Download Chapter 0: Indices, Surds and Logarithms, Quadratic Functions

Chapter 0: Indices, Surds and Logarithms, Quadratic Functions, Remainder and
Factor Theorems, Basic Functions and their Graphs
1. Law of Indices
br  bs  br s
b r  c r   bc 
br
 br s
bs
1
b  r
b
m
n
b  n bm 
r
r
 b
n
b   b 
r s
br  b 
 
bs  c 
s r
 brs
r
m
2. Rules of Surds
m  n  mn
m
m

n
n
3. Laws of Logarithms
log m xy  log m x  logm y
log m x k  k log m x
x
 log m x  log m y
y
1
log a b 
logb a
log m
logb x
(change of base formula)
logb a
log a a  1 and log a 1  0
log a x 
4. Roots of Quadratic Equations
b 2  4ac  0 : 2 real and distinct roots
b 2  4ac  0 : repeated roots
b 2  4ac  0 : no real roots
5. Remainder Theorem
b
When a polynomial f ( x ) is divided by (ax  b) , where a  0 , the remainder is f ( )
a
6. Factor Theorem
b
For a given polynomial f ( x ) , f ( )  0  ( ax  b) is factor of f ( x ) , where a  0
a
Chapter 1: Functions, Inverse Functions and Composite Functions
1. Vertical Line Test
A relation f is a function if and only if any vertical line x  k , where k  Domain of f,
cuts the graph of f at exactly one point.
2. Horizontal Line Test
F is one-one if and only if any horizontal line y = k, where k  Range of f, cuts the graph
of f at exactly one point. (Therefore the inverse function of f exists.)
3. Inverse Functions
D f 1  R f
R f 1  D f
4. Composite Functions
Dgf  D f
To determine if fg exists:
Rg  ...
D f  ...
Since Rg  D f , fg exists.
Chapter 2: Graphing Techniques
1. Simple Transformations
Transformation
For the equation of the Equation
image after transformation
transformation
Translation in the positive Replace y by y – a
y = f(x) + a
y-direction by a units
Translation in the negative Replace y by y + a
y = f(x) – a
y-direction by a units
Translation in the positive Replace x by x – a
y = f(x – a)
x-direction by a units
Translation in the negative Replace x by x + a
y = f(x + a)
x-direction by a units
Stretch parallel to the y-axis Replace y by y/a
y = af(x)
by factor a (with x-axis
invariant)
Stretch parallel to the x-axis Replace x by x/a
y = f(x/a)
by factor a (with y-axis
invariant)
Reflection in the x-axis
Replace y by -y
y = -f(x)
Reflection in the y-axis
Replace x by -x
y = f(-x)
after
2. Graphs of Rational Functions
[IN MF15]
ax  b
q
y
Express in the form of y  p 
cx  d
cx  d
d
Vertical asymptote: x  
c
a
Horizontal asymptote: y  p 
c
2
r
ax  bx  c
Express in the form of y  px  q 
y
dx  e
dx  e
e
Vertical asymptote: x  
d
Horizontal asymptote: y  px  q
3. Modulus Functions
y = |f(x)|
y = f(|x|)
Keep part of the graph of y = f(x) for which f(x)  0
Reflect the part of the graph of y = f(x) for which f(x) < 0 about the x-axis
Keep the part of the graph of y = f(x) for which x  0
Remove the part of the graph of y = f(x) for which x < 0
Reflect the part of the graph of y = f(x) for which x > 0 about the y-axis
4. Circle
(x – a)2 + (y – b)2 = r2, where r > 0
Mark center and radius, x and y-axis must have the same scale, note the position of the
circle with respect to the origin
5. Ellipse
( x  a ) 2 ( y  b) 2

