60737914 mathematics s1
TRANSCRIPT
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SIGNIFICANT FIGURES
1,013X105 = range 1,0125X105 ~1,0134X105
1 5 1,61 OFF TO 3sf
1 2 3 4 figures1,5 1 6 2x102
1,52x102 is 3sf
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Surds
m
mn mnn
a b a b
a a
bb
a a a
v !
!
! !
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2 2 2 2 2
2 2 2
3 3 2 2 3
3 3 2 2 3
2 2
3 3 2 2
2 2
2
2
3 3
3 3
3 3 ( )( )
a b a ab b a b
a b a ab b
a b a a b ab b
a b a a b ab b
a b a b a ba b a b a ab b
a b a b a ab b
! { !
! !
! !
!
factorising
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QUADRATIC EQUATIONS
y=ax2+bx+c
e.g. y=2x2+6x-20 A=2 2 1 2 1 1 2 2 1
. X or X or X or X
C=-20 2 -10 4 -5 4 -5 -4 5
(2x-4)(1x+5)=y
X=4/2=2 and x=-5/1=-5
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The formula
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Reducible QUADRATIC Equation
ay2k+byk+c=0
X can substitute yk
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m
n mn
m nm n
m nm n
nm mn
m m m
a a
a a aa a a
a a
ab a b
!
v !z !
!
!
1
0
1
1
1
m m
mm
m m
a aa
a a
a a
a b
b a
!!
!
! !
Indices / exponents
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log
ln
log log log
log log log
log log
y
a
y
a a a
a a
n
a a
y x x a
y x x e
mn m n
m
a m nn
m n m
! !
! !
!
! !
log
1
log 1
log 1 0
log log
log log
a
a
a
x
x
x
a a
a
a x
a a
x y x y
!
!
!
!
! !
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2
sec
2 2
2
1
2
1 1sin
2 2
360 2
0,23.0,23 180 13, 2
1
sin2
tor
segment
major segment segment
Area r
s
r
Area r r
Area r Area
rad
eg rad
Ar
ea ab C
U
U
U U
T
T
T
!
!
!
!
r !
! rv ! r
( !
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2 2 2
2 2
2 2
2 2
cos : 2
sin :sin sin sin
sin cos(90 ) cos(2 )
cos sin(90 )
tan cot(90 )
sin cos 1
sec tan 1
cos cot 1
ine rule a b c bcC osA
a b ce rule
A B C
ec
U U T U
U U
U U
U U
U U
U U
!
! !
! r !
! r
! r
! !
!
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sin( ) sin cos sin cos
sin( ) sin cos sin coscos( ) cos cos sin sin
cos( ) cos cos sin sintan tan
tan( )
1 tan tantan tan
tan( )1 tan tan
A B A B B A
A B A B B A
A B A B A B
A B A B A B
A BA B
A BA B
A BA B
!
! !
!
!
!
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sin sin 2 sin cos
2 2
sin sin 2 cos sin2 2
cos cos 2 cos cos2 2
cos cos 2 sin sin2 2
A B A BA B
A B A BA B
A B A BA B
A B A BA B
!
!
!
!
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1
sin cos sin sin2
1cos sin sin sin
21cos cos cos cos
2
1sin sin cos cos2
A B A B A B
A B A B A B
A B A B A B
A B A B A B
!
!
!
!
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2 2
2
2
2
2 2 sin cos sin sincos 2 cos sin
2 tantan 2
1 tan1
sin 1 cos 22
1cos 1 cos 2
2
sin A A A A A
A A A
AA
A
A A
A A
! !
!
!
!
!
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Test 1 summary
2 2 2 2 2
2 2 2
3 3 2 2 3
3 3 2 2 3
2 2
3 3 2 2
2 2
2
sec
2 2
2
2
2
3 3
3 3
3 3 ( )( )
1
2
1 1 sin2 2
tor
segment
major segment segm
a b a ab b a b
a b a ab b
a b a a b ab b
a b a a b ab b
a b a b a b
a b a b a ab b
a b a b a ab b
Area r
s
r
Area r r
Area r Area
U
U
U U
T
! {
!
!
! !
!
!
!
!
!
!
2 2 2
360 2
0,23.0,23 180 13,2
1sin
2
cos : 2
sin :sin sin sin
ent
rad
eg rad
Area ab C
ine rule a b c bcC osA
a b ce ruleA B C
T
T
r !
! rv ! r
( !
!
! !
log
1
log
ln
log log log
log log log
log log
log 1
log 1 0
log log
log log
a
y
a
y
a a a
a a
n
a a
a
a
x
x
x
a a
y x x a
y x x e
mn m n
ma m n
n
m n m
a
a x
a a
x y x y
! !
! !
!
!
!
!
!
!
!
! !
1
0
1
1
1
mm
n mnn
m
n mn
m nm n
m nm n
nm m n
m m m
m m
mm
m m
a b a b
a a
bb
a a a
a a
a a a
a a a
a a
a b a b
a a
a
a a
a a
a b
b a
v !
