Revision doesn’t just happen – you have to

plan it!

Mathematics + Extension 1 Content Summary

30 minute revision sessions Step 1: Get ready

• Sit somewhere quiet. • Get your books, pens, papers and calculator. • Turn off your mobile phone, your internet. • Choose a specific revision topic. • Start the clock!

Step 2: What can you recall? (5 minutes)

• Before you look at your notes, try to remember the key idea. • Try to write a summary without looking at your note : write a

few sentences, draw diagrams, give examples.

Step 3: Review the content (5 minutes) • Review your notes, look at the text books. • Update your revision note if necessary.

Step 4: DO a practice question (or three) (15 minutes)

• Do an easier and a harder question – check your answers. Step 5: Reflect and update your plan (5 minutes)

• Ask yourself: What did I learn? What do I still need to work on? • Update your revision note to reflect your learning. • Decide if you want to do another 20 minutes on this topic.

If so, allocate a time to return to the topic : use one of your catch-up dates, or mark an empty time period.

Stop after 30 minutes. Colour two sunflower seeds. Give yourself an immediate reward for doing 30 minutes revision.

Cover photo: Astrolabe by Andrés David Aparicio Alonso CC-BY-NC-2.0 http://www.flickr.com/photos/adapar/2562290656/

You are free to share, copy, or modify this work for non-commercial purposes so long as you:

(i) Attribute the source : enzuber (ii) Share all derived works under a similar CC license.

Revision 23U Content Summary v2013.1 This resource is designed for teachers and senior high school students. enzuber@gmail.com exzuberant.blogspot.com For my students: please use our class edmodo not email.

This work is licensed Creative Commons CC-BY-NC-SA. http://creativecommons.org/licenses/by-nc-sa/3.0/

Mathematics & Mathematics Extension 1

1 Basic arithmetic and algebra

2 Plane Geometry

2e Circle Geometry

3 Probability

4 Functions

5 Trigonometric ratios

6 Linear functions

7 Series and applications

8 Tangents and derivatives

9a The quadratic function

9b Locus and the parabola

10 Geometry of the derivative

11 Integration

12 Exponential and logarithmic functions

13 Trigonometric functions

14 Applications of calculus to the physical world

15E Inverse functions

16E Polynomials

17E The binomial theorem

18E Permutations and combinations.

01a Arithmetic

0101 Convert repeating decimals to fractions.

0102 Surd operations and identities.

0103 Rationalise the denominator.

01b Algebra

0104 Factorisation by grouping in pairs.

0105 Difference of squares.

0106 Factorise sum and difference of cubes.

0107 Factorise quadratic trinomials (monic and non-monic).

0108 Algebraic fractions.

01c Equations

0109 Solve linear equations and inequations.

0110 Solve quadratic equations and inequations.

0111 Completing the square (monic and non-monic).

0112 Simultaneous equations in 2 & 3 variables (linear + quadratic).

0113 Solving absolute value equations and inequations.

0114 Solve simple exponential equations 83𝑥+1 = 42𝑥 .

0115 Solve inequations with unknowns in the denominator.

𝑓(𝑥) = 𝑎 𝑓 𝑥 = 𝑎 𝑜𝑟 − 𝑓 𝑥 = 𝑎

𝑓(𝑥) < 𝑎 −𝑎 < 𝑓(𝑥) < 𝑎

𝑓(𝑥) > 𝑎 𝑓 𝑥 > 𝑎 𝑜𝑟 − 𝑓 𝑥 > 𝑎

Type 2 : | f(x) | compared g(x)

Solve 2𝑥 + 1 = 3𝑥 − 2

2𝑥 + 1 = 3𝑥 − 2 −(2𝑥 + 1) = 3𝑥 − 2 or

but have to check 3𝑥 – 2 is positive for each solution!

Type 1 : | f(x) | compared to a number

Absolute value equalities and inequalities

Inequalities with pronumeral in the denominator

3

𝑥 − 2≥ 4, 𝑥 ≠ 2

3

𝑥 − 2𝑥 − 2 2 ≥ 4 𝑥 − 2 2

3 𝑥 − 2 ≥ 4 𝑥 − 2 2

So 2 < 𝑥 ≤11

4

0 ≥ (𝑥 − 2)[4 𝑥 − 2 − 3]

0 ≥ 4 𝑥 − 2 2 − 3(𝑥 − 2)

0 ≥ (𝑥 − 2)(4𝑥 − 11)

Don’t expand out! Easier to factorise later.

Must exclude 𝒙 = 𝟐 ‼

02a Plane Geometry

0201 Properties of angles made by transversals crossing parallel lines.

0202 Ratio of intercepts.

0203 Angle sum of a triangle, quadrilateral and polygon.

0204 Exterior angle of a triangle.

0205 Exterior angles of a polygon.

0206 Congruent triangle proofs.

0207 Similar triangle proofs.

0208 Properties of parallelogram, rectangle, square, rhombus.

0209 Area of parallelogram, trapezium and rhombus.

02e Circle Geometry

0220 Terminology: parts of the circle. Segment and a sector.

0221 Problems involving angles at the centre compared to angles at the circumference.

0222 Problems involving angles in the same segment.

0223 Problems involving equal chords, chords equidistant from the centre, product of intercepts of two chords.

0224 Problems involving cyclic quadrilaterals.

0225 Problems involving tangents and radii, equal tangents from an external point, circles tangential to each other.

0226 Problems involving tangents and angles in alternate segment.

0227 Problems involving tangent and secant from external point.

Image: CC-BY-NC Thomas Hawk. http://www.flickr.com/photos/thomashawk/156398361/

3 Probability

0301 Random experiments, equally likely outcomes.

0302 Sample spaces and event spaces.

0303 Venn diagrams, mutually exlusive and non-mutually exclusive events.

0304 Probability of OR and AND events.

0305 Multi-stage events and probability tree diagrams.

A 2 3 4 5 6 7 8 9 10 J Q K

A 2 3 4 5 6 7 8 9 10 J Q K

A 2 3 4 5 6 7 8 9 10 J Q K

A 2 3 4 5 6 7 8 9 10 J Q K

4 suits: spades, hearts, clubs, diamonds.