 1 , where h,k > 0
h2
k2
Symmetrical about lines x = a and y = b
Scale of both axes should be the same
Mark center, “horizontal” and “vertical” radius
6. Parabola
(y – a)2 = k (x – b), (x – a)2 = k (y – b) where k  0
Mark line of symmetry: y = a and x = a respectively
Mark turning point
7. Hyperbola
( x  a ) 2 ( y  b) 2

 1 , where h,k > 0
h2
k2
Symmetrical about x-axis and y-axis
Mark two oblique asymptotes and intercepts
Chapter 3: Equations and Inequalities
1. Rules of Inequalities
a > b and c > 0  ac > bc
a
> 0  ab > 0
b
a > b and c < 0  ac < bc
a
< 0  ab < 0
b
Chapter 4a: Differentiation I – Techniques of Differentiation and Limits
1. Important Results and Rules
dy d
 (c )  0
dx dx
dy d
 (mx)  m
If y = mx where m is a constant, then
dx dx
dy d n
 ( x )  nx n 1
If y  x n where n is a constant, then
dx dx
d
d
d
 f ( x)  g ( x)    f ( x)    g ( x) 
dx
dx
dx
d
d
d
Product rule:
 f ( x) g ( x)   g ( x)   f ( x)    f ( x)   g ( x)  
dx
 dx

 dx

d

d

g ( x)   f ( x)    f ( x)   g ( x)  
d  f ( x) 
 dx

 dx

Quotient rule:


2
dx  g ( x) 
 g ( x) 
If y = c where c is a constant, then
Chain rule: If y is a function of u and u is a function of x, then
dy
1

dx dx
dy
2. Exponential and Logarithm Functions
d f ( x)
e   f '( x)e f ( x )

dx
d
f '( x)
 ln[ f ( x)] , f ( x)  0 
dx
f ( x)
3. Trigonometric Functions
d
 sin f ( x)   f '( x) cos  f ( x) 
dx
d
 cos f ( x)    f '( x) sin  f ( x) 
dx
d
 tan f ( x)   f '( x) sec2  f ( x) 
dx
dy dy du


dx du dx
d
 cos ecf ( x)    f '( x) cos ec  f ( x)  cot  f ( x) 
dx
d
 sec f ( x)   f '( x) s ec  f ( x)  tan  f ( x) 
dx
d
 cot f ( x)    f '( x)co s ec 2  f ( x) 
dx
*Above results hold for x measured in radians only. If x is in degrees, then convert x to
x
radian
180
4. Inverse Trigonometric Functions
d
f '( x)
sin 1 f ( x)  
, x 1

dx
1  [ f ( x)]2
d
f '( x)
cos 1 f ( x)   
, x 1

dx
1  [ f ( x)]2
d
f '( x)
tan 1 f ( x)  
,x

dx
1  [ f ( x)]2
5. Higher Order Derivatives
d n y d  d n1 y 
 

dx n dx  dx n1 
6. Implicit Differentiation
d
d
dy
 g ( y)    g ( y)  
dx
dy
dx
7. Parametric Differentiation
dy dy dt
 
