
unitary
Ftw)
:
f
:
-
unitary :
Ftw)
f- (f) e-
If
l
¥f
Flw)=
-
4- ✗ 2)
,
w
=
0.5
re
÷
¥
change
2
to
0.5497
→
Assume
Jwtdt
2- =5
"
-
choices
"
g-
.
n :O
#
!
-
'5ty
,
e
r(wso-
=
+
jsino )
I
( 1- ✗ 2)
cos
@
-
sit )dt
jf(
+
l
-
1- ✗2)
( Atj B)
sin
)
test
use ratio
fast )dt
l
A
=
!
.
G- ✗ 2)
20
=
error
dt
°
I
Recall :
(f) e-
math
→
,
jwt
e
'
=
"
0
↳
dt
=3
a
,
I
-
Assume
"
5,
to
¥
-
"
=
(1-1)
to
n
=
>
( sina.CH
5
=
25
non
n
L
:
B
n
Un
( 1)
"
x
-
=
2h
un
roo
-
"
"
UNH
lim
=
-11
"" + "
"
0.5187
→
If L
>
1
diverges
,
um ,
C- 1)
=
✗
nt 2
L< 1
1=1
y
converges
,
test fails
,
NWDE TABLE
Assume
↳
=
121T
i.
-
(
✗
-
=
i(
g
0.5
(w
is
f
-
f-(w )
now
'
)
(
0.5W
)dw
+
f
-
g
'
sin
( 9+01141-04
B
A
→
(
0.5W
)dw
)
L
=
Iim
✗
0
,
=
25
1
30
1
i.
;
100
1
→
(
✗
•
-1
L>
um ,
,
=
100
,
step
L= 1
: test fails
1
L=fC×)
f- (x)
,
,
un
LL 1
,
-
1=1
:
end
ftx)
25
A
L
,
fails !
If test
-
¥ ( Atj B)
start
✗
)
j
:| "u
text
choices
go
the variable
cos
9tw7(4tw
on
FL
e+Jw×dw
8
'
depends
-
Use
•
Assume
=
Fourier :
For inverse
fcx)
✗
becomes
0
for
✗
becomes
0
for
✗
:fz<
h
✗
<
=
:
-
V2
FL
converges
diverges
test fails
,
→
→
diverges !
diverges !
K
does not include
endpoints
-5
-
REFRESHER
A
coefficient
:
Matrix
B
-
/ ALIBI
=
10
3
10
8
-2
9
8
I
cos
=
0
(
'
-
0
cost
IAIIBI
)
In terms
of
A
B
-
84.32°
=
-10
,
adjoint / adjugatei
classical
A-
A
*
det( A) A
=
*
Cofactor :
.
"
↳
signed
III. ¥ ]
minor
-
-
(real)
47
11
40
152
-180
24
14
=
-
adjugale
-
komplex) adjoint
to
-44
c-
-
-
transpose
→
transpose of complex
conjugate
(A*)T
Radius of
curvature
-
-
C
of cofactor
:
11
152
24
40
-180
14
47
-44
-10
of flx) :
In terms
µ
✗
,
2)
'
+
y
y
,
%
µ
(✗
,
/
"
In terms
R
of
'
r
2)
/ r2t2r ?rr
'
r
'
✗
(t ) and
'
+
y'
'
y
2)
"
'
+
yct )
'
✗
y
:
3/2
/
polar :
(r 't
=
'
✗
312
"
7
'
""
e-
✗
=
i
,
y
=
✗ cost
'
cot 20
,
,
Y' Sino
✗
=
aotasino
=
=
✗
✗
=
-
GAY
'
cot 20=0-0
,
Bxy Cy
1-
✗ cos
✗
sin
✗
(E-
'
×
b.
f' ( x )
:
use
table
2×-6
70
-
y
or :
2×76
to
=
:O
'
:
Caltech
'
20
'
y
Cause
acoso
0
=
0
=
fan 20
=
:
-
,
,
✗
y
( ( atacoso )'t Casino)
✗ >
3
45°
=
45°
-
45° +
=
tan 90° -10
→
y' sin 45°
y' 45°
cos
E-
=
'
×
-
¥× 't
:
'
'
f' 1×1=1×-2)e×
f- 1×1
'
)( EG-
'
'
✗
+
Ey
"
×
-
2-4 y
'
12
)
'
=
=/
y
( -2/-2)
x
✗
i.
y
e'
=
I
'
-
:
'
=L
'
✗
y
'
(0
e×
+
1×-2)e×te×te×=
"
I
Ey
E- y
EY
-
,
2)
=
0
=
'
)%
Ilatacoso )(awso) fasino-kac.int ) /
asina
:
"
asino
R=
=L
xy
.
y
acoso
B
<
tan
'
at a cost
"
AI
a-
:
y
'
X' Sino ty 'cos0
"
!
-
✗
( x 2) e. +20=0
-
xe×=0
REFRESHER
wronskianofflxl.GG/),hCx)
W=
45=3
gcxl
hlx)
f- Cx )
g. Cx )
h'(x )
"
f- (x)
g
4s t Lar
. .
.
f- 1×1
'
✗
,
A
2ñr
4
h'' 1×7
"(× )
-
3
=
i.
2Ir
=
(¥
'
ITRZ
t
-
£1T#
IF
+
21¥ tzitr )( £1T )
-
1.3197ft
=
d=rt
300
kph
9£
,
0.5km
An
=f
s
>
An
zookph
1-
y
✗
For horizontal
vertical
.
asymptotes
terms
Arrange
asymptote
degree
:
✓
top
=
top
bottom
>
:
degree
degree
bottom
H.A.
:
No
4th
-
It
4(¥h)h
=
204dL
=
dt
y=o
dt
leading
dh
terms
-
It
H.A.