13 cards per suit.

52 cards in one deck.

𝑆 = { 1, 2, 3, 4, 5, 6 }

𝐸 = { 4, 5, 6 }

𝑃 4 𝑜𝑟 5 𝑜𝑟 6 = ?

H

T

H

T

H

T

HH

HT

TH

TT

Start 1st throw 2nd throw Outcome

𝑆 = { HH,HT, TH, TT }

0 ≤ 𝑃 𝐸 ≤ 1

𝑃 𝐸 =𝐸

𝑆

Complementary events P 𝐸 = 1 − P(𝐸)

𝑃 𝐴 𝐨𝐫 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 𝐚𝐧𝐝 𝐵 )

Addition rule

𝑃 𝐴1𝐴2 … 𝐴𝑛 = 𝑃 𝐴1 × 𝑃 𝐴2 ×⋯× 𝑃(𝐴𝑛)

Multiplication rule for independent multi-stage events

04 Functions

0401 Function concepts: relations and functions, domain and range, function notation.

0402 Find the implicit domain of a function eg: 𝑓 𝑥 = 25 − 𝑥2.

0403 Simplifying composite function expressions ( eg: f(g(2)) given f(x) and g(x) ).

0404 List and sketch 10 different groups of functions.

0405 Prove algebraically if a function is odd, even or neither.

0406 Given a graph f(x), can draw f(x)+c, f(x+c), f(x)-c, f(x-c), kf(x), f(kx), -f(x), f(-x).

0407 Finding the centre and radius of 𝑥2 + 2𝑥 + 𝑦2 − 5𝑦 − 2 = 0.

0408 Finding the vertical asymptotes of rational functions.

0409 Evaluate limits as x → ∞

0410 Graphing regions.

Meat-A-Morphosis watch it on YouTube

Piecewise-defined Functions

𝑓 𝑥 =

𝑥2 + 1 𝑥 < 1

1 − 3𝑥 𝑥 > 1

3 𝑥 = 1

If 𝑓 −𝑥 = 𝑓 𝑥 ⇒ 𝑓(𝑥) even

If 𝑓 −𝑥 = −𝑓 𝑥 ⇒ 𝑓(𝑥) odd

lim𝑥→∞

1

𝑥= 0

Function Translation and Reflection

05 Trigonometric Ratios

0501 Exact ratios for 0,30,45,60,90 degrees.

0502 Sine rule : the ambiguous case (obtuse angles)

0503 Cosine rule.

0504 Sine area rule.

0505 3D problems.

0506 Unit circle concepts.

0507 Solving equations in form 3sin2x = 1 -360°≤x≤360°.

0508 Pythagorean trig identities.

0509 Proving trig identities.

0510 Compound angle formulae.

0511 Double angle formulae.

0512 t-transformation.

0513 Auxiliary angle transformation.

0514 The general solution.

𝑎

sin 𝐴=

𝑏

sin 𝐵 =

𝑐

sin 𝐶

Ambiguous case

𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos𝐶

cos𝐶 =𝑎2 + 𝑏2 − 𝑐2

2𝑎𝑏

𝐴 =1

2𝑎𝑏 sin𝐶

065o

𝑥

𝑦

𝑥

Trig problems – including 3D

Draw the 3D situation, then draw every triangle.

Add variables where needed. Write all the equations you

might need. Then solve.

Complementary Angles

cos𝜃 = sin (90° − 𝜃)

cosec𝜃 = sec (90° − 𝜃)

cot𝜃 = tan (90° − 𝜃)

Reciprocals

sec𝜃 =1

cos𝜃

cot 𝜃 =1

tan𝜃

Ratios

tan𝜃 =sin 𝜃

cos𝜃

Pythagorean Identities

sin2 𝜃 + cos2 𝜃 = 1

tan2 𝜃 + 1 = sec2 𝜃

1 + cot2 𝜃 = cosec2 𝜃

LHS = RHS Look different at first,

but they are the same.

Artist: Ian MsQuee

Trigonometric Identities

cot 𝜃 =cos𝜃

sin 𝜃

cosec 𝜃 =1

sin𝜃

Compound Angle Formulae

cos(𝐴 + 𝐵) = cos𝐴 cos𝐵 − sin𝐴 sin𝐵

cos(𝐴 − 𝐵) = cos𝐴 cos𝐵 + sin𝐴 sin𝐵

sin(𝐴 + 𝐵) = sin𝐴 cos𝐵 + cos𝐴 sin𝐵

sin(𝐴 − 𝐵) = sin𝐴 cos𝐵 − cos𝐴 sin𝐵

tan(𝐴 + 𝐵) =tan𝐴+ tan𝐵

1 − tan𝐴 tan𝐵

tan(𝐴 − 𝐵) =tan𝐴− tan𝐵

1 + tan𝑎 tan𝑏

sin(2𝐴) = 2 sin𝐴 cos𝐴

cos(2𝐴) = cos2 𝐴 − sin2 𝐴

tan(2𝐴) =2tan𝐴

1 − tan2 𝐴

= 1 − 2 sin2 𝐴

= 2cos2𝐴 − 1

Double Angle Formulae

𝒕 = 𝐭𝐚𝐧𝜽

𝟐

tan 𝜃 =2𝑡

1 − 𝑡2

sin 𝜃 =2𝑡

1 + 𝑡2

cos 𝜃 =1 − 𝑡2

1 + 𝑡2

t is undefined when 𝜃 = 180° Check for this solution.