!
! !
!
v !
z !
!
!
!
!
!
!
!
2 2
2 2
2 2
si n cos(90 )
cos si n(90 )
ta n cot(90 )
si n cos 1
sec tan 1
cos cot 1
si n( ) sin cos si n cos
si n( ) sin cos si n cos
cos( ) cos cos si n si n
cos( ) cos cos si n si n
tta n( )
ec
A B A B B A
A B A B B A
A B A B A B
A B A B A B
A B
U U
U U
U U
U U
U U
U U
! r
! r
! r
!
!
!
! !
!
!
!
an ta n
1 ta n ta n
ta n ta nta n( )
1 ta n ta n
1cos cos cos cos
2
1si n si n cos cos2
1si n cos si n si n
2
1cos si n si n si n
2
si n si n 2 si n cos2 2
si n si n 2 co
A B
A B
A BA B
A B
A B A B A B
A B A B A B
A B A B A B
A B A B A B
A B A BA B
A B
!
!
!
!
!
!
!
2 2
2
2
2
s si n2 2
cos cos 2 cos cos2 2
cos cos 2 sin si n2 2
2 2 sin cos
cos 2 cos si n
2 t anta n 2
1 ta n
1si n 1 cos 2
2
1cos 1 cos 2
2
A B A B
A B A B
A B
A B A BA B
sin A A A
A A A
AA
A
A A
A A
!
!
!
!
!
!
!
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Test1 Summary B
1cos cos 2 cos cos cos cos cos cos
2 2 2
1sin sin 2 sin cos sin cos sin sin
2 2 2
1sin sin 2 cos sin cos sin sin sin2 2 2
cos cos 2 sin sin sin sin2 2
A B A B A B A B A B A B
A B A B A B A B A B A B
A B A B A B A B A B A B
A B A B A B A B
! !
! !
! !
!
1cos cos
2 A B A B!
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Test 2 Start
Trigonometry
Curve sketching
Sine function: Amplitude, frequency, phase
Co-ordinate system -Chapter 7
Cartesian co-ordinates
Polar co-ordinates
Complex numbers -Chapter 8
Imaginary numbers
Operations
Argand
Modulus-argument or polar form
De Moivres Theorem 1
De Moivres Theorem 2
Co-ordinate Geometry and graphs -Chapter 9
Formulas for distance, midpoint of line
Gradient and inclination of line Angle between 2 lines
Straight line forms
Identify and sketch
Straight line p.96
Parabola p.98 Hyperbola
Rectangular p.99
Central p.100
Ellipse p.101
Circle
Standard p.102
Off centre p.103
Logarithmic p.21
Exponential p.14
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Assignment 1 p55-59 Trigonometry
Curve sketching
Sine function: Amplitude, frequency, phase
sin
2 12
. .
sin
sin .
sin .
1 2
sin 2 .
y A
f Tt T f
t angle in rad
A Amplitude
y A t
y A t leads phase
y A t lags phase
xv f
T
t xy A leads phase
T
U
U T
[ T
U [
[
[ E
[ E
TP P E
P
T
P
!
! ! ! !
! !
!
!
! p
! n
! ! @ !
! p
Q SI units
A=amplitude in metersVelocity=m/s
f=frequency in Hz
T is period in seconds
T
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+y
+x
-y
-x
POLAR /Modulus-argumentCOORDINATES (r;)
2 2
1 1
tan tan
cos
sin
r x y
y jy
x x
x r
y r
U
U
U
!
! !!
!
Eg. (2;3)
2
3 (2;3)
2
3
+x
+y
r
2 3z j!
Both x and y are
positive thus it isin the 1st quadrant
Both x and y are
positive thus it is
in the 1st quadrant
If both x and yare negativethus it is in the3rd quadrant180+
If x negativeand y positivethus it is in the2nd quadrant180-
If y negative
and x positive
thus it is in the
4th quadrant
360-
r=modulus =absolute value
=argument=amplitude
Rectangular
/CARTESIAN
COORDINATES (x;y)
2 3z j!
(2;-3)
Conjugate of Z is z
=180
90 1
180 2T T!
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Imaginary number
1. .
0 . 0
.
1
1 .
n m m n
mn n m
n
n
ja j j a
j
j j jj j
j
ja a
! !
!
! !
!
!
0
1
2
3 2 1
4
2 71 0 8 4
1 2 3 4 3 0 3
1
1 1 1
1 .
( 1) ( 1) 1
1
( )
j
j jj
j j j j j
j
j j
j j j j
!
!! !
! ! !
! !
! !
!
!
2
2
2
2 2
4
2
( 2 ) ( 2 ) 4 (1) ( 2 )2 (1)
2 4 1 41
2 2 2
1 21 1
2
y x x
b b a cx
a
x
x
x j
!
s !
s !