If x = f(t) and y = g(t), then
dx dt dx
Chapter 4b: Differentiation II – Applications of Differentiation
1. Strictly Increasing and Strictly Decreasing
dy
0
Strictly increasing:
dx
dy
0
Strictly decreasing:
dx
2. Concave Upwards and Concave Downwards
d2y
Concave upwards:
0
dx 2
d2y
Concave downwards:
0
dx 2
3. Determining Stationary Points
x
dy
dx
a-ve
+ve
a
0
or
0
a+
-ve
a+ve
a
0
a+
-ve
a-ve
a
0
a+
+ve
+ve
Tangent to curve
or
Nature of stationary
point
d2y
If
dx 2
If
d2y
dx 2
Stationary point of
inflexion
Maximum turning
point
Minimum turning
point
 0 , then (a, f(a)) is a maximum turning point
xa
 0 , then (a, f(a)) is a minimum turning point
xa
4. Graph of f’(x)
Stationary points on f(x)  y = 0 on f’(x)
Maximum / minimum gradient on f(x)  maximum / minimum points on f’(x)
5. Graph of y 
1
f ( x)
y = f(x)
x-intercept (vertical asymptote) at x = h
y-intercept at y = k, k  0
maximum (minimum) point at (a,b), b  0
horizontal asymptote at y = q, where q  0
f(x)  
f(x) = 1
f(x) > 0 [f(x) < 0]
increasing (decreasing) in (a,b)
1
f ( x)
vertical asymptote (x-intercept) at x = h
1
y-intercept at y = y  , k  0
k
1
minimum (maximum) point at ( a, ) , b  0
b
1
horizontal asymptote at y  , where q  0
q
1
0
f ( x)
1
1
 =1
f ( x) 1
1
> 0 [f(x) < 0]
f ( x)
decreasing (increasing) in (a,b)
y
6. Graph of y 2  x
Sketch y = f(x)
Consider f(x)  0. Square root the y-values to obtain y 
Reflect y 
f ( x)
f ( x) in the x-axis to obtain y   f ( x)
7. Relationship between Gradients of Tangent and Normal to a Curve
If A (a, f(a)) is a point on the graph of y = f(x), then
 The gradient of the tangent at A is f’(a), and
1
 The gradient of the normal at A is 
, f '(a)  0
f '(a)
8. Equations of Tangents and Normals to a Curve
If A (a, f(a)) is a point on the graph of y = f(x), then
 Equation of tangent to the curve at A is
o y – f(a) = f’(a)(x – a)
 Equation of the normal to the curve at A is
1
o y – f(a) = 
(x – a), provided f '(a )  0
f '(a)
9. Connected Rates of Change
 Denote each changing quantity by a variable
 Find the equations relating the variables
 Use the chain rule to link up the derivatives
 Write down the values for the variables and rates given
 Solve for the unknown rate
Chapter 5a: Integration Techniques
1. Basic Rules of Indefinite Integral
[f(x)  g(x)] dx = f(x) dx  g(x) dx
kf(x) dx = k  f(x) dx, where k is a constant, k  0
d
 dx f ( x)dx = f(x) + c
d
 f ( x)dx  = f(x)
dx 
2. Basic Properties of Definite Integral
b
b
a
a
 f ( x)dx   F ( x)
 F (b)  F (a)
a
 f ( x)dx  0
a
b
a

f ( x)dx    f ( x)dx
b
c
a

a
b
b
f ( x)dx   f ( x)dx   f ( x )dx , where c is such that a  c  b
a
c
b
b
b
a
a
  f ( x)dx  g ( x)dx    f ( x)dx     g ( x)dx 
a
b
b
a
a
 k  f ( x)dx   k   f ( x)dx  , where k is any constant, k  0
3. Integration of Standard Functions
 f '( x)  f ( x)
n

 f ( x) 
dx 
n 1
n 1
 c, n  1
f '( x)
dx  ln f ( x)  c
f ( x)
 f '( x)e dx  e  c
 f '( x) cos  f ( x)  dx  sin  f ( x)   c
 f '( x) sin  f ( x)  dx   cos  f ( x)   c
f ( x)
f ( x)
 f '( x) sec  f ( x)  dx  tan  f ( x)   c
2
f '( x)

a  [ f ( x)]
2
a
2
2
dx  sin 1
f ( x)
 c [IN MF15]
a
f '( x)
1
f ( x)
dx  tan 1
 c [IN MF15]
2
 [ f ( x)]
a
a
4. Trigonometric Formulae
sin 2 x  cos 2 x  1
tan 2 x  1  sec 2 x
cot 2 x  1  cos ec 2 x
sin 2 x  2sin x cos x
cos 2 x  2 cos 2 x  1  1  2sin 2 x  cos 2 x  sin 2 x
5. Partial Fractions
[IN MF15]
px  q
A
B


