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Additional Maths Notes (20 Oct 2014)

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Table of Contents

Ex 1.1 Simultaneous Equations ................................................. 5

Solve a Pair of Linear & Non-linear Eqns .............................. 5

Form Relation ................................................................................... 5

Ex 1.2 Sum and Product of Roots .............................................. 5

Sum & Product of Roots ................................................................ 5

Form a Quadratic Equation from its Roots ........................... 5

Useful Formulae ............................................................................... 5

Prove Identities involving Roots ............................................... 5

Ex 1.3 Discriminant ........................................................................ 5

Complete the Square ...................................................................... 5

Sketch Quadratic graphs .............................................................. 6

Discriminant & Nature of Roots/Number of x-intercepts/Number of Intersections ....................................... 6

Conditions for ax2 + bx + c to be always positive or negative ............................................................................................... 6

Ex 1.4 Quadratic Inequalities ..................................................... 6

Solve Quadratic Inequality .......................................................... 6

Solve Simultaneous Inequalities ............................................... 6

Form Quadratic Inequality from Solution ............................. 6

Ex 2.1 Surds ...................................................................................... 6

Surds Properties .............................................................................. 6

Simplify Surds ................................................................................... 7

Rationalize Denominator ............................................................. 7

Solve Surds Equation ..................................................................... 7

Method of Difference ..................................................................... 7

Ex 2.2 Indices ................................................................................... 7

Law of Indices ................................................................................... 7

Ex 2.3 Index equations .................................................................. 7

Equality of Indices .......................................................................... 7

Different Types of Manipulation ............................................... 8

Ex 2.4 Exponential Functions ..................................................... 8

Sketch Exponential Functions .................................................... 8

Ex 3.1 Polynomials and Identities ............................................. 8

Definition of Polynomials ............................................................ 8

Multiply Polynomials .................................................................... 8

Find Unknown(s) in an Identity ............................................... 8

Ex 3.2 Division of Polynomials ................................................... 8

Long Division .................................................................................... 8

Division Algorithm ......................................................................... 9

Ex 3.3 Remainder Theorem ......................................................... 9

Remainder Theorem ..................................................................... 9

Ex 3.4 Factor Theorem .................................................................. 9

Factor Theorem ............................................................................... 9

Sum/Difference of Cubes ............................................................. 9

Ex 3.5 Cubic Polynomials and Equations ............................... 9

Factorize Cubic Expressions ...................................................... 9

Form Cubic Polynomial ................................................................ 9

Ex 3.6 Partial Fractions ................................................................. 9

Break into Partial Fractions ....................................................... 9

Cover-up Rule ................................................................................. 10

Compare Coefficients .................................................................. 10

Juggling ............................................................................................. 10

Proper & Improper fraction ..................................................... 10

Ex 4.1 Modulus Functions and their Graphs ...................... 10

Modulus Definition ...................................................................... 10

Modulus Properties ..................................................................... 10

Solve Modulus Equations .......................................................... 10

Sketch y = f(|x|) ............................................................................ 10

Ex 4.2 Power Graphs ................................................................... 11

Sketch Power Graphs .................................................................. 11

Ex 5.1 Binomial Expansion of ( + ) ............................... 11

Factorial ............................................................................................ 11

Combination ................................................................................... 11

Use Pascals Triangle ................................................................... 11

Expand (1 + b)n ............................................................................ 11



Binomial Theorem Cross-applications ................................ 12

Ex 5.2 Binomial Expansion of ( + ) ................................12

Expand (a + b)n ............................................................................ 12

Use Tr+1 ............................................................................................ 12

Ex 6.1 Mid-point of a Line Segment .......................................12

Distance Formula ......................................................................... 12

Gradient ........................................................................................... 12

Find line ........................................................................................... 12

Use point on line/curve ............................................................. 13

Ratio of Diagonal Segments ..................................................... 13

Use Vectors ..................................................................................... 13

Find Intersection .......................................................................... 14

Mid-point Formula ...................................................................... 14

Ex 6.2 Parallel Lines .....................................................................14

Angle of Inclination ..................................................................... 14

Parallel Lines .................................................................................. 14

Collinearity ..................................................................................... 15

Find Parallel Line ......................................................................... 15

Ex 6.3 Perpendicular Lines .......................................................15

Perpendicular Lines .................................................................... 15

Find Perpendicular Line ............................................................ 15

Find Perpendicular Bisector .................................................... 15

Ex 6.4 Areas of Triangles and Quadrilaterals .....................15

Shoelace Formula ......................................................................... 15

Ex 7.1 Introduction to Logarithms .........................................15

Logarithm Definition .................................................................. 15

Special Log Values ........................................................................ 15

Convert between Log & Index Form .................................... 16

Ex 7.2 Laws of Logarithms ........................................................16

Laws of Logarithm ....................................................................... 16

Ex 7.3 Logarithmic Equations ..................................................16

Equality of Logarithms ............................................................... 16

Solve Log Equations .................................................................... 16

Ex 7.4 Log and Eqns of the form = ..............................16

Solve ax = b .................................................................................... 16

Solve Index Equations ................................................................ 16

Ex 7.5 Logarithmic Graphs ........................................................17

Draw Logarithmic Graphs ........................................................ 17

Ex 8.1 Reducing Equations to Linear Form ........................17

Linearize .......................................................................................... 17

Form Non-linear Equation ....................................................... 17

Equate Coordinates ...................................................................... 17

Ex 8.2 Linear Law ......................................................................... 17

Linearization ................................................................................... 17

Gradient & Y-intercept ............................................................... 17

Scale.................................................................................................... 17

Graphical Reading ........................................................................ 17

Intersection ..................................................................................... 18

Ex 9.1 Graphs of Parabolas of the Form = ............. 18

Sketch y2 = kx ............................................................................... 18

Ex 9.2 Coordinate Geometry of Circles ................................ 18

Circle Equation .............................................................................. 18

Circle Equation Cross-applications ....................................... 19

Ex 10.1 Triangle Theorems ............................................................ 20

Use Line Addition and Subtraction ........................................ 20

Angle Properties of Line(s) ...................................................... 20

Angle Properties of Triangles .................................................. 20

Congruency Tests ......................................................................... 20

Similarity Tests .............................................................................. 20

Mid-point Theorem ...................................................................... 21

Ex 10.2 Quadrilaterals Theorems ........................................ 21

Definition & Properties of Quadrilaterals ........................... 21

Prove Quadrilaterals ................................................................... 22

Ex 10.3 Circles Theorems ........................................................ 22

Angle Properties of Circle .......................................................... 22

Chord Properties of Circle ......................................................... 22

Tangent Properties of Circle .................................................... 22

Ex 11.1 Trigo Ratios of Acute Angles .................................. 22

Special Angles ................................................................................. 22

Convert between Degrees and Radians ............................... 23

Complementary s ....................................................................... 23

Supplementary s ........................................................................ 23

Identify Quadrant ......................................................................... 23

Find Basic Angle ........................................................................ 23

Find General Angle ................................................................... 23

Use .................................................................................................. 23

Ex 11.2 Trigo Ratios of any Angles ...................................... 23

Trigo Function Definition .......................................................... 23

Use in Quadrant(s) .................................................................. 23

Reciprocal Identities ................................................................... 24

Negative Angles ............................................................................. 24

ASTC Rule ......................................................................................... 24



Solve Trigo Eqn f(x) = k ........................................................... 24

Ex 11.3 Trigo Graphs .................................................................25

Solve Trigo Eqn f(x) = k by Graph ........................................ 25

Range of Sine & Cosine ............................................................... 25

Find Unknowns of Trigo Function af(bx) + c ................... 25

Sketch Trigonometric Functions ........................................... 25

Use Symmetrical/Cyclical Nature of Trigo Graphs ........ 26

Inverse Trigo Function .............................................................. 27

Ex 12.1 Simple Identities .........................................................27

Questions involving Identities ................................................ 27

Ratio Identities .............................................................................. 27

Pythagorean Identities ............................................................... 28

Square Root of Trigo Function f(x) ....................................... 28

Ex 12.2 Further Trigo Eqns .....................................................28

Simplify to Tangent Eqn ............................................................ 28

Factorize Trigo Eqn ..................................................................... 28

Solve Trigo Eqn f(ax + b) = k ................................................. 28

Ex 13.1 The Addition Formulae ............................................28

Addition Formulae ....................................................................... 28

Ex 13.2 The Double Angle Formulae ...................................29

Double Formulae ...................................................................... 29