! s ! s
! s ! s
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Operations Complex numbers
1
2
1 2
1 2
1 2
1 2/
z a jb
z c jd
Addition z zSubtraction z z
Multiplication z z
Division z z
!
@ @
@
@
g
2 2
( ) ( )
( ) ( )
( ) ( )
( ) ( )
T
T
T
T
z a c j b d
z a c j b d
z ac bd j bc ad
ac bd j bc ad zc d
!
!
!
!
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Argand diagram+j
-j
+R
2j
5
2(5 2 ) 5 2 5 2j z j j j j j ! ! !
-2
5j
Parallelogram law1 2 (2 3 ) (2 )
(2 2) (3 1)
4 4
z z j jzt j
zt j
! !
!
2
j3
1j
4+j4
j4
4
Polar or Modulus Argument cos sinz r rcis r jU U U U! ! !
10 20 10 cos 20 sin 20
10 cos 20 10 sin 20
9, 3 3, 4
z cis j
z j
z j
! r ! r r
! r r
!
Rectangular form
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Moivres theorem1
1 2 1 2 1 2
1 11 2
2 2
1 1 1
n n
z z r r
z r
z r
z r n
U U
U U
U
!
!
!
Change rectangular to Polar
Square roots = 2
Cube roots =3
Cubic roots = 3
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Moivres theorem2
2; 0;1;2;3;.... 1n
k
kz r k n
n
U T! !
? A
1
3 3
1
3
3
1
2
3
8 120
8 120
120 3608 ; 0,1, 2
3
120 0 360 120
2 2 2 40 ; 03 3
120 1 360 4802 2 2 160 ; 1
3 3
120 2 360 8402 2 2 280 ; 2
3 3
z z
z
kz k
z k
z k
z k
! ! r
! r
r r! !
r r r
! ! ! r !
r r r! ! ! r !
r r r! ! ! r !
120*
120* 120*
? A
3
1
3
8 120
8 120
z
z
! r
!r
1 2 40z ! r2
2 160z ! r
3 2 280z ! r
4
4
2 2
1
1 0
1 1 0
( 1) 0 1 1
0tan
1
z
z j
r
jU T
!
! !
! ! !
! ! ? A
4
114 44
1
4
1
1
21 0,1,2,3
4
3 5 71 ;1 ;1 ;1
4 4 4 4
z
z z
kz k
z
T
T
T T
T T T T
!! !
! !
!
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Moivres theorem2
3
3 33
2 2 1
1 0
1 1 0
01 0 ; tan 0
1
x
x j
jr U
!
! !
! ! !
5 2
2
2
1
5
5 5
(1 )
(1 ) 24
(1 ) 2 24 2
5 22
2222 2 , 0,1,2,3,4
5 10 5
z j
j
j
z
kk
z k
T
T T
T
TT
T T
!
!
! !
!
! ! !
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Exponential form
1 2( )
1 2 1 2 1 21 2
j
j
z r re
z z r r e r r
U
U U
U
U U
! !
! !
inRAD
1
4
2 2
2 2 22
tan42
2 24
j
z j
r
j
z eT
TU
T
!
! !
! !
! !Principle value
-
Principle value of ln(5 210 )r
5
6
(5 210 )
5 210 360
5 150 .............210 360 150
5 .................210 0 210
5ln ln 5
6
ln 1.61 2, 62 ..
j
z
z k
z
z e
z j
z j pri value
T
T
! r
!
! r !
! !
!
! !
210-360 or 210-0
Which ever is smaller than 180
Convert to RAD and then to exponent
And ln to calculate principle vale
5ln ln 5
6
5ln 1.61 ..6
z j
z j pri value
T
T
!
! !
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Geometry Types of triangles
Side Lengths
Equilateral = all sides have the same length
Isosceles = two sides have equivalent length.
scalene = all sides have unequal lengths.
Triangles largest angle:
right= an angle is a right angle (measure 90
degrees or /2 radians).
obtuse = obtuse angle larger than 90 degrees
acute = all angles are acute (smaller than a right
angle)
Types of qua
drilaterals
Rectangular = 4 right angles and opposite sides
are equal but not all 4.
Square = 4 right angles and 4 equal sides
Parallelogram = 2 opposite sides are equal and
parallel and no right angle..
Rhombus = Parallelogram with 4 equal sides and
no right angle.
Trapezium = 1 pair of opposite sides are parallel
Kite = adjacent sides are same lengths
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Coordinate geometry
22( ) B A B AAB x x y y!
( ; )2 2
B A B A
AB
x x y yM
!
1tan B AAB
B A
y ym
x x
!
1;AB CDm m perpendicular !
1tan1
CD DG
CD DG
m m
m m
!
; AB EF m m parallel!
A
B
MAB
y
x
C
DE
F
Angle between 2 lines
G
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Equations
y mx c!
Straight line:
Standard form
Gradient-Point formula
B A
A A
B A
y y
y y x xx x
! B AB A
y ym
x x
!