(ax  b)(cx  d ) (ax  b) (cx  d )
px 2  qx  r
A
B
C



2
(ax  b)(cx  d )
(ax  b) (cx  d ) (cx  d ) 2
px 2  qx  r
A
Bx  C

 2 2
2
2
(ax  b)( x  c ) (ax  b) ( x  c )
1
1  xa
dx 
ln 
  c, x  a
2
a
2a  x  a 
1
1 ax
 a 2  x 2 dx  2a ln  a  x   c, x  a
x
2
6. Substitution
x2

x1
u2
f ( x)dx   F (u )
u1
dx
du
du
7. By Parts
dv
du
 u dx dx  uv  v  dx dx
Choose u based on LIATE
Chapter 5b: Applications of Integration
1. Fundamental Theorem of Calculus
b
 f ( x)dx  F (b)  F (a)
a
2. Parametric Equations
t q
b
 dx 
a ydx  t p y  dt dt
3. Volume of Solid of Revolution
Rotation about x-axis:   y 2 dx
Rotation about y-axis:   x 2 dx
Chapter 7: Arithmetic and Geometric Progression
1. Arithmetic Progression
un  a  (n  1)d
n
Sn  [2a  (n  1)d ]
2
2. Geometric Progression
un  ar n 1
a(1  r n )
Sn 
,r 1
1 r
a
S 
, r 1
1 r
Chapter 8: Summation of Series and Mathematical Induction
1. Properties of the Σ Notation
n
 n

[
af
(
r
)]

a

  f (r ) 
r m
 r m

n

n


n

 [af (r )  bg (r )]  a   f (r )   b   g (r ) 
r m
r m
r m
2. Standard Results
n(n  1)
2
n
Sum of AP:
r 
r 1
n
Sum of GP:
 rk 
k 1
r (1  r n )
1 r
n
1
r 2  n(n  1)(2n  1) *must always start from r=1 [IN MF15]

6
r 1
2
1
n

r  n 2 (n  1) 2   (n  1)  *must always start from r=1 [IN MF15]

4
2

r 1
n
3
3. Mathematical Induction
 Let Pn be the statement “…”
 Show that P1 is true
 Prove that the implication Pk  Pk 1 is true
o Assume Pk is true, prove that Pk 1 is true
o Pk 1 is true whenever Pk is true
Since P1 is true, by Mathematical Induction, Pn is true for n 


Chapter 10: Binomial Expansion
1. Binomial Expansion for Positive Integral Index
n  ,
(a  b) n  a n   1n  a n 1b   n2  a n  2b 2  ...   nn 1  ab n 1  b n ,
For
   r !(nn! r )! , r 
n
r
n

Cr  nCn r , r 
0
Cr 1  Cr 
n 1
(r  1) term:
n
r
n
n
th

0
,0  r  n

Cr , r 
 a
, 0  r  n [IN MF15]
nr
b
,1  r  n
r
2. Binomial Expansion for Rational Index
n(n  1) 2
n(n  1)...(n  r  1) r
(1  x) n  1  nx 
x  ... 
x  ... [IN MF15]
2!
r!
*Condition: |x| <1
where
Chapter 11: Power Series
1. Maclaurin’s Theorem
f ( x)  f (0)  f '(0) x 
f ''(0) x 2 f '''(0) x3
f ( n ) (0) x n

 ... 
 ... [IN MF15]
2!
3!
n!
2. Small Angle Approximations
For all small angles, positive or negative, we have
(kx)2
cos kx  1 
sin kx  kx
tan kx  kx where x is measured in radians
2
Chapter 12A: Vectors I
T
1. Vector Algebra
Triangle: PQ  QR  PR
S
R
Parallelogram: PQ  PS  PR
Polygon: PQ  QR  RS  ST  PT
ST  OT  OS
ab  ba
(a + b) + c = a + (b + c)
(   )a =  (a)   (a)
a  a
( )a =  ( a)
 (a + b) = a + b
2. Fundamental Results
Parallel vectors: b  a for some  
Unit vectors: a  a aˆ
P
Q
\{0}
Collinearity: AB   AC , with A as common point
Non-parallel and non-zero vectors: a  b for some  ,   , then     0
a  b
Ratio theorem: If OP  r , then r 
. If P is the mid-point of AB, then