Half Formulae ............................................................................ 29

Ex 13.3 The R-Formulae ..........................................................29

R-Formulae ..................................................................................... 29

Ex 14.1 The Derivative and its Basic Rules .......................29

Derivative as Gradient ............................................................... 29

Power Rule ...................................................................................... 30

Constant Multiple Rule .............................................................. 30

Sum/Difference Rule .................................................................. 30

Differentiation from First Principles .................................... 30

Ex 14.2 The Chain Rule .............................................................30

Chain Rule ....................................................................................... 30

Ex 14.3 The Product Rule ........................................................30

Product Rule ................................................................................... 30

Ex 14.4 The Quotient Rule ......................................................31

Quotient Rule ................................................................................. 31

Ex 15.1 Tangents and Normals..............................................31

Find Tangent .................................................................................. 31

Find Normal .................................................................................... 31

Tangent Properties ...................................................................... 31

Normal Properties ....................................................................... 31

Ex 15.2 Increasing and Decreasing Functions ................ 31

Increasing/Decreasing function ............................................. 31

Ex 15.3 Rates of Change ........................................................... 31

Rate of Change ............................................................................... 31

Quantity & Constant Rate .......................................................... 31

Ex 15.4 Connected Rates of Change .................................... 32

Connected Rates of Change ...................................................... 32

Ex 16.1 Nature of Stationary Points .................................... 32

Stationary Point/Value ............................................................... 32

1st Derivative Test ....................................................................... 32

2nd Derivative Test ...................................................................... 32

Ex 16.2 Maxima and Minima .................................................. 32

Maxima/Minima ............................................................................ 32

Ex 17.1 Derivatives of Trigo Functions .............................. 32

Derivatives of Trigonometric Functions ............................. 32

Ex 17.2 Derivatives of Exponential Functions ................ 32

Derivatives of Exponential Functions .................................. 32

Ex 17.3 Derivatives of Log Functions ................................. 33

Derivatives of Log functions ..................................................... 33

Use Logarithmic Differentiation ............................................. 33

Ex 18.1 Indefinite Integrals .................................................... 34

Integral Rules ................................................................................. 34

Find Integral from Derivative .................................................. 34

Find Curve from Derivative ...................................................... 34

Integrals of Power Functions ................................................... 34

Ex 18.2 Definite Integrals ........................................................ 35

Definite Integrals .......................................................................... 35

Definite Integrals Rules .............................................................. 35

Integrals of Modulus Functions .............................................. 35

Ex 18.3 Integrals of Trigo Functions ................................... 35

Integrals of Trigonometric Functions .................................. 35

Ex 18.4 Integrals of Exponential Fns & 1/x ..................... 35

Integrals of Exponential Functions ....................................... 35

Integrals of 1/x & 1/(ax + b) ................................................... 35

Ex 19.1 Area by Integration ................................................... 36

Area by integration ...................................................................... 36

Ex 19.2 Area bounded by Curves ......................................... 36

Strategies to find area bounded by curves ......................... 36

Ex 20.1 Kinematics .................................................................... 36

Kinematics Relation ..................................................................... 36

Implications of Kinematics Statements ............................... 37



Distance ............................................................................................ 37

Appendix 1 Geometric Formulae .............................................38

2D Shapes ........................................................................................ 38

3D Shapes ........................................................................................ 38

Appendix 2 Trigonometric Identities .....................................39

Appendix 3 Calculus Formulae .................................................40

Differentiation ............................................................................... 40

Integration ...................................................................................... 40

Additional Math Notes (20 Oct 2014)



Step 1: Subject variable in linear eqn Step 2: Substitute it into non-linear eqn

Step 1: Assign variables

Step 2: Form relation between variables

Step 1: Simplify to ax2 + bx + c = 0

Step 2: State roots

Step 3: Find SOR/POR

Sum of roots = + = b

a

Product of roots = =c

a

Applications Evaluate expressions involving its roots

e.g. find 2

+

2

Given context e.g. the heights of two men satisfy 40x2 138x + 119 = 0. Without solving the

equation, find the average height of these two

men. Average height =+

2

Find roots and unknowns e.g. the equation x2 4x + c = 0 has roots which

differ by 2. Find the value of each root and c.

To prove existence of positive & negative root, use

< 0

Convert to quadratic equation in y by substitution

e.g. x2

3 2x1

3 + 3 = 0 has roots &

sub y = x1

3:

y2 2y + 3 = 0 has roots 1

3 & 1

3

Step 1: State roots

Step 2: Find SOR/POR

Step 3: Form equation

x2 (SOR)x + (POR) = 0

2 + 2 = ( + )2 2

= ( )2

( )2 = ( + )2 4

4 + 4 = (2 + 2)2 2()2

To form useful equations (to be substituted),

(i) is a root of ax2 + bx + c = 0, a2 + b + c = 0 (1)

(ii) Multiply n to (1)

(iii) Apply power of n to (1)

Given that is a root of the equation x2 = x 3, show that (i) 3 + 2 + 3 = 0 (ii) 4 + 52 + 9 = 0

(i) is root, 2 = 3 (1)

(1) : 3 = 2 3 (2) sub (1) into (2): 3 = ( 3) 3 3 = 2 3 3 + 2 + 3 = 0 (shown)

(ii) (1)2: 4 = ( 3)2

4 = 2 6 + 9 4 = 2 6(2 + 3) + 9 [use (1) to make the subject] 4 = 2 62 18 + 9 4 = 52 + 9 4 + 5 9 = 0 (shown)

x2 + kx = (x +k

2)2 (

k

2)2

Solve a Pair of Linear & Non-linear Eqns

Ex 1.1 Simultaneous Equations

Form Relation

Sum & Product of Roots

Ex 1.2 Sum and Product of Roots

Form a Quadratic Equation from its Roots

Useful Formulae

Solution

Question

Prove Identities involving Roots

Complete the Square

Ex 1.3 Discriminant




Step 1: Express as y = a(x h)2 + k

Step 2: Obtain turning point (h, k)

Step 3: Determine or shape from a

Step 4: Sub x = 0 to find y-intercept

Step 5: Sub y = 0 to find x-intercept Note: x = h is the line of symmetry e.g.

To find x1, 2+x1

2= 5 x1 = 8

Step 1: Simplify to ax2 + bx + c = 0

(by substituting line into curve)

Step 2: Use relation between b2 4ac & nature of roots/x-intercepts/intersections

Discriminant Nature of roots

No. of x-intercepts/ intersections

b2 4ac > 0 2 distinct 2 b2 4ac = 0 2 equal 1 (tangent) b2 4ac 0 2 1 or 2 (meet) b2 4ac < 0 0 0

ax2 + bx + c > 0 for all x a > 0, b2 4ac < 0

ax2 + bx + c < 0 for all x a < 0, b2 4ac < 0 Step 1: Simplify inequality to ax2 + bx + c vs 0

Step 2: Use 2 conditions (i) a > 0 or a < 0 (ii) b2 4ac < 0

Step 1: Simplify to ax2 + bx + c vs 0, a > 0 Step 2: Factorize Step 3: Draw sign diagram (Arrange roots & alternate signs with + at left)

Step 4: Find range of x

f(x) < g(x) < h(x) f(x) < g(x) and g(x) < h(x)

Step 1: Split into 2 inequalities using and

Step 2: Solve each inequality

Step 3: Take intersection of both solutions

x1 < x < x2 k(x x1)(x x2) < 0

x < x1 or x > x2 k(x x1)(x x2) > 0

For a > 0 and b > 0,

a b = ab

a

b =

a

b

a a = a Notation

For xn ,

n index

x radicand

radical sign or radix or root symbol

xn

surd (if xn is irrational)

Note: For xn and x < 0,

Even n results in non-real number

Odd n results in real number

e.g. 4 does not exist but 83

exists

2 (5, 3)

1

= 2 + +

Sketch Quadratic graphs

Discriminant & Nature of Roots/Number of x-intercepts/Number of Intersections

Conditions for ax2 + bx + c to be always positive or negative

1 2 + +

Solve Quadratic Inequality

Ex 1.4 Quadratic Inequalities

Solve Simultaneous Inequalities

Form Quadratic Inequality from Solution

Surds Properties

Ex 2.1 Surds




Factor out largest square number

e.g. 45 = 9 5 = 35 Prime factorize (for more challenging numbers)

e.g. 540 = 22 33 5

= 2 31.5 5

= 2 33 5

= 615

1

a

a

a=

a

a

1

ah + bk

1

ah bk=

1

a2h b2k

Square both sides

a = b a = b2

Equate rational & irrational terms

a + bk = c + dk a = c, b = d

Note: Check the answer mentally by substituting it into

the original equation.