The t- transform

𝑎 sin 𝑥 + 𝑏 cos 𝑥 = 𝑹 sin (𝑥 + 𝜶)

𝑹𝟐 = 𝒂𝟐 + 𝒃𝟐

𝐭𝐚𝐧𝜶 =𝒃

𝒂

The auxiliary angle transform

tan(2𝐴) =2tan 𝐴

1 − tan2 𝐴

When solving equations such as 5 sin𝑥 + 2 cos 𝑥 = 4

Transforms

06 Linear Functions and Lines

0601 Distance and midpoint formula.

0602 Dividing an interval in the ratio m:n (internal and external).

0603 Point-gradient and two-point form to derive equations of a line.

0604 General form: relationship between coefficients for parallel and perpendicular lines.

0605 Perpendicular distance from a line to a point.

0606 Finding equation of a line passing through a point and the intersection of two other lines (k-method).

0607 Angle between two intersecting lines.

René Descartes (1596-1650)

Gradient

𝑥1, 𝑦1 and 𝑥2, 𝑦2 is

𝑦 = 𝑚𝑥 + 𝑏 gradient is 𝑚, y-intercept is 𝑏

𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 gradient is

Parallel lines: 𝑚1 = 𝑚2

Perpendicular lines: 𝑚1𝑚2 = −1 or

Points and Distances

𝑑2 = 𝑥2 − 𝑥12 + 𝑦2 − 𝑦1

2

Midpoint of an interval =

Dividing an interval in ratio 𝑚: 𝑛 = ( 𝑛𝑥1 +𝑚𝑥2𝑚+ 𝑛

,𝑛𝑦1 +𝑚𝑦2𝑚+ 𝑛

)

( 𝑥1 + 𝑥2

2,𝑦1 + 𝑦2

2 )

Distance from a line to a point 𝑎𝑥1 + 𝑏𝑦1 + 𝑐

𝑎2 + 𝑏2

Lines

Point gradient form

𝑦 − 𝑦1𝑥 − 𝑥1

= 𝑚

Two point form 𝑦 − 𝑦1𝑥 − 𝑥1

=𝑦2 − 𝑦1𝑥2 − 𝑥1

Intercept form 𝑥

𝑎+𝑦

𝑏= 1

Line through point of intersection of two other lines:

𝑎1𝑥 + 𝑏1𝑦 + 𝑐1 + 𝑘 𝑎2𝑥 + 𝑏2𝑦 + 𝑐2 = 0

General form

𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0

Gradient intercept form

𝑦 = 𝑚𝑥 + 𝑏

tan𝜃 =𝑚1 −𝑚2

1 +𝑚1𝑚2 Acute angle between two lines:

𝑚 =𝑦2 − 𝑦1𝑥2 − 𝑥1

−𝑎

𝑏

𝑚1 = −1

𝑚2

7 Series and Applications

0701 Arithmetic series : formula for the nth term and sum of n terms.

0702 Geometric series: formula for the nth term and the sum of n terms.

0703 Limiting sum of geometric series when |r| < 1.

0704 Applications to annuities and repayments.

Sigma notation 𝑓(𝑘)

𝑞

𝑘=𝑝

𝑞 − 𝑝 + 1 terms.

𝐴𝑛 = 𝑃 1 + 𝑟 𝑛 Compound Interest

Annuities

Watch out for ‘birth of grandchild’ problems ! Paying at the start of the period will give more interest.

You can think of each payment as compounding on its own. The first payment will compound for 𝑛 − 1 periods. The last payment won’t have any interest (if paid at the end of the period).

𝑆 = 𝑃 + 𝑃 1.01 + 𝑃 1.01 2 +⋯+ 𝑃 1.01 𝑛−1

eg: 𝑛 payments of $𝑃 at 10% per period, paid at the end of each period:

𝑆 = 𝑃 1.01 + 𝑃 1.01 2 +⋯+ 𝑃 1.01 𝑛

=𝑃 1.01𝑛 − 1

1.01 − 1

=(𝑃 1.01 1.01𝑛 − 1

1.01 − 1

= 𝑃 1.01 + 𝑃(1.01)(1.01) + ⋯+ 𝑃(1.01) 1.01 𝑛−1

Repayments Imagine as two accounts: a loan account compounding, offset by payment account which is an annuity. When the two accounts are equal the loan is paid off.

Arithmetic sequences (AP)

𝑇𝑛 = 𝑎 + 𝑑 𝑛 − 1

𝒏th partial sum 𝑆𝑛 =𝑛

2𝑎 + 𝑙

𝑆𝑛 =𝑛

2[2𝑎 + 𝑛 − 1 𝑑]

𝑇𝑛 − 𝑇𝑛−1 = 𝑑 for all 𝑛 > 1

𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … , 𝑎 + 𝑑(𝑛 − 1)

Geometric sequences (GP)

𝒏th term

𝒏th partial sum

𝑇𝑛 = 𝑎𝑟𝑛−1

𝑇𝑛𝑇𝑛−1

= 𝑟 for all 𝑛 > 1

𝑆𝑛 =𝑎 𝑟𝑛 − 1

𝑟 − 1

𝑆𝑛 =𝑎 1 − 𝑟𝑛

1 − 𝑟

𝑆∞ =𝑎

1 − 𝑟 if and only if 𝑟 < 1 limiting sum (|𝒓| < 𝟏)

𝑎, 𝑎𝑟, 𝑎𝑟2, 𝑎𝑟3, … . 𝑎𝑟𝑛−1

𝒏th term

0.99999 = 0.9 + 0.91

10+ 0.9

1

10

2

+⋯ =0.9

1 − 0.1= 1.0

7e Series: Induction

0705 Proof by mathematical induction - proving a formula.

0706 Proof by mathematical induction - proving divisibility.

0707 Proof by mathematical induction - proving inequalities.

Photo:Paul Stevenson CC-BY-2.0 http://www.flickr.com/photos/pss/1359407958/

The Principle of Mathematical Induction

• We can make any statement 𝑆𝑛 we like based on a positive integer n.

• If we can prove 1. The statement is true for a starting value 𝑛0, 2. Statement 𝑆𝑘 true implies 𝑆𝑘+1 is true, (𝑘 > 𝑛𝑜)

• then the Principle of Mathematical Induction says the statement 𝑆𝑛

is true for all positive integer values of 𝑛 ≥ 𝑛0 .

When we write a proof “by mathematical induction”, we prove the two conditions in Step 1 and Step 2 are met, then we appeal to the Principle of Mathematical Induction in Step 3 to complete the proof.

Establish the first specific case. Usually 𝑛 = 1.

Assume true for 𝑛 = 𝑘.

Prove this means the statement is true for 𝑛 = 𝑘 + 1.

Appeal to the principle of mathematical induction.