Perpendicular=mabxmcd=-1=90
Bisector=Midpoint
Inclination =angle in degres
Gradient=angle in tan-1
Parallel=mab=mcdQuadrilateral=4sides
Parallelogram=opposite sides are parallel
Diagonals=
2 2
4 12 1 1
2 1
x yx y
y x
!@ ! !
!
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2
2
2
2 2
1. int
2. min _ 4
0 _
0 1_
0 2_
3.max_ _min_
0 max_0 min_
; _ _2 4
4. 00
4 4
2 2 2
y ersept c
discri ant b ac
image roots
root
roots
or tp
a a xa
bx y axis of symmetry
a a
rootsy
ax bx c
b b ac b b acx
a a a
!
( !
( @
( ! @
( @
! ! !
!
(! ! |
!
!
s ! ! s
p
f
pf
Q
Parabola:2y ax bx c!
2
4; ;
2 4
b b acx y
a a
!
2 4
2
b b acx
a
s !
( 0)y x c! !
2 y ax bx c!
min 0;... ..max
dytp solve x
dx! !
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Hyperbola : rectangular
0; . 2 , 4 _
0; 1 , 3 _
. 2 ;
kxy k y
xk negative nd rd quadrant
k positive nd rd quadrant
O nearest k x k y k
! @ !
@ !
@ !
! v @ ! !
p
f
+
+
-
-
OA
B
Hyperbola: central2 2 2
2 2 2
2 2
1;........ . .
;
x y yleft and right
a b b
x a a y b b
! @
! ! ! !b
-b
a-a
xx
y
y
y
y
xx
-3 3
2 2 2
2 2 21;....... . .
y x yup and down
b a b ! @
2 2
2 2
2 2
2 2
2 9 18
2 9 18
2 9 18
18 18 18
19 2
x y
x y
x y
x y
!
!
!
!2
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ellipse 2 22 2
1x y
a b
!
-a a
b
-b
M N
2 2
.2
aM N origin
M c a b
! !
! !
m nm+n=2a
M and N is foci points
2 2
2 2
2 2
9 25 225
9 25 225
225 225 225
125 9
25 5; 9 3
x y
x y
x y
a b
!
!
!
! ! ! !
If a =b then its a circle
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Circle Standard form
Circle at
origin
2 2 2
2 2 2
2 2
; 0
x y r
x r x r x r y
y r
!
! @ ! ! !
!
Q
Standard form
Circle off
origin
2 2 2 x a y b r !
2 2
2 2
2 2
2 2
2 2
2 2 2
2 2 12 8 6 0
2 6 4 3 0
6 4 3
6 9 4 4 3 9 4.......( . )
6 9 4 4 16
3 ( 2) 4
3; 2 4
x y x y
x y x y
x x y y
x x y y complete square
x x y y
x y
x y r
!
z @ !
!
!
!
!
! ! @ !
(3; 2)
3
-2
2 2
2
22
2 2
6 4 3 0
6 3 0
6 6 4 1 34
2 2 1
6 6 4 1 3 6 6 12
2 1 2 2
483 3 6.92 9.92.... 3.92
2
x y x y
x x
b b acx
a
x
!
!
s s ! !
s ! ! s
! s ! s !
Taking half of x and square root it adding
on LHS and RHS
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Exponential
xy a!
a>1
1
0>a
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Ellipse2 2
2 21
x y
a b !
Graph summary2 2
2 21
x y
a b !
Circle: origin2 2 2x y r !
Circle: non-origin 2 2 2( ) ( ) x a y b r !
a
b
Hyperbola Centralx y k!
Hyperbola Rectangular
; 0x
y a a! "
exponential
; 0x
y a a! logarithmic
log ; 0ay x a! "
log ; 0a y x a!
) (
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Test 2
summary
2 2
2 2: 1x yhypC
a b !
1 2 2 2
( ) ( )ac bd j bc ad z z
c d
z !
2
; 0;1; 2; 1.k
n n kz r k n
n
U T! !
(cos sin ) ;jz r j r re radUU U U U! ! ! !
ln ln ( 2 ); ; ,z r j k radU T U T T ! ! p
1 1 2
1 2
tan1
m m
m m
!
222 1 2 1( )AB x x y y!
1 2 1
2 1
tanAB
y ym
x x
!
2
2
2
1. int
2. min _ 4
0 _
0 1_
0 2_
3.max_ _min_
0 max_
0 min_
; /2 4
4. 0
0
4
2
y ersept c
discri ant b ac
image roots
root
roots
or tp
a
a
b x y as
a a
roots y
ax bx c
b b acx
a
!
( !
( @
( ! @( @
!
!
(! ! |
!!
s !
p
p
:
2
hypR xy k
OA OB k
!
! ! v
2 2
2 2: 1
x yellips
a b !
2 2 2: ( ) ( )CircleN x a y b r !
xy a!logay x!