e.g. 6 5x = x 6 5x = x2 x2 + 5x 6 = 0 (x + 6)(x 1) = 0 x = 6 or x = 1 (rej) When x = 1,

LHS = 6 5 = 1 RHS = 1 LHS RHS

If you cannot simplify to a = b or a + bk = c + dk, consider solving surds equation by substitution

e.g. 2x + 3x + 1 = 0

sub u = x:

2u2 + 3u + 1 = 0

Step 1: Break each term into partial sums

Step 2: Arrange partial sums vertically

Step 3: Cancel diagonally

a0 = 1

an =1

an

am

n = (an )

m= am

n

(am)n = amn

When you multiply/divide terms, identify common base/power

e.g. 313309

13

2723

(common base is 3)

e.g. (a3 + b2 + b3

) (a3 + b2 b3

)

(common power is 1

3)

When you add/subtract terms, identify highest common factor e.g. 8x+2 34(23x)

= (23)x+2 2 17(23x)

= 23x+6 17(23x+1) (HCF is 23x+1)

= 23x+1(25 17)

= 23x+1(15)

Note: Equations involving even power functions may have multiple solutions

e.g. x4 = 16 x = 2 or x = 2

ax = an, for a > 0, a 1 x = n

Simplify Surds

Rationalize Denominator

Solve Surds Equation

Method of Difference

am an = am+n

am

an = amn

an bn = (ab)n

an

bn = (

a

b)n

Same Base

Same Power

Law of Indices

Ex 2.2 Indices

Equality of Indices

Ex 2.3 Index equations




Manipulate Key Words Simplify Complex to simple Express In terms of Evaluate Find numerical value Show Work towards distinct characteristic Solve Equation Given Consider rearranging given equation.

y = ax, a > 1 (slopes up)

y = ax, 0 < a < 1 (slopes down)

Note: y = ax 0 for 0 < a < 1

Polynomials must have

non-negative power

integer power

Expand using strategic alignment

Find coefficient using selective multiplication

Substitute Compare coefficients Tip: Sub values of x that makes a factor zero e.g. a(x 2) + b = 5 3x

sub x = 2: a(0) + b = 5 3(2) b = 1

compare x: a = 3

Step 1: Surface out hidden terms and express polynomial

in powers of decreasing integers

Repeat step 2-5 until Deg(R) < Deg(divisor)

Step 2: Divide

Step 3: Multiply

Step 4: Subtract

Step 5: Bring down e.g. (3x2 2x + 5) (x + 2)

Different Types of Manipulation

1

1

Sketch Exponential Functions

Ex 2.4 Exponential Functions

Definition of Polynomials

Ex 3.1 Polynomials and Identities

e.g. Find coefficient of x2 in (x2 + x + 1)(x2 + 2x + 3)

e.g. (x + 1)(x2 + x + 1) = x3 +x2 +x +x2 +x +1

Multiply Polynomials

Find Unknown(s) in an Identity

Long Division

Ex 3.2 Division of Polynomials




Quotient Q(x) Divisor g(x) Dividend f(x) Remainder R(x)

Dividend = Divisor Quotient + Remainder

Dividend

Divisor = Quotient +

Remainder

Divisor

Deg(Dividend) = Deg(Divisor) + Deg(Quotient)

Deg(Remainder) < Deg(Divisor)

For quadratic divisor, f(x) (px2 + qx + r) R(x) = ax + b f(x) = (px2 + qx + r)Q(x) + ax + b

Polynomial f(x) (ax + b) R = f (b

a)

Tip: Insert value of x that makes the divisor zero Note: If polynomial is not given, use division algorithm

Polynomial f(x) has factor (ax + b) f (b

a) = 0

Tip: Insert value of x that makes the factor zero

a3 b3 = (a b)(a2 ab + b2)

Step 1: Guess factor (factor thm)

Step 2: Compare x3

Step 3: Compare x0

Step 4: Compare x2

Step 5: Compare x (optional)

Given 1 distinct root x1,

f(x) = k(x x1)3

Given 2 distinct roots x1 and x2,

f(x) = k(x x1)2(x x2) or k(x x1)(x x2)

2 Given 3 distinct roots x1, x2 and x3,

f(x) = k(x x1)(x x2)(x x3)

Step 1: Convert to proper fraction

Step 2: Factorize denominator

Step 3: Break into partial fraction forms

Step 4: Solve for unknowns Cover-up rule Substitution Compare coefficients

Denominator Form

ax + b A

ax+b

(ax + b)2 A

ax+b+

B

(ax+b)2

x2 + c2 Ax+B

x2+c2

Division Algorithm

Remainder Theorem

Ex 3.3 Remainder Theorem

Factor Theorem

Ex 3.4 Factor Theorem

Sum/Difference of Cubes

(x )

(x )(px2 + + )

(x )(px2 + + r)

(x )(px2 + qx + r)

(x )(px2 + qx + r)

Factorize Cubic Expressions

Ex 3.5 Cubic Polynomials and Equations

Form Cubic Polynomial

Break into Partial Fractions

Ex 3.6 Partial Fractions




To solve for unknown with linear factors,

Step 1: Insert root Step 2: Cover up linear factor (that becomes zero) Step 3: Equate unknown (highest power)

e.g. f(x)

(xx1)(xx2)2=

A

(xx1)+

B

(xx2)+

C

(xx2)2

insert x1, cover up (x x1) and equate A

x = x1: f(x)

( )(xx2)2|x=x1

= A

insert x2, cover up (x x2) and equate C

x = x2: f(x)

(xx1)( )2|x=x2

= C

Step 1: Clear fractions by multiplying denominator

Step 2: Simplify to polynomial of descending power

Step 3: Compare coefficients

e.g. x2+2x+15

x(x2+3) =

5

x+

Bx+C

x2+3

x2 + 2x + 15 = 5(x2 + 3) +(Bx + C)x

= 5x2 + 15 +Bx2 + Cx

= (5 + B)x2 + Cx + 15

Compare coefficients: x2: 1 = 5 + B B = 4 x: C = 2

x2+2x+15

x(x2+3)=

5

x+

4x+2

x2+3

Step 1: Copy denominator to numerator Step 2: Multiply to match term with highest power Step 3: Add to balance

e.g.

4x2 + 3

x2 2

(x2 2)

x2 2 copy (2 2) to the to numerator

4(x2 2)

x2 2

Multiply 4 to match term with highest power

=4(x2 2) + 11

x2 2 add 11 to numerator to balance

= 4 +11

x2 2

Divide each term in numerator by denominator

Proper Fraction: Deg(Numerator) < Deg(Denominator) Improper Fraction: Deg(Numerator) Deg(Denominator)

|x| = {x x 0

x x < 0

Tip: Use given condition to determine if |f(x)| = f(x) or f(x)

e.g. Given a > 2, simplify |3 2a|

a > 2

2a < 4

3 2a < 1

< 0

|3 2a| = (3 2a) = 2a 3

|a| = |a|

|ab| = |a||b|

|a

b| =

|a|

|b|

|an| = |a|n

|a|2 = |a2| = a2

Tip: differ is the trigger word to use modulus

e.g. A differs from B by 10 |A B| = 10

|f(x)| = g(x) f(x) = g(x) or f(x) = g(x) Note: Check the answer by mentally substituting it into the original equation. Tip: Consider squaring both sides to remove mod |f(x)| = k

[f(x)]2 = k2

Step 1: Sketch y = f(x)

Step 2: Reflect negative part of f(x) in x-axis

Cover-up Rule

Compare Coefficients

Juggling

Proper & Improper fraction

Modulus Definition

Ex 4.1 Modulus Functions and their Graphs

Modulus Properties

Solve Modulus Equations

Sketch y = |f(x)|




For y = axn, a > 0

Integer

Integer Negative Positive

Even Odd Even Odd ( 1)

y =1

x2 y =

1

x y = x2 y = x3

Rational

Rational n < 0

( at rate)

0 < n < 1

( at rate)

n > 1

( at rate)

n! = n (n 1) 2 1

= n (n 1)!

0! = 1

To divide between factorials,

Step 1: Expand bigger factorial till smaller factorial

Step 2: Strike out smaller factorials

e.g. 7!