1
r  (a  b) .
2
A

P
a

r
O
b
B
3. Properties of Vectors
Cartesian form: r  xi  yj  zk
 x1 
 
Matrix form:  y1 
z 
 1
 x1   x2 
   
Equality of vectors: a  b   y1    y2   x1  x2 , y1  y2 , z1  z2
z  z 
 1  2
 x1   x2   x1  x2 
    

Addition: a  b   y1    y2    y1  y2 
z  z  z z 
 1  2  1 2 
 x1   x2   x1  x2 
    

Subtraction: a  b   y1    y2    y1  y2 
z  z  z z 
 1  2  1 2 
 x1    x1 
  

Multiplication: a    y1     y1  for any  
 z  z 
 1  1
Distance between two points: AB  ( x2  x1 ) 2  ( y2  y1 ) 2  ( z2  z1 ) 2
4. Scalar Dot Product
a.b = a b cos (considering acute angle)
a.b = b.a
a.(b  c) = (a.b)  (a.c)
 (a.b) = (a).b = a.(b)
a.b  0 if 0    90 and a.b  0 if 90    180
a.a  a 2 or a  a.a
A
Perpendicular vectors: a.b  0
 x1   x2 
  
a.b   y1  .  y2   x1 x2  y1 y2  z1 z2
z  z 
 1  2
Length of projection: OP  OP  a.bˆ 
a
a.b
b
O
*Cosine rule: c = a + b - 2 ab cos  = a + b - 2a.b
2
2
2
2
2
b P
B
5. Equation of a Straight Line
 b1 
 x   a1 
 
 


Vector form: r   y    a2     b2 
 z  a 
b 
   3
 3
 x  a1  b1

Parametric form:  y  a2  b2 ,  
 z  a  b
3
3

x  a1 y  a2 z  a3


Cartesian form:
b1
b2
b3
If b1 =0, line is parallel to yz-plane.
Determining if point lies on line: equate point to equation of line and see whether able to
solve for unique value of lamda
Determining foot of perpendicular:
 Foot lies on horizontal line
 Vertical line which the point lies on DOT direction vector of horizontal line = 0
 Solve simultaneous equations
Perpendicular distance from point to line
 Find length of projection to line. Then apply Pythagoras’ Theorem
 Find foot of perpendicular from point to line. Then find distance.
Characteristics of a pair of lines
 Parallel: direction vector of one line multiple of the other line
 Not parallel but intersect: unique value of lamda and miu that satisfies the
equations of both lines when l1  l2 . Once the unique value is found, can
substitute it into the equation of one line to find the point of intersection
 Not parallel and do not intersect: no unique value
Angle between two lines: cos  
b1 .b 2
b1 b 2
Chapter 12b: Vectors II
1. Vector Product
a  b   a b sin   nˆ
a  b  (b  a)
a  b  b  a  a b sin   a b
a  b  c  a  b  a  c
  a  b    a   b  a   b 
Parallel: a b  a  b  0
Perpendicular: a  b  a  b  a b nˆ
 x1   x2   y1 z2  z1 y2 
    

Cartesian notation: a  b   y1    y2    ( x1 z2  z1 x2 ) 
z  z  x y  y x

 1  2  1 2 1 2 
1
1
*Area of triangle (sine rule): a b sin   a  b
2
2
*Area of parallelogram: a  b
A
Perpendicular distance from point to line: a  b
2. Equation of a Plane
 b1 
 c1 
 x   a1 
 
 
 