Every induction proof must have three clear parts:

So long as your proof is well set out, long essays and special incantations are not required – just a clear statement of what has been shown and your appeal to the principle of mathematical induction. Here is a simple version: “Proven for 𝑛 = 2, and that true for 𝑛 = 𝑘 implies true for 𝑛 = 𝑘 + 1, therefore by the principle of mathematical induction, the statement is true for all 𝑛 ≥ 2”

Type 1: Proving the formula for a series

Prove that 4𝑛 + 1

𝑛

𝑖=1

= 𝑛 2𝑛 + 3 for all 𝑛 ≥ 1

Type 2: Proving divisibility

“Prove by mathematical induction that 34𝑛 − 1 is divisible by 80 for all 𝑛 ≥ 0”

Type 3: Proving inequalities

Useful technique: Show RHS – LHS > 0.

“Prove by mathematical induction that 𝑛2 < 2𝑛 for all integers 𝑛 ≥ 5”

Classic types of induction proofs

Useful technique: write the statement as 34𝑛 − 1 = 𝑚80, 𝑚 an integer.

Setting out a proof by mathematical induction

08 Tangents and Derivatives

0801 The concept of a gradient function.

0802 Differentiation from first principles.

0803 The power rule.

0804 The chain rule.

0805 The product rule.

0806 The quotient rule.

0807 Equations of tangents and normals to the curve at a given point.

Isaac Newton 1642 - 1727

Gottfried Leibniz 1646 - 1716

I invented calculus.

No I did!

𝒅𝒚

𝒅𝒙 I call it I call it 𝒚′

𝑓′(𝑥) = lim∆𝑥→0

𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥

∆𝑥

𝑑𝑦

𝑑𝑥=𝑑𝑦

𝑑𝑢×𝑑𝑢

𝑑𝑥

Tangent gradient: 𝑚𝑇 = 𝑓′(𝑥)

Normal gradient: 𝑚𝑁 = −1

𝑚𝑇= −

1

𝑓′ 𝑥

𝑦′ = 𝑢𝑣′ + 𝑣𝑢′

𝑦 = 𝑎𝑥𝑛 ⇒ 𝑦′ = 𝑎𝑛𝑥𝑛−1

ℎ 𝑥 = 𝑓 𝑥 + 𝑔 𝑥 ⇒ ℎ′ 𝑥 = 𝑓′ 𝑥 + 𝑔′(𝑥)

ℎ 𝑥 = 𝑎𝑓 𝑥 ⇒ ℎ′ 𝑥 = 𝑎𝑓′ 𝑥

𝑦′ =𝑣𝑢′ − 𝑢𝑣′

𝑣2

Differentiation from first principles

The Chain Rule

The Product Rule

The Quotient Rule

The Power Rule

Basic properties

Equation of the tangent and normal to a point on a function

09a Quadratic Function

0901 Problems involving the discriminant.

0902 Working with positive definite, negative definite, indefinite.

0903 Problems involving sum and product of roots.

0904 Problems involving Quadratic Identity Theorem.

0905 Equations reducible to quadratics.

𝒂 𝒙 − 𝒉 𝟐 + 𝒌 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 expand

complete the square

Three points completely determine a quadratic function.

𝑦 = 𝑎(𝑥 − 𝛼)(𝑥 − 𝛽)

Completing the square for nonmonic quadratics

3𝑥2 + 𝑥 − 5

= 3 𝑥2 +1

3𝑥 −

5

3

= 3 𝑥 +1

6

2

−1

6

2

−5

3

= 3 𝑥 +1

6

2

−61

36

= 3 𝑥 +1

6

2

−61

12

Axis of symmetry

∆ = 𝑏2 − 4𝑎𝑐 The discriminant:

Positive definite: always positive, so no roots ∆ < 0 and 𝑎 > 0 Negative definite: always negative, so no roots ∆ < 0 and 𝑎 < 0 Otherwise: Indefinite.

𝛼𝛽 =𝑐

𝑎 𝛼 + 𝛽 = −

𝑏

𝑎

Sum and Product of Roots

Quadratic Identity Theorem

If 𝑓 𝑥 = 𝑎1 𝑥2 + 𝑏1𝑥 + 𝑐1,

𝑔 𝑥 = 𝑎2𝑥2 + 𝑏2𝑥 + 𝑐2

and 𝒇 𝒙 ≡ 𝒈 𝒙 , then 𝒂𝟏 = 𝒂𝟐, 𝒃𝟏 = 𝒃𝟐, 𝒄𝟏 = 𝒄𝟐.

Pattern: 2+ + 𝑘 = 0

Equations reducible to quadratics

9𝑥 − 4 3 𝑥 + 3 = 0

𝑥𝟒 + 15𝑥2 − 36 = 0

𝑥 = −𝑏

2𝑎

2 cos2 𝑥 − 3 cos 𝑥 + 1 = 0

09b Locus of the Parabola

0910 Solving basic locus problems (circles, bisectors, etc).

0911 Locus description of a parabola and derivation of equation from the locus.

0912 Four standards orientations of the parabola.

0913 Translation of the parabola from vertex (0,0) to vertex (h,k) - in the four standard orientations.

0914 Problems converting between geometric and cartesian descriptions of the parabola with vertex at (h,k).

0914 Parametric representations of curves - conversion between parametric and Cartesian form.

0915 Parametric representation of the parabola.

0916 Equations of the chord of a parabola and properties of the focal chord.

0917 Equations of tangents to the parabola (parameteric and Cartesian).

0918 Equation of the chord of contact.

0919 Special properties of the parabola (including the reflection property, focal chords and their tangents).

0920 Locus problems involving parabolas.

Photo by Kevin Dooley http://www.flickr.com/photos/pagedooley/1918813544/ CC-BY-2.0

𝑥 = 2𝑎𝑡 𝑦 = 𝑎𝑡2

𝑥2 = 4𝑎𝑦 𝑥2 = −4𝑎𝑦

𝑥 = 2𝑎𝑡 𝑦 = −𝑎𝑡2

𝑦2 = 4𝑎𝑥

𝑦 = 2𝑎𝑡 𝑥 = 𝑎𝑡2

𝑦2 = −4𝑎𝑥

𝑦 = 2𝑎𝑡 𝑥 = −𝑎𝑡2

Focus point at: 𝑆(0, 𝑎)

Directrix at: 𝑦 = −𝑎

𝑆𝑃 = 𝑃𝑄 Locus constraint:

𝟒𝒂

𝒂

𝒂

𝑺(𝟎, 𝒂)

𝑽

𝒚 = −𝒂

𝑷(𝒙, 𝒚)

𝑸(𝒙, −𝒂)

directrix chord focal chord latus rectum

See the video Locus Pocus (3 mins) on You Tube for a fun demonstration of common loci

http://www.youtube.com/watch?v=qLj55p7BQGY

Crop circles resulting from Centre Pivot Irrigation Locus: all points ≤ a fixed distance from a point.