5!=

765!

5!= 7 6 = 42

(n+1)!

(n2)!=

(n+1)n(n1)(n2)!

(n2)!= (n + 1)n(n 1)

(nr) =

n!

(nr)!r!

(n0) = 1

(n1) = n

(n2) =

n(n1)

2

(n3) =

n(n1)(n2)

3!

(n4) =

n(n1)(n2)(n3)

4!

To create table,

n Binomial coefficients 0 1 1 1 1 2 1 2 1

Step 1: Insert 1 at the sides

Step 2: Form numbers inside triangle by adding the 2 numbers above it

Step 3: Use formula

(1 + b)n = (1st coeff)b0 +(2nd coeff)b1 ++ (last coeff)bn

(1 + b)n = (n0) b0 +(

n1) b1 ++ (

nr) br ++ (

nn) bn

= 1 +nb ++ (nr) br ++ bn

Sketch Power Graphs

Ex 4.2 Power Graphs

Factorial

Ex 5.1 Binomial Expansion of (1 + b)n

Combination

Use Pascals Triangle

Expand (1 + b)n




App 1: Multiply selectively

App 2: Substitute value/terms

App 3: Compare coefficients

(i) Expand (2 x) (1 +1

2x)

8 in ascending powers of x, as

far as the term in x3.

(ii) Hence estimate the value of 1.9 (1.05)8

Multiply selectively

(i) (2 x) (1 +1

2x)

8

= 2 +8x +14x2 +14x3

x 4x2 7x3 +

= 2 +7x +10x2 +7x3 + Substitute values (ii) 1.9 (1.05)8 = (2 0.1) (1 + 0.5)8

= [2 (0.1)] [1 +1

2(0.1)]

8

= 2 +7(0.1) +10(0.1)2 +7(0.1)3 + [sub x = 0.1 into (i)]

2.807

The first three terms in the expansion, in ascending powers of x of (1 + 2x)n are 1 + 16x + ax2. Find n and a.

Compare coefficients (1 + 2x)n = 1 + 2nx + 2n(n 1)x2 + (by Binomial Thm) 1 + 16x + ax2 (given) Compare x: 2n = 16 n = 8

Compare x2 2n(n 1) = a Sub n = 8: 2(8)(8 1) = a a = 112

(a + b)n = (n0) an0b0 +(

n1) an1b1 ++ (

nn) annbn

= 1 +nan1b ++ bn

Tr+1 = (nr) anrbr

To find particular term,

Step 1: Simplify to (a + b)n

Step 2: Use Tr+1 Pull out x

Step 3: Find r Equate power

middle term r =n

2

constant power = 0

Step 4: Insert r into Tr+1

(x1 x2)2 + (y1 y2)2

m =y1 y2x1 x2

If y-intercept is not given,

Step 1: Find point

Step 2: Find gradient

Step 3: Find line y y1 = m(x x1)

If y-intercept is given,

Step 1: State y-intercept

Step 2: Find gradient

Step 3: Find line y = mx + c

Note: For m = 0, horizontal line y = c For m , vertical line x = a

Solution

Question

= (2 x)(1 +4x + 7x2 + 7x3 + ) [by Binomial Thm]

Solution

Question

Binomial Theorem Cross-applications

Expand (a + b)n

Ex 5.2 Binomial Expansion of (a + b)n

Use Tr+1

Distance Formula

Ex 6.1 Mid-point of a Line Segment

Gradient

Find line




(x1, y1) lies on y = f(x)

y1 = f(x1) (form eqn)

(x1, f(x1)) (express coordinates in only 1 variable)

e.g. (2,1) lies on y = kx 2 1 = k(2) 2

2k = 3

k =3

2

e.g. A(a1, a2) lies on y = 2x

A(a1, 2a1)

By similar triangles, A: B (diagonal) = C:D (horizontal) = E: F (vertical)

AB = AO + OB

= OB OA

Given A is (0,9) & C is (6,3) and AB: BC = 2: 1, find the coordinates of B.

OB = OA +AB

= OA +2

2+1AC

= OA +2

3(OC OA )

= OA +2

3OC

2

3OA

=1

3OA +

2

3OC

=1

3(09) +

2

3(63)

= (45)

B(4,5)

A is (0,6), B is (2, 2) and D is (2, 2). AB is parallel to DC and AB: DC = 1: 2. Find the coordinates of C

OC = OD +DC

= OD +2AB

= OD +2(OB OA )

= OD +2 [(88) (

06)]

= OD +2(82)

= (2

2) +(

164

)

= (182

)

C is (18,2)

Use point on line/curve

Ratio of Diagonal Segments

Solution

Question

Solution

Question

Use Vectors

A

B

C E

F

D

(6,3)

(0,9)

2

1

(0,6)

(2,2)

(8,8)




A is (2,4) and B is (6,10). ACB and MDB are 90. AC:MD = 3: 1. Find the coordinates of M

ACB ~ MDB. AB

MB =

AC

MD (corr. sides or ~ s)

=3

1

= 3

AB = 3MB

AM:MB = 2: 1

OM = OA +AM

= OA +2

3AB

= OA +2

3(OB OA )

=1

3OA +

2

3OB

=1

3(33) +

2

3(69)

= (57)

M is (5,7)

Solve a pair of equations

M = (x1 + x2

2,y1 + y2

2)

To find endpoint,

Step 1: Denote endpoint

Step 2: M = (x1+x2

2,y1+y2

2)

Step 3: Equate coordinates To find curve traced by mid-point,

Step 1: Find midpoint M

Step 2: Let M = (x, y)

Step 3: Equate coordinates

Step 4: Connect x & y

Shapes Implications Parallelogram ABCD

MAC = MBD

iso. ABC with AB = AC

MBC = Foot of from A to BC

Circle with diameter AB

MAB = Centre

A is (2,4) and B is (6,10). AC:MD = 2: 1. Given the diagram below, find the coordinates of M.

M = MAB = (2+6

2,4+10

2) = (4,7)

m = tan (wrt positive x axis)

l1 l2 m1 = m2

Solution

Question

Find Intersection

A B

C D M

A

B C

A B

Solution

2

1

(2,4)

(6,10)

Question

Mid-point Formula

Angle of Inclination

Ex 6.2 Parallel Lines

Parallel Lines

3

1

(3,3)

(6,9)




If A, B & C are collinear, mAB = mAC (any two line segments)

Step 1: Find point/y-intercept

Step 2: Find gradient (m1 = m2)


y = mx + c

l1 l2 m1. m2 = 1

m1 =1

m2

Shapes Implications Rhombus ABCD

mAC mBD

P is equidistant from A and B

AB intersects P

Step 1: Find point/y-intercept

Step 2: Find gradient (m1 =1

m2)


y = mx + c

Step 1: Find mid-point (MAB)

Step 2: Find gradient (1

mAB)

Step 3: Find bisector (AB)

y y1 =1

mAB(x x1)

Tip: To find the line equidistant to points A & B, find the perpendicular bisector of AB If the 2 points have the same x or y-coordinate, bisector = average of the other coordinates

Area of polygon

=1

2|x1 x2 xn x1y1 y2 yn y1

|

=1

2[(sum of products )

(sum of product )] Coordinates should be in anti-clockwise order to have

positive output. On the contrary, if coordinates are in

clockwise order, the output is negative.