4
,
Mls
0.25
=
In # =/
"
=
.
.
v
'
•
y
X
=
✗
-
:
✗
=
-1
=
I
4
,
y
=L
-
du
-0.5
=
6ft
12in
×
↳ find dh_
at
when D= 2
"
or
hi
2
C2
C
f
-1g fts
✓
b
>
B
a
↳
c.
=
62+82
and
+
b
flx)
'
I
.
24%7=24%9-+216%7
C
A
=
=
10
2in
lqftls
10%7=61%1-81--4 )
dc
Td
=
ftls
+ (a)
f' G)
-
=
(9×21-4)
(3×3+4×+8)
f- (x)
f- (b)
=
=
-
-
'
f- (a)
'
( x 2)
-
( b- 2)
ddtf
=
-0.5
=
It
=
¥
3×3+4×+8
=
-
¥0
d=h
¥43
=
dh
"
1ft
3¥
cu
→
'g( 1T¥)h
:
"
✓
and
10h2
=
6=2011.2 )d_h
H.A.is ratio of both
:
degree
is
tzbhl tzbhcsl
-
=
dv
↳ 1. 2m
Izh
b=
-
V
< bottom
=
?
to lowest
kph
¥ by
i
I
top
13.86
§
±
✗ t
2+500
(300-200)
a
8
:
highest
from
70
=
-1
=
✗
=
=
at
( denominator )
1-1--0
✗
=
1T¥
asymptotes :
For vertical
✗
=
ds_
¥
I
¥
=
0.5km
Xs%=X×¥
¥=±s
¥
y=
=O
0.21
=
s2
✗
Qtr
1-
-
=
r
4
Caltech
use
=
'
W-XZcosxtzxsinx-XZsinxt2xcosxp.mg
↳
s
3- 20h
s
XZCOSX
Zsinx
A
=
h2dh_
dt
(2)
=
'
d¥
-0.1592
in
REFRESHER
"
( ttx )
f- Cx)=
"
'
f- G)
=
( ttx)
-
-
-3
"
f- (x)
I
211 + )
'
3 cost
:
=-3 Sint
y
21T
f
S=
( 3 cost )2+f3sint ) dt
'
0
Third term
:
2
2
2
'
×
2
x
=
I
S
2
✗
✗
=
61T
=
!
Let
y=
f- (x )
In / I
=
'
f- G)
=
I
G Caltech
-
X
-
j?
X
to
✗
3
)
l
-
=L
a
=
'"
f- (X)
get
-13×3
3
✗
:
-1-3
coefficient is
3.5631
ls
=
<
-2
SA
=
2ñr
.
S
I
=
↳
sinx
odd !
→
f- (x )
✗
=
§:X
-
+
1+(3×2)
'
DX
0
¥
>
f 21T¥
¥
-
✗
7
=
3.5631
Fourth term :
✗
-
7
Conic
z=ei°
Iz /
→
r
=
e.
fdz
ill
>
ei
i%
-
=
-
It
(
i
'T
I
-
-
=
-
Bxyt Cy
Biu
+
4AC < 0
>0
ziddz
2--1
i
:
,
-
:
+
Dxt
COI
+
2yZ
EytF=o
D
?§
ellipse
>
+
O
'
EZU
Ty
+
F=o
-4111111=-4
parabola
hyperbola
i
=
=
-
l
:
=O
i
2
Iti )t
-
+
-
2×2y
12¥
-
Zo
ANU
132
e
-
1- LIT
=
=
2-
Ax't
2×2
=
e.
DE :
1
Z
=
and 0=0
r=1
:
Z
I
-
l
-
I
=
-
l
l
-
l
-
=L
,
:-# i
=
i
i
=
-
-
l
Iti
22h
21T
Yrms
ffltsint )Ñt
=
211--0
ro
=
2
yyz
t
y
to
024
to
zyz
negative !
-549=0
where
y
20
REFRESHER
dd¥
Assume
✗
6×2-3×2
=
¥
compare with given
same
pA
to
calc
get y
,
,
a
B
D( 2.33 )
1
get ¥
It i
b)
-
,
must
,
mul 2
.
I ± i
Hi
c)
d)
m
^
.
-
I ±
i
114,2
12
-
ti
,
-2 ±
µ
i
probability
y
c. em
=
case 2 :
m,
y
=
case 3
:
y
+8m 't 8m
bpfx
+4=0
't
me
=
"
+
Gem
=
=
=
m
emt ( c. + cat )
m
a ±
=
bi
eat ( c. cosbttczsinbt )
:
m5
A.
3m
-
0 mill 2
"
-
.
B. 0mn12
+
0,
D.
0 Mul
.
.
2
-
I mul 3
I
I mul 3
C.
max
,
,
3ms m2
"
mud
2
✗
I mul 3
.
,
=/
A of
>
2-
=
A of
<
2-
=
A of ( 2- to Zo )
I
Z
2
✗
region
-
.
,
,
0
✗
.
2
=
Z
z,
↳
RCZ )
Plz )
to'z e-
dx
,
plz )
-
RC Z)
-
Qlz)
-
+
plz)
}
pfz)
QC
-
z
)
:
same
0.995
or
or
thing
99.5%
MODE
-
mean
STAT
→
Qcz)
=
'
:( %)
→
-
Ac
right
side
left side
16
14.2
=
0=2.3
> 16
=p ( z >
=/ Mrs
,
0.9901
=
✗
mul 2
✗
case 1 !
)
=o
✗
+4ms
Rez )
.
a)
m
as
2-
Zo
Substitute
"
value
region
y
and 0=3
1.1
=
anµCZ
(
)
R( ¥ )
)
t
3)
16-14.2
2.
+
0.38633
P( ✗ <
+
12
)
Plz < 122k¥)
Pf -33 )