Find these on Google Earth.

𝑆𝑃2 = 𝑃𝑄2

𝑥2 + 𝑦2 − 2𝑎𝑦 + 𝑎2 = 𝑦2 + 2𝑎𝑦 + 𝑎2

𝒙𝟐 = 𝟒𝒂𝒚

𝑥 − 0 2 + 𝑦 − 𝑎 2 = (𝑦 − −𝑎 )2

chord: gradient at 𝑃 2𝑎𝑝, 𝑎𝑝2 = 𝑝 tangent: 𝑦 = 𝑝𝑥 − 𝑎𝑝2

gradient at 𝑃 𝑥0, 𝑦0 = chord: tangent: 𝑥𝑥0 = 2(𝑦 + 𝑦0)

𝑥02𝑎

chord of contact 𝑥𝑥0 = 2(𝑦 + 𝑦0)

𝑃 𝑥𝑜 , 𝑦0

Exams questions are likely to ask you to derive one or more of these so no need to memorise them

Cartesian form 𝑥2 = 4𝑎𝑦

Parametric form 𝑥2 = 2𝑎𝑝, 𝑦 = 𝑎𝑝2

𝑦 =1

2𝑝 + 𝑞 𝑥 − 𝑎𝑝𝑞

focal chord : 𝑝𝑞 = −1

4𝑎𝑦 = 𝑥 𝑥0 + 𝑥1 − 𝑥0𝑥1

focal chord : 𝑥0𝑥1 = −4𝑎2

“Beam us up Scotty!” by Rising Damp CC-BY http://www.flickr.com/photos/66126733@N04/6335265385/

Note also the symmetry in the chord equations. Any equation you derive should be symmetric. Why?

The Reflection Property

Tangents at focal chord endpoints meet at directrix, at right angles

Use the equation of the tangent at P to find point M. Show that FM = FP Hence isosceles Δ

M

Use pq = -1 to show tangents (gradient p, gradient q) are ⊥ Solve intersection of tangents to show meet on 𝑦 = −𝑎

Image: Artist's impression Square Kilometre Array. Swinburne Astronomy http://en.wikipedia.org/wiki/File:SKA_dishes_big.jpg. You could be working on this massive project in the next few years!

10 Geometry of the Derivative

1001 Meaning of the first derivative : increasing, decreasing and stationary points.

1002 The concept of local maximum and local minimum (compared to global max/min).

1003 Testing the first derivative to determine the nature of stationary points.

1004 Meaning of the second derivative: concavity, points of inflexion.

1005 Testing the second derivative to determine the nature of stationary points and inflexions.

1006 Curve sketching using the first and second derivative.

1007 Curve sketching of rational polynomials.

1008 Optimisation problems - finding maximum and minimum values.

Photo: Six Flags El Toro by Amy Loves Yah CC-BY-2.0 http://www.flickr.com/photos/amylovesyah/4846346877/

𝑓′ 𝑥 > 0 Increasing

𝑓′ 𝑥 < 0 Decreasing

𝑓′ 𝑥 = 0 Stationary

Second derivative 𝒇’’(𝒙) : a measure of concavity

𝒇′′ 𝒙 < 𝟎 concave down

𝒇′′ 𝒙 > 𝟎 concave up

Could be a point of inflexion.

NEEDS TESTING

𝒇′′ 𝒙 = 𝟎

𝒇′ 𝒙 = 𝟎 3 types of stationary points

maximum turning point

minimum turning point

a horizontal point of inflexion

3

First derivative 𝒇’(𝒙) : a measure of gradient

𝑉 = 𝑥(20 − 2𝑥)(15 − 2𝑥) 𝑑𝑉

𝑑𝑥= 0

problem algebraic model optimisation

You must prove a stationary point is the desired min or max – don’t assume it is or you will lose marks. Also check boundary conditions.