1. Shape
2. Critical points
3. Equation
4. Identify
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Test3 start
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Long division
3 2
21
0 4
xx
x x
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Partial fractions, Linear factors in the denominator, none repeated
3 4
( 1)( 2) 1 2
3 4 ( 2) ( 1)
_ 1
x A B
x x x x
x A x B x
let x
!
!
! @
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Partial fractions, Linear factors in the denominator, some repeated
2
2 2
2 2
1
( 1) ( 1) ( 1) ( 1) ( 1)1 ( 1)( 1) ( 1) ( 1)
_ 1
x A B C
x x x x xx A x x B x C x
let x
!
!
!
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Binomial Theoremn is not a negative integer
2 32 3
1 ( 1) ( 1)( 2)( ) ...
2! 3!
n n
nn n nn n a b n n n a b
a b a na b b
!
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Limits definition first principle2
2 2 2
2 2 2
2 2 2
2
( )( ) ( ) ( ) 2
( ) ( ) ( 2 ) ( )
( ) ( ) 2
( ) ( ) 2 (2 1)
lim (2 1)'( )
0
lim(2 1)
0
2 0 1
2 1
f x y x xf x h x h x h x xh h x h
f x h f x x xh h x h x x
f x h f x x xh h x h x x
f x h f x xh h h h x h
h x hf x
h h
x hh
subsitude
x
x
! ! ! !
!
!
! !
!
p
p
!
!
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Limits Limits 0/0 type (determine by substitution)
2
2
2
lim 3 2
1 1
1 3 2 01 1
_
lim 3 2 lim ( 1)( 2)
1 1 1 1lim
( 2)1
1 2 1
x x
x x
substitute
thus factorise
x x x x
x x x x
thus xx
p
!
!
p p
p
! !
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Limits Limits / type (determined by dividing each
by highest xn
2
3
2 3
3
lim 3 2
1
_ _ _
3 1 2
lim
11
0 0 00
1 0
n
x x
x x
divide by highest x
x x x
x x
p g
p g
! !
2
2
2
2
lim 5 1
2 2
_ _ _
15
lim
22
5 0 5
2 0 2
n
x
x x
divide by highest x
x
xx
p g
p g
! !
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Differentiations
1
2
2
0
ln
1ln '
1log 'ln
sin ' cos
cos ' sin
tan ' sec
cot ' cos
sec ' sec .tan
cos ' cos .cot
1'
2
n n
x x
x x
a
y k y
y kx y nkx
y e y e
y a y a a
y x yx
y x yx a
y x y x
y x y x
y x y x
y x y ec x
y x y x x
y ec x y ecx x
y x y
x
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
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Differentiation rules
Sum and Difference Rule
Product Rule
Quotient Rule
Chain Rule
( ) ( ) ' '( ) '( )y f a g b y f a g b! s p ! s
( ) ( ) ' ( )( ') ( ')( )y a x b x y b a b a! v p !
2( ) ( )( ') ( ')( )'( )a x b a b ay yb x b
! p !
[ ( )] ' ' [ '] ( ')y y u v y y u v! p ! g g
L ith i Diff ti ti
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Logarithmic Differentiation
ln ln ln1
ln
1ln 1 ( ' ' )
1ln 1
(ln 1)
ln 1
x
x
x
y x
y x x xdy dy
y ydx dx y
x x ba b ax
dyx
dx y
dyy x
dx
dy x xdx
!
! !
! p
!
!
!
!
g
Q
2
2
2
2
2
2
2
tan 1
cos
tan 1ln ln
cos
ln ln(tan 1) ln( cos )
ln ln tan ln ln 1 ln ln cos
1 1ln ln tan ln( 1) ln ln cos
2 2
1 sec 2 11
tan 2( 1) 2
x
x
x
x
x e xy
x x
x e xy
x x
y x e x x x
y x e x x x
d d y x x x x x
dx dx
d x x
d x y x x
!
!
!
!
!
!
2
2
2 2
2
sin
cos
sec 11 tan
tan ( 1) 2
tan 1 sec 11 tan
tan ( 1) 2cos
x
x
x x
d x xy x
d x x x x
d x e x x xx
d x x x xx x
!
!
li i iff i i
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Implicit Differentiation
sin 2
sin 2
cos ( ) 2
1 cos 2
2
1 cos
y y x
d d y y x
dx dx
dy dy d y x
dx dx dx
dy ydx
dy
dx y
!
!
!
!
!
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Higher Order Differentiation
Derivative of a derivative
which will be used at the McLaren series
2 2 3 3( 1) ( 1)( 2)
n nb b
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Summary
1
2
2
' 0
'
'
' ln
1ln '
1log '
ln
sin ' cos
cos ' sin
tan ' sec
cot ' cos
sec ' sec . tan
cos ' cos .cot
1'
2
n n
x x
x x
a
y k y
y kx y nkx
y e y e
y a y a a
y x yx
y x yx a
y x y x
y x y x
y x y x
y x y ec x
y x y x x
y ec x y ecx x
y x yx
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p ! ! p !
