Revision doesn’t just happen – you have to

plan it!

Mathematics 2U Content

enzuber 2014

2014 v2

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 30 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 Checklist Mathematics 2U v2014.1 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 2U

1 Basic arithmetic and algebra

2 Plane 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

1a Arithmetic

0101 Convert repeating decimals to fractions.

0102 Surd operations and identities.

0103 Rationalise the denominator.

1b Algebra

0104 Factorisation by grouping in pairs.

0105 Difference of squares.

0106 Factorise sum and difference of cubes.

0107 Factorise quadratic trinomials (monic and nonmonic).

0108 Algebraic fractions.

1c 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 .

2a 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.

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.

𝑆 = { 1, 2, 3, 4, 5, 6 }

𝐸 = { 4, 5, 6 }

𝑃 4 𝑜𝑟 5 𝑜𝑟 6 = ?

Start 1st throw 2nd throw Outcome

𝑆 = { HH,HT, TH, TT }

𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)

4 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

5 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.

0506 Unit circle concepts.

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

0508 Pythagorean trig identities.

0509 Proving trig identities.

6 Linear Functions and Lines

0601 Distance and midpoint formula.

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).

René Descartes (1596-1650)

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.

Arithmetic sequences (AP)

𝑆𝑛 =𝑛

2𝑎 + 𝑙 𝑆𝑛 =

𝑛

2[2𝑎 + 𝑛 − 1 𝑑]

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

Geometric sequences (GP)

𝑆𝑛 =𝑎 𝑟𝑛 − 1

𝑟 − 1 𝑆𝑛 =

𝑎 1 − 𝑟𝑛

1 − 𝑟

𝑆∞ =𝑎

1 − 𝑟 if and only if 𝑟 < 1

𝑎, 𝑎𝑟, 𝑎𝑟2, 𝑎𝑟3, … . 𝑎𝑟𝑛−1

8 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

𝒅𝒚

𝒅𝒙 I call it I call it 𝒚′

𝑓′(𝑥) = lim∆𝑥→0

𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥

∆𝑥

9a 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

9b 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).

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

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)

𝑥2 = 4𝑎𝑦 𝑥2 = −4𝑎𝑦 𝑦2 = 4𝑎𝑥 𝑦2 = −4𝑎𝑥

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.

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 or a “really flat” min/max.

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.

𝒇 𝒙 Differentiate

𝒇′(𝒙) Integrate

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....

log 𝑎𝑏 = log 𝑎 + log 𝑏

log𝑎

𝑏= log 𝑎 − log𝑏

log 𝑎𝑏 = 𝑏 log 𝑎

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

𝑑

𝑑𝑥ln 𝑥 =

1

𝑥

1

𝑥𝑑𝑥 = ln 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.

𝒔𝒊𝒏𝒙 𝒅𝒙

14 Applications of Calculus to the Physical World

1401 Rates of change problems.

1403 Exponential growth and decay dP/dt = kP.

1405 Velocity and acceleration in terms of time - differentiation with respect to time.

1406 Distance given v(t), velocity given a(t) - integration with respect to time.

One sunflower seed = 15 minutes of revision

Design: enzuber 2012. Original concept: Peter Garside Photo: CC-BY-NC Fir0002/Flagstaffotos

One sunflower seed = 15 minutes of revision

Design: enzuber 2012. Original concept: Peter Garside Photo: CC-BY-NC Fir0002/Flagstaffotos

One sunflower seed = 15 minutes of revision

Design: enzuber 2012. Original concept: Peter Garside Photo: CC-BY-NC Fir0002/Flagstaffotos