Use modulus if unsure anti-clockwise or clockwise

Zero area implies points are collinear

App 1: To find angle, use area =1

2ab sin C

App 2: To find height ( distance from point to line),

use Area =1

2(base)(height)

A logarithm must have

(i) base > 0

(ii) base 1

(iii) arg > 0

loga a = 1 Same base & argument results in output of 1

loga 1 = 0 argument of 1 results in output of 0

Collinearity

Find Parallel Line

A B

C D

P A

B

Perpendicular Lines

Ex 6.3 Perpendicular Lines

Find Perpendicular Line

Find Perpendicular Bisector

Area of quadrilateral 1

2|x1 x2 x3 x4 x1y1 y2 y3 y4 y1

|

=1

2(x1y2 + x2y3 + x3y4 + x4y1

x2y1 x3y2 x4y3 x1y4)

Area of triangle

=1

2|x1 x2 x3 x1y1 y2 y3 y1

|

=1

2(x1y2 + x2y3 + x3y1

x2y1 x3y2 x1y3)

Shoelace Formula

Ex 6.4 Areas of Triangles and Quadrilaterals

Logarithm Definition

Ex 7.1 Introduction to Logarithms

Special Log Values




x = loga y y = a

x

Step 1: Identify base

Step 2: Connect base to opp. side of eqn

Step 3: Switch form keeping base

e.g. Log Index

Index Log

loga x = loga n x = n

Step 1: Use laws of log

Step 2: Remove log

Equality of log Change to index form

Step 3: Check log conditions

Note: Use substitution u = loga x if you cannot simplify to log =

To solve ax = b, log both sides

Method 1: ax = an x = n

Method 2: ax = b (log both sides)

Method 3: Convert to log form

Method 4: Substitution Step 1: Use laws of indices to simplify to ax = an or b

When multiply/divide terms, identify common base/power

When add/subtract terms/identify highest common factor

Step 2: Remove base

Equality of indices (ax = an x = n) Log both sides (ax = b) Convert to log form

If you cannot simplify to ax = an or b, use substitution u = ax

e.g. 9(3x)2 + 1 = 10(3x)

e.g. x3

2 8x3

2 = 7

ax = y x = loga y

Base is a. Connect base a to y

Switch from index to log form. Keep the base a, therefore argument is x.

loga y = x y = ax

Base is a. Connect base a to x

Switch from log to index form. Keep the base a, therefore power is x.

Convert between Log & Index Form

Product Law loga xy = loga x + loga y

Quotient Law logax

y = loga x loga y

Power Law loga xr = r loga x

Change-of-Base Law

loga b =logc b

logc a =

1

logb a

Laws of Logarithm

Ex 7.2 Laws of Logarithms

Equality of Logarithms

Ex 7.3 Logarithmic Equations

Laws of log Action

Change-of-base law convert to common base

Power law move coefficient to power

Product law/ Quotient law

combine to single log

Solve Log Equations

Solve ax = b

Ex 7.4 Log and Eqns of the form ax = b

Solve Index Equations




y = loga x, a > 1 (slopes up)

y = loga x, 0 < a < 1 (slopes down)

Note: For base > 1, there is an inverse relation between base & rate of increase For a > b > c > 1,

Contains x & y

Contains constants

X & Y m & c

e.g. if ax2 + by3 = 1, then y3 = a

bx2 +

1

b

i.e. Y = y3, X = x2, m = a

b, c =

1

b

e.g. if y = eb+x

a , then ln y =1

ax +

b

a

i.e. Y = ln y , X = x,m =1

a, c =

b

a

To find unknowns,

Step 1: Linearize to axes variables

Step 2: Equate gradient & Y-intercept or use points on line

(whichever is given)

Step 1: Find point/gradient/ Y-intercept

Step 2: Form linear equation Y = mX + c (If Y-intercept is given) Y Y1 = m(X X1) (If Y-intercept is not given)

Step 3: Form non-linear eqn by replacing X & Y with axes variables

The first and second coordinates are not necessarily x and y respectively!

If C(9, 8) lies on the graph of yx against x, find the value of y corresponding to c.

Equate 1st coordinate: x = 9 Equate 2nd coordinate: yx = 8

y9 = 8

y = 8

3

Step 1: Simplify to Y = mX + c Step 2: Complete table

Step 1: State 2 points:

(i) On y-axis (ii) Halfway-down

Step 2: Equate gradient & Y-intercept

Step 1: Estimate Y-intercept

Y1 = mX1 + c c = Y1 mX1

Step 2: State domain & range Step 3: Find X & Y interval

X-Interval =XlastX1st

10

Y-Interval =YlastY1st

12

(Round down to 1, 2, 25 or 5) Step 4: State X & Y scale

Step 1: Simplify to (X or Y)

Step 2: Identify point

Step 3: Equate (Y or X) & solve for desired variable Note: Graphical reading is reliable only within the data range (interpolation) & not reliable outside the data range (extrapolation)

1

1

= log = log

= log

1

Draw Logarithmic Graphs

Ex 7.5 Logarithmic Graphs

Linearize

Ex 8.1 Reducing Equations to Linear Form

Form Non-linear Equation

Solution

Question

Equate Coordinates

Linearization

Ex 8.2 Linear Law

Gradient & Y-intercept

Scale

Graphical Reading




Step 1: Work towards 2 curves on each side

Step 2: Plot 2nd curve & use intersection

y2 = kx, k > 0

Note: y = 0 is the line of symmetry

Given the graph y2 = 2x, draw a suitable line to solve x2 8x + 9 = 0.

x2 8x + 9 = 0 x2 6x + 9 = 2x (x 3)2 = 2x y = x 3 or y = (x 3)

Standard form (x a)2 + (y b)2 = r2

General form x2 + y2 + 2gx + 2fy + c = 0

Centre (a, b) = (g,f)

Radius r = g2 + f2 c

Note: It appears to be a counter-intuitive convention that g comes before f in the formula

Trigger/Setup Action

2 points Find bisector of chord where centre lies on

Centre & point Use distance formula to find radius.

Diameter Use midpoint formula

Touches horizontal/vertical line

Sketch graph. Deduce coordinates, centre, radius or point on circle. (see example)

Right angle triangle drawn

Use Pythagoras Theorem

0,1 or 2 intersections

Use discriminant.

Touches another circle

Connect centres with a line.

Line is tangent to circle

Identify right angle (tan rad) Find normal.

If the centre cannot be found from the approaches above (or only 1 coordinate can be deduced), use given information about centre (if any)

e.g. centre C(h, k) lies on line y = f(x) C is (h, f(h))

e.g. centre C(h, k) is 6 units away from point A(1,2)

(h 1)2 + (k 2)2 = 6 insert parameters into (x h)2 + (y k)2 = r2 and

solve for unknowns by elimination.

Intersection

Solution

Question

Sketch y2 = kx

Ex 9.1 Graphs of Parabolas of the Form y2 = kx

Circle Equation

Ex 9.2 Coordinate Geometry of Circles




Examples of sketching graph to deduce information

Touch axis

Given: centre (3, 2), touches x-axis

Deduce: radius = 2

Cut axis

Given: Cuts y-axis at 2 and 5

Deduce: y coordinate of centre

=1+(5)

2= 3

Touch line(s)

Given: Touches x = 2 & x = 8

Deduce: radius =82

2= 3

x coordinate of centre

=2+8

2= 5

Pythagoras Theorem

Find length of PT, given radius is 13.

Idea: Find PC by distance formula

PT = PC2 CT2 (Pythagoras thm) Find AC

Idea: AC = r2 AB2 (Pythagoras thm)

Solve System of Equations

To find circle equation given 3 points on the circle,

insert the points into general form of circle

Complete the Square

Convert general form to standard form

Use Discriminant Find number of intersections between line & circle

(you can also compare the perpendicular distance

with the radius to determine the number of points of

intersection)

Find unknown c in line eqn given line is tangent to

circle

Find Intersection Point

Find point on circle

Find point of contact between tangent & normal

Find centre where line through centre meets

perpendicular bisector of chord

Use Distance Formula Find radius

To check if point A lies within circle, compare distance between A and centre with the radius

Use Midpoint Formula

Given that A(2,3) and B(4,5) are points on the circle,

the find the centre.

Given A is (2,3), the centre C is (4,5) and AB is the

diameter of the circle, find the point B

Find line Find tangent/normal at point of contact

e.g. Find AB

Find bisector

Whenever two points on circle are given, consider

finding the perpendicular bisector. The perpendicular

bisector of the chord passes through the centre of the

circle

Use Properties of Circle (refer to Ex 10.3)

bisector of chord

tan rad

(3, 2)

2

(1, 3)

1

5

= 2 = 8

3

(5, 2)

A C 2

6 r = 5

B

P(9,2)

T

C(2, 1)

Circle Equation Cross-applications

A B

P(3,10) 210

C(1, 4) AB




AB + BC = AC

AC AB = BC

Given AB = CD, prove AC = BD

AB = CD AB + (BC) = CD + (BC) AC = BD

s in line opp. int. corr. alt. Prove straight lines by s in line = 180 Prove parallel lines by int., corr. & alt.

s in = 180

ext. = sum of

int. opp. s

iso. eq.

Prove equal sides/angles using iso. & eq.

SSS SAS AAS RHS 3 eq. sides

2 eq. sides, 1 included

2 eq. s, 1 corr. sides

1 rt , 1 eq. hyp, 1 eq. side

Note: Order of Points matter e.g. ABC XYZ is not the same as

ACB XYZ Prove equal sides/angles using congruent s

SSS SAS AA

3 sides 2 sides, 1 included

2 eq.