! p !
! p !
2
( ) ( )( ') ( ')( )'
( )
a x b a b ay y
b x b
! p !
2 2 3 31 ( 1) ( 1)( 2)( ) ...
2! 3!
n nn n n nn n a b n n n a ba b a na b b
!
3 4
( 1)( 2) 1 2
3 4 ( 2) ( 1)
_ 1
x A B
x x x x
x A x B x
le
tx
!
!
! @
2
2 2
2 2
1
( 1) ( 1) ( 1) ( 1) ( 1)
1 ( 1)( 1) ( 1) ( 1)
_ 1
x A B C
x x x x x
x A x x B x C x
let x
!
!
!
2
2 2 2
2 2 2
2 2 2
2
( )
( ) ( ) ( ) 2
( ) ( ) ( 2 ) ( )
( ) ( ) 2( ) ( ) 2 (2 1)
lim (2 1)'( )
0
lim(2 1)
0
2 0 1
2 1
f x y x x
f x h x h x h x xh h x h
f x h f x x xh h x h x x
f x h f x x xh h x h x xf x h f x xh h h h x h
h x hf x
h h
x hh
subsitude
x
x
! !
! !
!
! ! !
!
p
p
!
!
2
2
2
lim 3 2
1 1
1 3 20
1 1
_
lim 3 2 lim ( 1)( 2)
1 1 1 1
lim ( 2)1
1 2 1
x x
x x
substitute
thus factorise
x x x x
x x x x
thus xx
p
!
!
p p
p
! !
2
2
2
2
lim 5 1
2 2
_ _ _
15
lim
22
5 0 5
2 0 2
n
x
x x
divide by highest x
x
x
x
p g
p g
! !
( ) ( ) ' ( )( ') ( ')( )y a x b x y b a b a! v p !
[ ( )] ' ' [ '] ( ')y y u v y y u v! p ! g g
( ) ( ) ' '( ) '( )y f a g b y f a g b! s p ! s
ln ln ln
1ln
1ln 1 ( ' ' )
1ln 1
(ln 1)
ln 1
x
x
x
y x
y x x x
dy dyy y
dx dx y
x x ba b ax
dyx
dx y
dyy x
dx
dy x xdx
!
! !
! p
!
!
!
!
g
Q
sin 2
sin 2
cos ( ) 2
1 cos 2
2
1 cos
y y xd d
y y xdx dx
dy dy d y x
dx dx dx
dyy
dx
dy
dx y
! !
!
!
!
T 4
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Test 4 start Differentiation Application
Critical points and curve sketching
Optimisation (minimum and maximum)
Linear and rotational motion
Integration
Rules
Techniques
Definite Integration
Integration Application (area) Matrice 3x3
Multiplication
Dependants
Cramers rule
Macclaurin
Formula
series
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Differentiation Curve and Critical points
2
2
2 2
2 2
2 2
2 2
......
int ; . 0,
. . 0
/ . 0
. . . . . 0
. . . ......( . . )
0
. ' . . . . 0. . .
. ' . . . . 0. .min
y
y ercept where x
x roots where y
dyt p
dx
dyget value of x of
dx
Put x in y to get y
d y
dx
dy d y d yput x in if then Max
dx dx dx
dy d y d yput x in if then
dx dx dx
!
! !
! !
!
!
!
!
2
2
2
2
.
. /
0
. . . . . 0
. . . ......( . . )
Inflexion t p
d y
dx
d yget value of x of
dx
Put x in y to get y
!
!
!
O ti i ti (M i i ti ^ d i i i ti )
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Optimisation (Maximisation>^ and minimisation
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Linear and rotational motion
'( )
''( )
s l
dSv f x
dt
dVa f x
dt
!
! !
! !3
1
2
50
. '( )
. ''( )
t
t
dv rad s f t
dt
da rad s f tdt
U
U[
[E
!
! ! !
! ! !
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Integrate Alg
? A
? A1
( ) ( )
( )
( ) ( )
( ) ( ) ( ) ( )
( )'( ) ( )
1
'( )ln ( )
( )'( )
1'( )
ln
nn
f x f x
f x
adx ax c
k f x dx k f x dx
f x g x dx f x dx g x dxf x
f x f x dx k cn
f xdx f x c
f xf x e dx e c
f x a dx ca
!
!
s ! s
!
!
!
!