11 Integration

1101 The Definite Integral : concepts and properties.

1102 The Fundamental Theorem of Calculus.

1103 Definite integral of polynomial functions.

1104 Area under the curve - using absolute values.

1105 Composite areas : between curves, across curves.

1106 The Indefinite Integral.

1107 Volumes of solids of revolutions about the x-axis, about the y-axis).

1108 Numerical methods: Trapezoidal Rule.

1109 Numerical methods : Simpson's Rule.

1110 Integration by substitution: indefinite and definite integrals.

𝒇 𝒙 Differentiate

𝒇′(𝒙) Integrate

𝑥𝑛𝑑𝑥 =𝑥𝑛+1

𝑛 + 1+ 𝐶, 𝑛 ≠ −1

𝑓 𝑥 𝑑𝑥𝑏

𝑎

= 𝐹 𝑏 − 𝐹 𝑎

𝑓 𝑥 𝑑𝑥𝑎

𝑏

= − 𝑓 𝑥 𝑑𝑥𝑏

𝑎

𝑘 𝑓 𝑥 𝑑𝑥𝑏

𝑎

= 𝑘 𝑓 𝑥 𝑑𝑥𝑏

𝑎

𝑓 𝑥 + 𝑔 𝑥 𝑑𝑥𝑏

𝑎

= 𝑓 𝑥 𝑑𝑥𝑏

𝑎

+ 𝑔 𝑥 𝑑𝑥𝑏

𝑎

𝑉 = 𝜋 𝑦2𝑑𝑥𝑏

𝑎

𝑉 = 𝜋 𝑥2𝑑𝑦𝑏

𝑎

About the x-axis

About the y-axis

Trapezoidal Rule

Simpson’s Rule

𝑓 𝑥 𝑑𝑥𝑏

𝑎

≈𝑏 − 𝑎

2𝑓 𝑎 + 𝑓 𝑏

𝑓 𝑥 𝑑𝑥𝑏

𝑎

≈1

6(𝑏 − 𝑎) 𝑓 𝑎 + 4𝑓(

𝑎 + 𝑏

2) + 𝑓 𝑏

𝑓′ 𝑥 𝑓 𝑥𝑛𝑑𝑥 =

𝑓 𝑥𝑛+1

𝑛 + 1+ 𝐶, 𝑛 ≠ −1 Reverse Chain Rule

lim∆𝑥→0

𝑓(𝑥)∆𝑥

𝑏

𝑎

= 𝑓 𝑥 𝑑𝑥𝑏

𝑎

𝐴 = 𝑓 𝑥 𝑑𝑥−1

−3

+ 𝑓 𝑥 𝑑𝑥2

−1

𝑎𝑥 + 𝑏 𝑛𝑑𝑥 =𝑎𝑥 + 𝑏 𝑛+1

𝑎 𝑛 + 1+ 𝐶 , 𝑛 ≠ −1

Volumes of Solids of Revolution Areas (unsigned)

Always sketch the situation first!

12 Exponential and Logarithmic Functions

1201 Properties of exponential functions.

1202 Properties of logarithms (log rules, including change of base rule).

1203 Properties of log functions.

1204 Differentiation of exponential functions.

1205 Integrating exponential functions.

1206 Differentiation log functions - using log rules to simplify.

1207 Integration of reciprocal functions.

Leonhard Euler (1707 – 1783) [pronounced ‘oiler’]

𝒆 = 2.7182818284590452353602....

𝑑

𝑑𝑥𝑒𝑓(𝑥) = 𝑓′(𝑥)𝑒𝑓(𝑥)

𝑑

𝑑𝑥𝑒𝑘𝑥 = 𝑘𝑒𝑘𝑥

log 𝑎𝑏 = log 𝑎 + log 𝑏

log𝑎

𝑏= log𝑎 − log 𝑏

log 𝑎𝑏 = 𝑏 log𝑎 log𝑎 𝑥 =log𝑏 𝑥

log𝑏 𝑎

log𝑎 𝑥 = 𝑏 means 𝑥 = 𝑎𝑏 log𝑏 𝑏𝑥 = 𝑥 𝑏log𝑏 𝑥 = 𝑥 for 𝑥 > 0

𝑑

𝑑𝑥𝑒𝑥 = 𝑒𝑥 𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝐶

𝑒𝑎𝑥+𝑏𝑑𝑥 =1

𝑎𝑒𝑎𝑥+𝑏 + 𝐶

𝑓′ 𝑥 𝑒𝑓(𝑥)𝑑𝑥 = 𝑒𝑓(𝑥) + 𝐶

𝑎𝑥 𝑑𝑥 =𝑎𝑥

log 𝑎+ 𝐶

𝑓𝑜𝑟 𝑎 > 0 𝑎𝑛𝑑 𝑎 ≠ 1

𝑑

𝑑𝑥𝑎𝑥 = (log𝑎) 𝑎𝑥

𝑑

𝑑𝑥log𝑥 =

1

𝑥

𝑑

𝑑𝑥log 𝑎𝑥 + 𝑏 =

𝑎

𝑎𝑥 + 𝑏

𝑑

𝑑𝑥log 𝑓 𝑥 =

𝑓′ 𝑥

𝑓 𝑥

𝑑𝑥

𝑥=

1

𝑥𝑑𝑥 = log 𝑥 + 𝐶

𝑓′ 𝑥

𝑓 𝑥𝑑𝑥 = log |𝑓 𝑥 | + 𝐶

13 Trigonometric Functions

1301 Radian measure.

1302 Arc length, area of sector, area minor segment using radians.

1303 Graphs of the six trigonometric functions in radians.

1304 Graphs of translated, scaled and reflected trigonometric functions.

1305 General solutions in radians.

1306 Small angle approximations for sin 𝑥 , cos 𝑥 , tan 𝑥 .

1307 Limit of sin 𝑥

𝑥 as 𝑥 → 0

1308 Differentiation of sin 𝑥 , cos 𝑥 , tan 𝑥 .

1309 Integration of sin 𝑥, cos 𝑥 and sec2 𝑥.

1310 Integration of sin2 𝑥 and cos2 𝑥.

𝒔𝒊𝒏𝒙 𝒅𝒙

sin𝜃 = sin𝛼 𝜃 = 𝑛𝜋 + −1 𝑛𝛼 in radians

cos𝜃 = cos𝛼 𝜃 = 2𝑛𝜋 ± 𝛼 in radians

tan𝜃 = tan𝛼 𝜃 = 𝑛𝜋 + 𝛼 in radians

The general solutions

𝑑

𝑑𝑥sin𝑥 = cos𝑥

𝑑

𝑑𝑥cos𝑥 = −sin 𝑥

𝑑

𝑑𝑥tan𝑥 = sec2 𝑥

Derivatives

lim𝑥→0

sin𝑥

𝑥= 1

A very special limit

𝑑

𝑑𝑥sin 𝐴𝑥 + 𝐵 = 𝐴 cos(𝐴𝑥 + 𝐵)

sin 𝑥 ≈ 𝑥 cos𝑥 ≈ 1 tan 𝑥 ≈ 𝑥

Small angle approximation

The following results are ONLY TRUE for 𝑥 IN RADIANS

sin 𝑥 𝑑𝑥 = −cos 𝑥 + 𝐶

cos𝑥 𝑑𝑥 = sin 𝑥 + 𝐶

sec2 𝑥 𝑑𝑥 = tan𝑥 + 𝐶

Integrals

cos(𝐴𝑥 + 𝐵) 𝑑𝑥 =1

𝐴sin(𝐴𝑥 + 𝐵) + 𝐶

sin2 𝑥 𝑑𝑥

cos2 𝑥 𝑑𝑥

Convert to cos 2𝑥 then integrate

14 Applications of Calculus to the Physical World

1401 Related rates of change problems.

1402 Exponential growth and decay dP/dt = kP.

1403 Exponential growth and decay - difference from a constant dP/dt = k(P-A).

1404 Velocity and acceleration in terms of time - differentiation with respect to time.

1405 Distance given v(t), velocity given a(t) - integration with respect to time.

1406 Distance given v(x) - integration with respect to x.

1407 Velocity given a(x) - integration with respect to x.

1408 Simple harmonic motion concepts & problems.

1409 Projectile motion concepts & problems.

Natural Growth and Decay

𝑑𝑃

𝑑𝑡= 𝑘(𝑃 − 𝑃0)

then 𝑃 𝑡 = 𝑃0 + 𝐴𝑒𝑘𝑡 is the general solution.