Given ABC ~ DEF, prove that AB DF = AC DE

Given ABC ~ DEF, prove that AB DF = AC DE

Whenever you encounter product of multiple line segments, consider using the property of similar triangles: ratios of corresponding sides are equal.

AB

AC

Identify which line segments in the above product correspond to the triangle ABC. AB and AC.

Take ratio AB

AC at the left. Note the

sequence. AB is 12 and AC is 13. 12 over 13.

AB

AC =

DE

DF

Use same sequence on the other triangle DEF at the right. 12 is

DE and 13 is DF. Take ratio DE

DF at

the right.

AB DF = AC DE [proven]

Cross multiply.

Given ABC ~ DEF & DE: EF = 1: 2, prove that

AB =1

2BC (or AB: BC = 1: 2)

AB

BC =

DE

EF

=1

2

AB =1

2BC

AB: BC = 1: 2

Solution

A B C D

Question

A B C

Use Line Addition and Subtraction

Ex 10.1 Triangle Theorems

a b a b

a b a

b a b

Angle Properties of Line(s)

a b c a b c a b a b a b

c

Angle Properties of Triangles

Congruency Tests

Solution

Question (Prove relation/ratio of line segments)

Solution

Question (prove product of sides)

Similarity Tests




Given ABC ~ EDC, BC: CD = 1: 2 & area of ABC = x, find the area of DEC

Area of DEC = (2

1)2x = 4x [use

A1

A2= (

l1

l2)2]

D = MAB, E = MAC

DE BC, DE =1

2BC

Kite

Quad. with two pairs of equal adjacent sides

s between unequal sides are equal (angle)

One diagonal bisects the other (diagonal)

Longer diagonal bisects s (diagonal)

Diagonals are (diagonal)

Note: Concave kite have interior s > 180 Trapezium

Quad. with exactly one pair of parallel sides

supplementary interior s Parallelogram

Quad. with two pairs of parallel sides

Opp. sides are equal (side)

Opp. s are equal (angle)

interior s are supplementary (angle)

Diagonals bisect each other (diagonal) Rectangle

Quad. with four right angles Opp. sides are parallel (side)

Opp. sides are equal (side)

Diagonals bisect each other (diagonal)

Diagonals are equal (diagonal) Rhombus

Quad. with four equal sides

Opp. sides are parallel (side)

Supplementary interior s (angle)

Diagonals bisect s (diagonal)

Diagonals are bisector of each other (diagonal) Square

Quad. with four equal sides & four right angles

Diagonals bisect angles (diagonal)

Diagonals are equal (diagonal)

Diagonals are bisector of each other (diagonal)

Solution

Question (Use ratio of area of similar triangles)

A D

B C

E

Mid-point Theorem

Definition & Properties of Quadrilaterals

Ex 10.2 Quadrilaterals Theorems

A

B C

D

E 1

2




Parallelogram

2 pairs of sides (definition)

2 pairs of equal & opp. sides (side)

1 pair of equal & sides (side)

2 pairs of equal opp. s (angle)

Diagonals bisect each other (diagonal) Rectangle

4 right s (definition)

Parallelogram + 1 right (angle) Rhombus

4 equal sides (definition)

Parallelogram + eq. adj. sides (side)

Parallelogram + bisecting diagonals (diagonal)

Parallelogram + diagonals (diagonal) Square

4 equal sides & 4 right s (definition)

Rectangle + eq. adj sides (side)

Rhombus + 1 right (angle) Trapezium

Parallel opposite sides (definition) Kite

2 pairs of equal adjacent sides (definition)

in

semicircle at centre = 2 at

circumference

s in same segment

s in opp. segment

bisector of chord passes through centre

Equal chords are equidistant from

centre Equal arcs results in equal chords

alt. segment

thm tan rad tangents

from ext. point

Table

0 30 45 60 90

0

6

4

3

2

sin 0 1

2 2

2

3

2 1

cos 1 32

2

2

1

2 0

tan 0 1

3 1 3

Triangle

Unit circle

Prove Quadrilaterals

a

a

b a b a

b

Angle Properties of Circle

Ex 10.3 Circles Theorems

A C B

O B

A X

Y C O

D

Chord Properties of Circle

a b O P

Q

R

Tangent Properties of Circle

45 60

30

60

1

1 2

2

1

3

Special Angles

Ex 11.1 Trigo Ratios of Acute Angles




rad = 180

To convert from degrees to radians, multiply

180

To convert from radians to degrees, multiply 180

Tip:

Track the unit conversion to avoid the mistake of

multiplying the wrong fraction

e.g.

60 = 60

180 =

1

3

[deg] = [deg] [rad]

[deg] = [rad]

60 = 60 180

=

10800

[deg] = [deg] [deg]

[rad] =

[deg]2

[rad]

sin(90 ) = cos

cos(90 ) = sin

tan(90 ) =1

tan

sin(180 ) = sin

cos(180 ) = cos

tan(180 ) = tan

Step 1: Add or subtract 360 until 0 360 Step 2: Use table

Angle Quadrant 0 < < 90 1

90 < < 180 2 180 < < 270 3 270 < < 360 4

Step 1: Add or subtract 360 until 0 360

Step 2: Use table

Quadrant 1 2 180 3 180 4 360

Quadrant 1 2 180 3 180 + 4 360

Step 1: Draw

Step 2: Find all 3 sides (by Pythagoras Thm)

Angles measured anti-clockwise from the positive x-axis

are positive.

On the contrary, angles measured clockwise from the

positive x-axis are negative.

Step 1: Identify quadrant

Step 2: Draw in quadrant

Step 3: Find coordinates

Convert between Degrees and Radians

Complementary s

Supplementary s

Identify Quadrant

Find Basic Angle

Find General Angle

Use

x

y r

sin =y

r

cos =x

r

tan =y

x

r = x2 + y2

Trigo Function Definition

Ex 11.2 Trigo Ratios of any Angles

Use in Quadrant(s)




Given that tan A =

5

12 and that tan A and cos A have

opposite signs, find the value of each of the following.

(i) sin(A)

(ii) cos(A)

(iii) tan (

2 A)

Thought Process

Step 1: Identify quadrants

tan A = 5

12< 0

2nd or 4th quad.

Observe that ratio for tan is negative. Tan is only positive in 1st or 3rd quad. Therefore, it is in 2nd or 4th quad.

tan A & cos A have opp. signs 3rd or 4th quad.

In 3rd quad., only tan is positive In 4th quad., only cos is positive Therefore, it is in 3rd or 4th quad.

4th quadrant Take overlap of above deductions. Therefore it is in 4th quadrant.

Step 2: Draw in quadrant

Draw in 4th quadrant.

Step 3: Find coordinates

tan A = 5

12=

y

x

tan A =y

x by definition.

y = 5,

Equate numerator, = 5. y-coordinate is negative in 4th quad.

x = 12, Equate denominator, = 12. x-coordinate is positive in 4th quad.

r = 122 + (5)2

= 13

Find hypotenuse r by Pythagoras Theorem

sin A =y

r=

5

13,

cos A =x

r=

12

13

Find other trigo ratios to serve as useful inputs. The rest of the question makes use of the 3 basic trigo ratios: sin A , cos A & tan A.

sec =1

cos

csc =1

sin

cot =1

tan

cos() = cos()

sin() = sin()

tan() = tan()

All trigo functions can be converted to trigo function of basic angle with positive or negative sign depending on ASTC rule. e.g. sin(210) = sin(30)