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Integrate Trig
2
2
'( ) cos ( )......................................sin ( )
( ) sin ( ).................................... cos ( )
'( ) sec ( ).................................... tan ( )
'( ) csc ( ).............
f x f x f x c
f x f x f x c
f x f x f x c
f x f x
? A
.................... cot ( )
'( ) sec ( ) tan ( )........................sec ( )
'( ) csc ( ) cot ( )..................... csc ( )
'( ) sec ( ).................................ln sec ( ) tan ( )
'( ) c
f x
f x f x f x f x c
f x f x f x f x c
f x f x f x f x c
f x
? A? A? A
sc ( ).............................. ln csc ( ) cot ( )
'( ) t an ( ).................................ln sec ( )
'( ) cot ( ).............................. ln csc ( )
f x f x f x c
f x f x f x c
f x f x f x c
Diff i i d I
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Differentiation and Integers
( ) ( )
( ) ( )
' 0
'
( ) ' '( )' '( )
' '( ) ln
'( )ln ( ) '
( )
'( )log ( ) '
( ) ln
sin ( ) ' '( ) cos ( )
cos ( ) ' '( ) sin ( )
ta
f x f x
f x f x
a
y k y
y ax y a
y kf x y kf xy ke y k f x e
y ka y k f x a a
f xy k f x y k
f x
f xy k f x y k
f x a
y k f x y k f x f x
y k f x y k f x f x
y k
! p !
! p !
! p !! p !
! p !
! p !
! p !
! p ! ! p !
! 2
2
n ( ) ' '( ) sec ( )
cot ( ) ' '( ) csc ( )
sec ( ) ' '( ) sec ( ). tan ( )
csc ( ) ' '( ) csc ( ).cot ( )
1'2
( ) ( ) '( ) '( )
( ) ( ) ( ) '( ) '( ) ( )
( )
( )
f x y k f x f x
y k f x y k f x f x
y k f x y k f x f x f x
y k f x y k f x f x f x
y x yx
f a f b f a f b
f a f b f b f a f b f a
f a
f b
p !
! p !
! p !
! p !
! p !
s ! s
!
? A ? A
2
1
( ) '( ) '( ) ( )
( )
( ) ' ( ) '( )n n
f b f a f b f a
f b
y k f x y nk f x f x
!
! p !
? A
? A1
( ) ( )
( )
( ) ( )
( ) ( ) ( ) ( )
( )'( ) ( )
1
'( )ln ( )
( )
'( )
1'( )
ln
' ( ) cos ( )................................ ......sin (
nn
f x f x
f x
adx ax c
kf x dx
kf x dx
f x g x dx f x dx g x dx
f xf x f x dx k c
n
f xdx f x c
f x
f x e dx e c
f x a dx ca
f x f x f x
!
!
s ! s
!
!
!
!
2
2
)
( ) s in ( )................................. ... cos ( )
'( ) sec ( )................................ .... tan ( )
'( ) csc ( )................................ . cot ( )'( ) sec ( ) tan ( ).........
c
f x f x f x c
f x f x f x c
f x f x f xf x f x f x
? A
? A
...............sec ( )
'( ) csc ( ) cot ( )..................... csc ( )
'( ) sec ( )................................ .ln sec ( ) tan ( )
'( ) csc ( ).............................. ln csc ( ) cot ( )
f x c
f x f x f x f x c
f x f x f x f x c
f x f x f x f x c
? A
? A
' ( ) tan ( )................................ .ln sec ( )
' ( ) cot ( ).............................. ln csc ( )
f x f x f x c
f x f x f x c
I h i
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Integrate chain
? A
? A
( ) ( )
( )( )
1
( ) ( )
'( ) ln
( ) ln
( )
'( ) ( ) 1
'( )ln ( )
( )
'( )log ( )( ) ln
'( )
'( ) csc ( ) ln csc ( ) cot (
f x f x
f xf x
nn
a
f x f x
k f x a adx k a c
ak a dx k c
f x a
f x
k f x f x dx k cn
f xk dx k f x c
f x
f xk dx k f x cf x a
k f x e dx k e c
k f x f x dx k f x f
!
!
!
!
!
!
!
? A)x c
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Integration Substitute method1
2
11
1 1 22 2
11
2
33 322
3
2
33
2 1 (2 1)1
. (2 1) 22
1 1 1
2 2 2
1 1 2
2 2 3 3
(2 1)
3 3
x dx x dxdu
let u x dx d udx
uu du u dx c
u uc u c c
xuc c
!
! @ ! @ !
v ! !
! !
!
g g
I t ti d th
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Integration area under the curve
? A
? A ? A 2
1( )1
'( ) ( )
a
n
b
a b
a
nf xn
b
f x f x dx
unit
!
!
Integrate examples
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Integrate examples'( )
ln ( )
( )
f xdx f x c
f x
! ? A2 22 2 2 2
'( )sin ( ) cos ( )
csc cot '( ) csc ( ) cot ( )
1 1 12 csc cot ( csc ) csc
2 2 2
f x f x dx f x c
x x xdx f x f x f x dx
x x x dx x c x c
!
!