𝑎 =𝑑

𝑑𝑥

1

2𝑣2

𝑎𝑑𝑥 =1

2𝑣2

Velocity and acceleration as a function of 𝒙

𝑣 =𝑑𝑥

𝑑𝑡

𝑑𝑡

𝑑𝑥=1

𝑣

𝑡 = 1

𝑣𝑑𝑥

Given 𝒗(𝒙) Given 𝒂(𝒙)

𝑑2𝑥

𝑑𝑡2= −𝑛2𝑥

Equation of simple harmonic motion (SHM)

𝑥 = 𝐴 𝑐𝑜𝑠(𝑛𝑡 + 𝛼) 𝑇 =2𝜋

𝑛

𝑑2𝑥

𝑑𝑡2= −𝑛2(𝑥 − 𝑥0)

𝑥 = 𝑥0 + 𝐴 𝑐𝑜𝑠(𝑛𝑡 + 𝛼)

𝑣2 = 𝑛2(𝐴2 − 𝑥2)

Projectile Motion

𝑥 = 0

𝑦 = −𝑔

At 𝑡 = 0, 𝑥 = 𝑉 cos𝜃

𝑦 = 𝑉 sin𝜃

𝑑2𝑥

𝑑𝑡2= −𝑛2𝑥

15E Inverse Functions

1501 Finding inverse relations algebraically (swap x and y) and graphically (reflection in y=x).

1502 Definition and properties of monotonic functions and their inverses.

1503 Restriction of the domain to make inverse functions.

1504 Definitions and graphs of the inverse trigonometric functions.

1505 Transformations of inverse trigonometric functions.

1506 Four special properties of the inverse trigonometric functions.

1507 Differentiating and integrating the inverse trigonometric functions.

𝒚 = 𝐬𝐢𝐧−𝟏 𝒙 means: 𝑥 = sin 𝑦 , −𝜋

2≤ 𝑦 ≤

𝜋

2

Domain:

Range:

−1 ≤ 𝑥 ≤ 1

−𝜋

2≤ 𝑦 ≤

𝜋

2

sin−1(−𝑥) = − sin−1 𝑥

sin−1 𝑥 is an odd function

sin−1 𝑥 is an increasing function

means: 𝑥 = cos 𝑦 , 0 ≤ 𝑦 ≤ 𝜋

Domain:

Range:

−1 ≤ 𝑥 ≤ 1

0 ≤ 𝑦 ≤ 𝜋

cos−1(−𝑥) = 𝜋 − cos−1 𝑥

cos−1 𝑥 is neither odd nor even.

cos−1 𝑥 is a decreasing function

𝒚 = 𝐭𝐚𝐧−𝟏 𝒙 means: 𝑥 = tan 𝑦 , −𝜋

2< 𝑦 <

𝜋

2

Domain:

Range:

All 𝑥 ∈ ℝ

−𝜋

2< 𝑦 <

𝜋

2

tan−1(−𝑥) = −tan−1 𝑥

tan−1 𝑥 is an odd function

tan−1 𝑥 is a increasing function

𝑑

𝑑𝑥tan−1

𝑥

𝑎=

1

𝑎2 + 𝑥2

𝑑

𝑑𝑥cos−1

𝑥

𝑎=

−1

𝑎2 − 𝑥2

𝑑

𝑑𝑥sin−1

𝑥

𝑎=

+1

𝑎2 − 𝑥2

sin−1 𝑥 + cos−1 𝑥 =𝜋

2

𝒚 = 𝐜𝐨𝐬−𝟏 𝒙

Remembering the graphs is the easier way to remember all these facts.

16E Polynomials

1601 Polynomials definitions and concepts.

1602 Algebraic properties of polynomials.

1603 Polynomial long division.

1604 Polynomial division transform P(x) = D(x)Q(x) + R(x).

1605 Using the remainder theorem to find values of unknown coefficients.

1606 Using the factor theorem to find zeros of a polynomial.

1607 Equating coefficients of congruent polynomials to find values of unknown coefficients.

1608 Sum and product of roots.

1609 Numerical methods for finding roots: halving the interval.

1610 Numerical methods for finding roots: Newton's method.

𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥

𝑛−1 + …+ 𝑎3𝑥3 + 𝑎2𝑥

2 + 𝑎1𝑥 + 𝑎0

𝑎0, 𝑎1, … . 𝑎𝑛 are constants and 𝑛 is a positive integer

2𝑥−4 + 𝑥3, 2𝑥32 + 𝑥3 Not a polynomial:

2𝑥2 + 𝑥3

𝑥 − 1,

𝑎𝑛 = 𝑏𝑛 , 𝑎𝑛−1 = 𝑏𝑛−1, … , 𝑎1 = 𝑏1, 𝑎0 = 𝑏0 If then

The Identity Theorem for Polynomials

≡ 𝐴(𝑥) 𝐵(𝑥)

𝑃 𝑥 = 𝐷 𝑥 𝑄 𝑥 + 𝑅(𝑥)

remainder quotient

divisor

degree 𝑅(𝑥) < degree 𝐷(𝑥) or 𝑅(𝑥) = 0

Polynomial Division

The Remainder Theorem If we divide 𝑃(𝑥) by (𝑥 − 𝑎), then the remainder will be 𝑃(𝑎). The Factor Theorem If (𝑥 − 𝑎) is a factor of 𝑃(𝑥), then the remainder will be 𝑃 𝑎 = 0. and the converse is true.