Quadrants method

Step 1: Find = f1(|k|) & identify quadrants

Step 2: State interval

Step 3: Find x using quadrants

5

12

r

Solution

Question Reciprocal Identities

Negative Angles

A S T C

sin is + all are +

tan is + cos is +

ASTC Rule

1 2

2 3 4 +

1 2

360 3 4 180 180 +

Solve Trigo Eqn f(x) = k




Graphical method

When = 0 or 90,

i.e. sin f(x) = 0,1

cos f(x) = 0,1

tan f(x) = 0

Step 1: State interval

Step 2: Find x using graph

y = sin x y = cos x y = tan x

1 sin x 1

1 cos x 1

y = sin x y = cos x

Min sin x = 1

at x = 270 cos x = 1 at x = 180

Max sin x = 1 at x = 90

cos x = 1 at x = 0, 360

Sine/Cosine Tangent

Amplitude A = |a| =maxmin

2

Period T =360

b

Axis c =max+min

2

= min + A = max A

Period T =180

b

Step 1: Simplify to y = af(bx) + c Step 2: Find amplitude & period

Sin/Cos Tan Amplitude |a| Nil

Period 2

b

b

Step 3: Complete table and sketch graph

Domain x1 x x2 Axis with Amplitude

y = c |a|

Shape sin/cos/tan

Cycle x2x1

T

y = sin x y = cos x y = tan x

0 sin x = 0 at x =0, 180, 360

cos x = 0 at x =90, 270

tan x = 0 at x =0, 180, 360

Min sin x = 1 at x = 270

cos x = 1 at x = 180

Nil

Max sin x = 1 at x = 90

cos x = 1 at x = 0, 360

Nil

1

1

180 90 270

360 1

1

180 90 270

360 1

1

180 90 270

360

Solve Trigo Eqn f(x) = k by Graph

Ex 11.3 Trigo Graphs

1

1

180 90 270 360

1

1

180 90 270 360

Range of Sine & Cosine

c A = |a|

T =360

b

A = |a|

max

min

c

T =180

b

Find Unknowns of Trigo Function af(bx) + c

1

1

180 90 270 360

1

1

180 90 270 360

180 90 270 360

Sketch Trigonometric Functions




Sketch y = 3(1 2 cos 4x) for 0 x 270

Step 1: Simplify to = () + y = 3(1 2 cos 4x)

= 3 6 cos 4x = 6 cos 4x + 3

Step 2: Find amplitude & period A = |6| = 6

T =360

4= 90

Step 3: Complete table and sketch graph

Domain 0 x 270 Axis with Amplitude

y = 3 6

Shape cos

Cycle 2700

90= 3

Follow the sequence from top down to sketch the graph.

Mark the endpoint of domain, 270.

Mark the axis 3. Add and subtract 6 to get max 9 and min 3.

Draw 1 cycle of negative cosine.

There are 3 cycles in total. Draw 2 more.

Symmetrical

Given & are roots of 3 cos x + 2 = 2 where 3 < k < 4. Find in terms of , given that <

x = is line of symmetry, +

2 =

= 2 Cyclical

Given that is the smallest positive root of the equation

2 cos 4x = 3.1 tan 2x, where 0 x 360, state the other roots in terms of .

Period = 90, x = , + 90, + 180, + 270

Solution

Question

270

270

3

9

3

270

3

9

3

270

3

9

3

1 = 3.1 tan 2

2 = 2 cos 4

90

= 45 = 135

180

2

2

Solution

Question

x

y

2

5

-1 2

= 3 cos + 2

Solution

Question

Use Symmetrical/Cyclical Nature of Trigo Graphs




Principal values

2 sin1 x

2

0 cos1 x

2< tan1 x 0

For decreasing function, dy

dx< 0

Applications Determine whether a function is increasing or

decreasing Find the range of values of x for which a function is

increasing or decreasing

dy

dt rate of change of y wrt t

Consider adding/subtracting between related rates

Water is entering a container at a constant rate of 5 cm3/s Water is leaking from the container at a constant rate of 1 cm3/s. Find the net rate of water flow into the container.

Net rate = 5 1 = 4cm3/s

Quantity = (constant rate) time

d

dx[f(x)

g(x)] =

g(x) f(x) f(x) g(x)

[g(x)]2

Square Bottom

Diff Top

Bottom

Diff Bottom

Top

Quotient Rule

Ex 14.4 The Quotient Rule

Find Tangent

Ex 15.1 Tangents and Normals

Find Normal

Normal Properties

Tangent Properties

Increasing/Decreasing function

Ex 15.2 Increasing and Decreasing Functions

Solution

Question

Rate of Change

Ex 15.3 Rates of Change

Quantity & Constant Rate




Step 1: Assign variables, state given & unknown rate

Step 2: y = f(x) (Form relation between variablesof given rate & unknown rate

)

(see appendix 1)

Step 3: dy

dx= f(x) (Find derivative)

Step 4: dy

dt=

dy

dxdx

dt (Use chain rule)

Step 5: Find rate at instant

dy

dx= 0

x a a a+ dy

dx sign + 0

max

x a a a+ dy

dx sign 0 +

min

x a a a+ dy

dx sign 0

inflexion

d2y

dx2< 0 max

d2y

dx2> 0 min

d2y

dx2= 0 inflexion

Step 1: Assign variables

Step 2: y = f(x) (Express variable to be max/minas a function of a single variable

)

(see appendix 1)

Step 3: Find dy

dx (Find derivative)

Step 4: Solve dy

dx= 0 (Find stationary value)

Step 5: Find d2A

dx2 and compare against 0

(Verify max/min)

d

dx(sin x) = cos x

d

dx(cos x) = sin x

d

dx(tan x) = sec2 x

d

dx[sin(ax + b)] = a cos(ax + b)

d

dx[cos(ax + b)] = a sin(ax + b)

d

dx[tan(ax + b)] = a sec2(ax + b)

Consider simplifying using trigonometric identities before differentiating

e.g. d

dx(2 sin x cos x) =

d

dx(sin 2x) = 2 cos x

d

dx(ex) = ex

d

dx(eax+b) = aeax+b

Consider simplifying using indices properties before differentiating

e.g. d

dx(e2x e13x) =

d

dx(e1x) = e1x

Connected Rates of Change

Ex 15.4 Connected Rates of Change

Stationary Point/Value

Ex 16.1 Nature of Stationary Points

1st Derivative Test

2nd Derivative Test

Maxima/Minima

Ex 16.2 Maxima and Minima

Derivatives of Trigonometric Functions

Ex 17.1 Derivatives of Trigo Functions

Derivatives of Exponential Functions

Ex 17.2 Derivatives of Exponential Functions




d

dx(ln x) =

1

x

d

dx[ln(ax + b)]=

a

ax+b

Consider simplifying by using laws of logarithm before differentiating Product law

e.g. d

dx(ln xex) =

d

dx(ln x + ln ex)

=d

dx(ln x + 1)

=1

x

Quotient law

e.g. d

dx[ln (

x

x2+1)] =

d

dx[ln x + ln(x2 + 1)]

=1

x+

2x

x2+1

Power law

e.g. d

dx[ln(4x 3)2] =

d

dx[2 ln(4x 3)]

= 2(4

4x3)

=8

4x3

Change-of-base law

e.g. d

dx(loga x) =

d

dx(ln x

ln a)

=1

ln a

d

dx(ln x)

=1

ln a(1

x)

=1

xln a

Step 1: Take natural log both sides Step 2: Simplify using laws of log Step 3: Differentiate It is useful when differentiating

functions of the form y = [f(x)]g(x), f(x) e complicated products or quotients

Differentiate y = 2x with respect to x

y = 2x

ln y = ln 2x Take ln both sides

ln y = x ln 2 Simplify using power law

Diff wrt x: Differentiate both sides wrt x 1

ydy

dx = ln 2

dy

dx = (ln 2)y

= (ln 2)2x Replace y with 2x

Find dy

dx if y = (2 + x2)(1 x3)4

y = (2 + x2)3(1 x3)4 (1)

ln y = ln[(2 + x2)3(1 x3)4]

= ln(2 + x2)3 + ln(1 x3)4

= 3 ln(2 + x2) +4 ln(1 x3)

Diff wrt x: 1

ydy

dx = 3

1

2+x2

d

dx(2 + x2) +4

1

1x3

d

dx(1 x3)

= 3 1

2+x2 2x +4

1

1x3 (3x2)

=6x

2+x2

12x2

1x3

dy

dx = (

6x

2+x2

12x2

1x3) y

= 6x (1

2+x2

2x

1x3) y

= 6x [(1x3)2x(2+x2)

(2+x2)(1x3)] y

= 6x [1x34x2x3

(2+x2)(1x3)] y

= 6x [14x3x3

(2+x2)(1x3)] y (2)

sub (1) into (2): dy

dx = 6x [

14x3x3

(2+x2)(1x3)] (2 + x2)3(1 x3)4

= 6x(1 4x 3x3)(2 + x2)2(1 x3)3

Derivatives of Log functions

Ex 17.3 Derivatives of Log Functions

Solution

Question

Solution

Question

Use Logarithmic Differentiation




f(x) g(x) dx = f(x) dx g(x) dx

af(x) dx = a f(x) dx

Integration is the reverse of differentiation

Given d

dx6x + 5 =

3

6x+5,

find 1

6x+5dx.