! !
g
g
2
1 1
1 ( 1)( 1)
1
( 1)( 1) ( 1) ( 1)
1 11 ( 1) ( 1) ;
2 2
1 11 2 2
( 1)( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1)
1 11 1 1 12 2
( 1) ( 1) 2 ( 1) 2 ( 1)
1 1ln( 1) ln
2 2
dx dx x x x
A B
x x x x
A x B x A B
A B A Bdx dx dx dx
x x x x x x x x
dx dx dx x x x x
x
!
!
! @ ! !
! ! !
!
!
1 ( 1) ( 1)( 1) ln ln
2 ( 1) ( 1)
x xx c c c
x x
! !
Maclaurin power series
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Maclaurin power series2siny x!
2 22
2 2
2 2
2
( ) sin sin ........... (0) sin 0 0'( ) 2 sin cos sin 2 .. '(0) sin 2 0 0
''( ) 2 cos .................. ''(0) 2 cos 0 2
'''( ) 4 sin ................ '''(0) 4 sin 0 0
''''( ) 8 cos .........
f x x x ff x x x x f
f x x f
f x x f
f x x
! ! ! !! ! ! !
! ! !
! ! !
!
g
2
...... ''''(0) 8 cos 0 8f ! !
2 3 4
( ) ( 0 ) '( 0 ) ''( 0 ) '' '( 0 ) ''''( 0 ) . . .2 ! 3 ! 4 !
x x xf x f f x f f f!
2 3 42s i n 0 0 ( 2 ) ( 0 ) ( 8 ) . . .
2 ! 3 ! 4 !
x x xx x!
3
2 2 4( 2 ) ( 8 )s i n 0 0 ( 0 ) . . .2 1 3 2 1 4 3 2 1
x x x x x
!
v v v v v v
2 2 41s i n . . .3
x x x!
M t i 5 2 8 1 4
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Matrices 5 2 83 4 0
1
x y z
x y z
x y z
!
!
!
5 1 2
3 4 1
1 1 1
8
0 tan
1
5 1 2 8
3 4 1 0
1 1 1 1
1 0 00 1 0
0 0 1
0 0 0
0 0 0
0 0 0
coefficient
cons t
augmented
unit
zero
!
!
!
!
!
11 12 13 8
21 22 23 0
31 32 33 1
.......2 2 det
11 12 1322 23 21 23 21 22
21 22 23 3 3det 11 12 1332 33 31 33 31 32
31 32 33
11(22 33 32 23) 12(21 33 31 23) 13(21 32 31 22)
augmented
a bad cb
c d
!
v ! !
! v !
8 12 1322 23 0 23 0 22
0 22 23 det 8 12 1332 33 1 33 1 32
1 32 33
8(22 33 32 23) 12(0 33 1 23) 13(0 32 1 22) x
x
x
x
! (
! !
! (
(! (
1 41 2 3
2 5 ,4 5 6
3 6
tA A
! !
? A ? A
1 2
2 5 6 2 3 6 251 1
2 1 5 2 6 ( 1) 6
2 2 5 3 6 (1) 25
1 3 *3 2 1 2
A B
! v ! !
! v v v !
! v v v !
v v ! v
Test 4 crip'( ) cos ( )................................ ......sin ( )f x f x f x c
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Test 4 crip
? A
? A
( )( )
( ) ( )
1
( ) ln
ln
1
1ln
1 lo gln
' csc ln csc co t
x x
f xf x
f x f x
n
n
a
e e c
aa c
f x a
a a a c
xx c
n
x cx
x cx a
y x x x c
!
!
!
!
!
!
!
2'( ) csc ( )................................ . cot ( )
'( ) csc ( ) cot ( )..................... csc ( )
'( ) csc ( ).........................
f x f x f x
f x f x f x f x c
f x f x
? A
? A
..... ln csc ( ) cot ( )
'( ) cot ( ).............................. ln csc ( )
f x f x c
f x f x f x c
( ) ( )
( ) ( )
' 0
'
( ) ' '( )
' '( )
' '( ) ln
'( )ln ( ) '
( )
'( )log ( ) '
( ) ln
sin ( ) ' '( ) cos ( )
cos ( ) ' '( ) sin ( )
ta
f x f x
f x f x
a
y k y
y ax y a
y kf x y kf x
y ke y k f x e
y ka y k f x a a
f xy k f x y k
f x
f xy k f x y k
f x a
y k f x y k f x f x
y k f x y k f x f x
y k
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! p !
! 2
2
n ( ) ' '( ) sec ( )
cot ( ) ' '( ) csc ( )
sec ( ) ' '( ) sec ( ). tan ( )
csc ( ) ' '( ) csc ( ).cot ( )
1'
2
f x y k f x f x
y k f x y k f x f x
y k f x y k f x f x f x
y k f x y k f x f x f x
y x yx
p !
! p !
! p !
! p !
! p !2 3 4
( ) ( 0 ) '( 0 ) ''( 0 ) '' '( 0 ) ''''( 0 ) . . .2 ! 3 ! 4 !
x x xf x f f x f f f!