Roots & Coefficients : cubic

𝛼𝛽𝛾 = −𝑑

𝑎

𝛼 + 𝛽 + 𝛾 = −𝑏

𝑎

𝛼𝛽 + 𝛼𝛾 + 𝛽𝛾 = +𝑐

𝑎

𝛼 + 𝛽 + 𝛾 + 𝛿 = −𝑏

𝑎

𝛼𝛽 + 𝛼𝛾 + 𝛼𝛿 + 𝛽𝛾 + 𝛽𝛿 + 𝛾𝛿 = +𝑐

𝑎

𝛼𝛽𝛾 + 𝛼𝛽𝛿 + 𝛼𝛾𝛿 + 𝛽𝛾𝛿 = −𝑑

𝑎

𝛼𝛽𝛾𝛿 = +𝑒

𝑎

Photo: Reaching Out. Yiping Lim CC BY-NC-ND 2.0 http://www.flickr.com/photos/yipinglim/420070032/

Roots & Coefficients : quartic

Newton’s Method

𝑥1 = 𝑥𝑜 −𝑓 𝑥0𝑓′ 𝑥𝑜

17E Binomial Theorem

1701 Expansion of 𝑎 + 𝑏 𝑛 using the binomial theorem.

1702 Proving Pascal's triangle identities.

1703 Finding the greater coefficient or greater term in a binomial expansion.

1704 Proving identities on the binomial coefficients.

Useful 𝑪𝒓 𝒏 values to know

𝐶0 = 𝐶𝑛 = 1𝑛𝑛

𝐶1 = 𝐶𝑛−1 = 𝑛𝑛𝑛

𝐶2 = 𝐶𝑛−2 =1

2𝑛(𝑛 − 1)𝑛 𝑛

𝐶2 5 = 𝐶2

4 + 𝐶1 4

Yang Hui’s 杨辉 triangle aka Pascal’s triangle.

1. Each row starts and ends with 1: 𝐶0𝑛 = 𝐶𝑛

𝑛 = 1

2. Each row is reversible: 𝐶𝑟𝑛 = 𝐶𝑛−𝑟

𝑛

3. The sum of each row is 2𝑛

4. The Addition Property: 𝐶𝑟𝑛+1 = 𝐶𝑟

𝑛 + 𝐶𝑟−1 𝑛 (1 ≤ 𝑟 ≤ 𝑛)

Can you prove these using the expansion 𝑎 + 𝑏 𝑛?

4 Important Identities

𝐶𝑟𝑛 =

𝑛 × 𝑛 − 1 ×⋯(𝑛 − 𝑟 + 𝟏)

1 × 2 × ⋯× 𝑟

𝑟 terms

𝑟 terms

The Binomial Theorem

𝐶𝑟𝑛 =

𝑛!

𝑟! 𝑛 − 𝑟 ! or

For 𝑛 ∈ ℕ, and 𝑟 = 0,1,2, … 𝑛 𝑥 + 𝑦 𝑛 = 𝐶𝑟𝑛 𝑥𝑛−𝑟𝑦𝑟

𝑛

𝑟=0

“Unrolling” a factorial: 𝑛! = 𝑛 𝑛 − 1 ! = 𝑛 𝑛 − 1 𝑛 − 2 ! = …

Zero factorial: 0! = 1

𝑡𝑟+1 = 𝐶𝑟𝑛 𝑥𝑛−𝑟𝑦𝑟

The general binomial term

where 𝑟 ∈ 0, 𝑛 .

Danger! The first term starts at 𝑟 = 0

For example, the 5th term has 𝑟 = 4

There are 𝑛 + 1 terms in total

𝑡5 = 𝐶4𝑛 𝑥𝑛−4𝑦4

Finding the largest coefficient: Write the ratio of successive terms, and find when the ratio is less than 1.

Identities using the binomial theorem: classic problem types Type 1: Substitution – look for alternating signs to choose between 𝑥 ± 𝑦 𝑛 Type 2: Differentiation – look for terms 𝑟 ∙ 𝐶𝑟

𝑛

Type 3: Integration – look for terms 1

1+r∙ 𝐶𝑟𝑛 . Watch out for +C – use x=0

to find the constant. Type 4: Equation coefficients. Type 5: Induction.

18E Permutations and Combinations

1801 The Fundamental Counting Principle.

1802 Permutations - counting ordered selections - with and without repetition.

1803 Permutations - counting with identical elements.

1804 Combinations - counting unordered selections.

1805 Applications of permutations and combinations to probability.

1806 Arrangements in a circle.

1807 Binomial probability.

Use this table backwards for differentiation!

The Standard Integrals as provided by the Board of Studies with your exam paper*

𝑥𝑛 𝑑𝑥 =1

𝑛 + 1𝑥𝑛+1 , 𝑛 ≠ −1, 𝑥 ≠ 0 if 𝑛 < 0

1

𝑥 𝑑𝑥 = ln 𝑥 , 𝑥 > 0

𝑒𝑎𝑥𝑑𝑥 =1

𝑎𝑒𝑎𝑥 , 𝑎 ≠ 0

cos𝑎𝑥 𝑑𝑥 =1

𝑎sin 𝑎𝑥, 𝑎 ≠ 0

sin𝑎𝑥 𝑑𝑥 = −1

𝑎cos𝑎𝑥, 𝑎 ≠ 0

sec2 𝑎𝑥 𝑑𝑥 =1

𝑎tan𝑎𝑥, 𝑎 ≠ 0

sec𝑎𝑥 tan𝑎𝑥 𝑑𝑥 =1

𝑎sec𝑎𝑥 , 𝑎 ≠ 0

1

𝑎2 + 𝑥2𝑑𝑥 =

1

𝑎tan−1

𝑥

𝑎, 𝑎 ≠ 0

1

𝑎2 − 𝑥2𝑑𝑥 = sin−1

𝑥

𝑎, 𝑎 > 0,−𝑎 < 𝑥 < 𝑎

1

𝑥2 − 𝑎2𝑑𝑥 = ln 𝑥 + 𝑥2 − 𝑎2 , 𝑥 > 𝑎 > 0

1

𝑥2 + 𝑎2𝑑𝑥 = ln 𝑥 + 𝑥2 + 𝑎2

NOTE: ln 𝑥 = log𝑒 𝑥 , 𝑥 > 0

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