Vector form:  : r   y    a2     b2     c2 
 z  a 
b 
c 
   3
 3
 3
Scalar product form:  : r.n  p
Cartesian form:  : n1 x  n2 y  n3 z  p
a
C
Foot of perpendicular from point to plane:
 Foot lies on equation of normal of plane
 Foot lies on plane OP.n  OC.n  p
 Solve simultaneous equations
Perpendicular distance from point to plane: (OA  OC ).nˆ
b K
B
Characteristics between line and plane:
 Parallel and lies on plane: direction vector of line DOT normal of plane = 0 +
point on line fits the equation of the plane
 Parallel but does not lie on plane: direction vector of line DOT normal of plane =
0 + point on line does not fit the equation of the plane
 Not parallel  intersects at a point: direction vector of line DOT normal of plane
does not = 0
o To find point of intersection: point lies on equation of line and plane. So
equate both equations together and solve for lamda and subsequently the
point of intersection
Angle between line and plane: sin  
Angle between two planes: cos  
b.n
b n
n1 .n 2
n1 n 2
Characteristics of two planes:
 Parallel: normal vector of one plane is a multiple of normal vector of the other
plane
 Intersect to yield common line of intersection: Let z = t. Then solve simultaneous
equations of both equations of planes
Characteristics of three planes:
 Intersect at single point: unique solution for x, y and z
 Intersect along one common line: solution for x and y expressed in terms of z
 Do not intersect at common point or line: no solution found
Chapter 13a: Complex Numbers I
1.Properties of Conjugate Pairs
( z * )*  z
Important: z  z*  2 Re( z )
Important: z  z*  2 Im( z )
zz*  x 2  y 2
z  z *  z is real
2. Properties of Complex Conjugates
( z1  z2 )*  z1*  z2*
( z1  z2 )*  z1*  z2*
( z1 z2 )*  ( z1* )( z2* )
(
z1 * ( z1* )
)  *
z2
( z2 )
3. Forms of Complex Numbers
Cartesian form: z  x  iy
Polar / Trigonometric form: z  r (cos   i sin  )
z  r  x2  y 2
z  z*
  arg( z )  
z 2  zz *
arg( z* )   arg( z )
Exponential form: z  rei
arg  z1 z2   1  2  arg( z1 )  arg( z2 )
z1 z2  r1r2  z1 z2
z
z1 r1
  1
z2 r2 z2
z 
arg  1   1  2  arg( z1 )  arg( z2 )
 z2 
4. Geometrical Effect of Multiplying 2 Complex Numbers
If P represents the point with complex number z in an Argand diagram, the effect of
multiplying z by r (cos   i sin  ) or re i is to rotate the line segment of OP through an
angle of  about O in the anti-clockwise direction followed by a scaling factor r.
When you multiply a complex number by i, you are shifting it

radians about the
2
origin.
5. De Moivre’s Theorem
If z  r (cos   i sin  ) , then z n  r n (cos n  i sin n )
If z  rei , then z n  r n eni
Need to add 2k to the angle to get complex roots
Chapter 13b: Complex Numbers II
1. Circle
z a  r
Locus is a circle with center a and radius r.
Indicate intercepts if obvious, indicate radius, check if circle passes through origin
2. Perpendicular Bisector
z  a  z b
Locus is the perpendicular bisector of the line segment joining the points a and b.
Draw perpendicular bisector with solid line but draw the line adjoining both points with
dashed line.
Show the perpendicular sign and indicate that the line adjoining both points have been
segmented exactly in half.
Label intercepts after getting Cartesian equation (if question asks you to determine this)
Check whether line passes through origin
3. Half-line
arg( z  a)  
Locus is a half-line starting at (but not including) a, and making an angle of  with the
positive real axis.
Remember to indicate that the starting point is exclusive (ie. draw a circle where the halfline starts), indicate argument, use dashed lines to represent positive real axis
For Cartesian equation, must indicate the domain of x (eg. x > 0)
Check whether line passes through origin