1

6x+5dx =

1

3

3

6x+5dx

=1

36x + 5 + c

Consider rearranging equation involving derivative.

Given d

dx(x ln x) = 1 + ln x,

find ln x dx.

d

dx(x ln x) = 1 + ln x

ln x =d

dx(x ln x) 1

ln x dx = [d

dx(x lnx) 1] dx

= x ln x x + c

Given d

dx(x cos x) = cos x x sin x,

Find x sin x dx

d

dx(x cos x) = cos x x sin x

x sin x = cos x d

dxx cos x

x sin x dx = (cos x d

dxx cos x) dx

= sin x x cos x + c

To form equations and solve unknowns, use given equations (unknowns are already present)

e.g. dy

dx= x2(x k)

use proportionality e.g. Gradient is proportional to f(x)

dy

dx= kf(x)

introduce arbitrary constants from integration

e.g. dy

dx= 2x + 1

y = x2 + x + c

use point on curve e.g. (1, 2) lies on y = f(x)

2 = f(1)

use gradient

e.g. at turning point, dy

dx= 0

xn dx =xn+1

n+1+ c

axn dx =axn+1

n+1 +c

(ax + b)n dx =(ax+b)n+1

a(n+1) +c

Note: The rules for above hold for all real values of n

except for n = 1

e.g. x1 dx x0

0+ c

but x1 dx = 1

xdx = ln|x| + c

Consider simplifying before integrating Multiply or divide

e.g. [x(x + 1)] dx = (x2 + x) dx =x3

3+

x2

2+ c

(2x2+4x

x) dx = (2x + 4) dx = x2 + 4x + c

(x2+2x

x1) dx = (x + 3 +

3

x1) dx (long division)

Use law of indices

e.g. x dx = x1

2 dx =x32

3

2

=2

3xx

1

x2dx = x2 dx =

x1

1+ c =

1

x+ c

Breaking into partial fractions

e.g. x

(x1)2dx =

1

x1+

1

(x1)2dx

= ln|x 1| 1

x1+ c

Integral Rules

Ex 18.1 Indefinite Integrals

Solution

Question

Solution

Question

Solution

Question

Find Integral from Derivative

Find Curve from Derivative

Integrals of Power Functions




f(x)b

a dx = F(b) F(a)

f(x)b

a dx = f(x)

a

bdx

f(x)b

adx = a

cf(x) dx + f(x)

b

cdx

f(x)a

adx = 0

Definite integrals can be equal

because they have equal area

under curve

e.g. 01x2dx = x2

0

1dx

|f(x)| dx = { f(x) dx if f(x) 0

f(x) dx if f(x) < 0

sin x dx = cos x + c

cos x dx = sin x + c

sec2 x dx = tan x + c

sin(ax + b) dx = 1

acos(ax + b) + c

cos(ax + b) dx =1

asin(ax + b) + c

sec2(ax + b) dx =1

atan(ax + b) + c

Consider simplifying using trigonometric identities before integrating e.g. tan2 x dx = sec2 x 1dx

= tan x x + c Use special angles for definite integrals of trigonometric function

e.g. cos x

2

3

dx = [sin x]3

2

= sin

2 sin

3

= 1 3

2

ex dx = ex + c

eax+b dx =1

aeax+b + c

Consider simplifying using indices properties before integrating e.g. e2x e13x dx = e1x dx

=e1x

1 +c

= e1x +c

1

xdx = ln|x| + c

1

ax+bdx =

1

aln|ax + b| + c

Consider breaking into partial fractions before integrating

e.g. 2x1

(x+1)(x+2)dx = (

5

x+2

3

x+1) dx

= 5 ln|x + 2| 3 ln|x + 1| + c

Definite Integrals

Ex 18.2 Definite Integrals

Definite Integrals Rules

Integrals of Modulus Functions

Integrals of Trigonometric Functions

Ex 18.3 Integrals of Trigo Functions

Integrals of Exponential Functions

Ex 18.4 Integrals of Exponential Fns & 1/x

Integrals of 1

x &

1

ax+b

1 1

= 2

Equal areas




Area = (Top Bottom)x2

x1

dx

Area = (Right Left)y2

y1

dy

There should be a pair of lines parallel to x or y-axis enclosing the region. If parallel to y-axis, integrate wrt x and vice versa. Consider finding geometric area without integration

Triangle area =1

2(base)(height)

Trapezium area =1

2(sum of bases)(height)

Given the the diagram at the right. Find area of (i) Region A (ii) Region B

(i) Area of Region A ()

=1

2(base)(height)

=1

2(1)(2)

= 1 unit2

(ii) Area of Region B (trapezium)

=1


=1

2(2 + 6)(3 1)

= 8 unit2

Axis Integrate wrt x or y-axis

Break Break into smaller shapes

Complement Subtract area

Find the area bounded by y = x2, y = 2 x and the x axis.

Method 1 (integrate wrt y-axis)

Area of region F

= (Right Left)y2y1

dy

= [(2 y) y]1

0dy

Method 2 (break)

Area of region G +Area of region H

= x21

0dx + (2 x)

2

1dx

Method 3 (complement)

Area of Area of region I

=1

2(2)(2) (2 x)

1

0dx

1 2

= ()

Top

Bottom

1

= ()

Left

2

Right

Solution

Question

Area by integration

Ex 19.1 Area by Integration

Question

= 2

= 2 2

(1,1)

Solution

F

=

= 2

1

G

= 2

= 2 1

H

2

= 2

= 2 2 1

I 2

Strategies to find area bounded by curves

Ex 19.2 Area bounded by Curves

v

v =ds

dt a =

dv

dt

v = a dt s = v dt

s a

Kinematics Relation

Ex 20.1 Kinematics

(1,2)

(3,6)

1 3

= 2




t is time after passing O s|t=0 = 0

Rest v = 0

Time to turn around v = 0

Max/min quantity 1st derivative = 0

Max/min dist. from O v = 0

Max/Min v a = 0

Total Distance = |v| dt

Average Speed =total distance

total time

Total Distance = |v|t2

t1

dt

(Total distance travelled in between t1 and t2)

Method 1 (using s-t graph)

Step 1: Let v = 0 to find t

Step 2: Find s for each t found

Step 3: Find s for start & end

Step 4: Draw s-t graph Method 2 (using v-t graph)

Step 1: Draw v-t graph

Step 2: Use distance = |v| dt

Step 3: Split at v = 0

Step 4: Remove modulus

|v| = {v for v 0

v for v < 0

Implications of Kinematics Statements

Distance




Triangle ( )

Triangle area =1

2(base)(height)

=1

2ab sin C

Isosceles triangle area =s23

4

Sine rule: sin a

A=

sinb

B=

sin c

C

Cosine rule: a2 = b2 + c2 2bc cos A

Pythagoras theorem : a2 + b2 = c2

Similar triangles: a

A=

b

B=

c

C

Trigonometric identities:

sin =o

h o = h sin

cos =a

h a = h cos

tan =o

a

Quadrilateral ( )

Square area = x2

Rectangle area = (base)(height)

Parallelogram area = (base)(height)

Rhombus area =1

2(product of diagonals)

Trapezium area =1


Kite area =1

2(product of diagonals)

Circle ( )

Circle area = r2

Circumference = 2r

Arc length = r = s

Area of sector =1

2r2

=1

2rs

Area of segment =1

2r2

1

2r2 sin

Circle properties (refer to Ex 10.3)

Prism

Prism volume = (base area)(height)

Cube volume = x3

Cube surface area = 6x2

Cylinder volume = r2h

Cylinder surface area = 2r2 + 2rh = 2r(r + h)

Pyramid

Pyramid volume =1

3(base area)(height)

Cone Volume =1

3r2h

Cone area (exclude base) = rl

where l = r2 + h2

Sphere

Sphere volume =4

3r3

Sphere area = 4r2

2D Shapes

Appendix 1 Geometric Formulae 3D Shapes




Special Angles

0 30 45 60 90

0

6

4

3

2

sin 0 1

2 2

2

3

2 1

cos 1 32

2

2

1

2 0

tan 0 1

3 1 3

Complementary Angles

sin(90 ) = cos

cos(90 ) = sin

tan(90 ) =1

tan

Trigonometric Function Definition

Reciprocal Identities

sec =1

cos

csc =1

sin

cot =1

tan

Negative Angles

cos() = cos()

sin() = sin()

tan() = tan()

Principal Values

2 sin1 x

2

0 cos1 x

2< tan1 x

Documents

Math 360 Notes