Math j1 v1.0 Final

8/4/2019 Math j1 v1.0 Final

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H2 Mathematics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved

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Foreword

I feel that amongst all Singapore students, many

of us may not have the privilege of receiving

quality education in the subject of mathematics

due to differing teaching pedagogies in various

institutions and teachers/mentors.

Despite my limited ability, I hope that these noteswill assist you in your learning journey for

mathematics, be it the A you are aiming for, or

to sustain your genuine interest in the subject.

It is also apparent many of us learn for the sake of

learning. Should you feel that I am actually

somewhat trustworthy, I strongly urge you to

consider why you are actually in school.

Ultimately, it was never about your interest, it is

only about the usefulness of the subject.

(Supposedly) In JC, I believe all readers can

already see how mathematics is useful in daily life.

Its like how the NASSA swimming test could be

really boring but could save your life should you

accidentally fall into the ocean.

In essence, learn math for a purpose.

With that understanding, I wish you all the best

for H2 Mathematics for your promotional exams.

Ang Ray Yan

Hwa Chong Institution (11S7B)

Disclaimers / Terms and Conditions

- Mathematics need practice. This note getsyou the U grade if you only read it.

- There might be errors. Please use somediscretion when reading through. This note is

definitely not the best.

- At A Levels now, it is assumed that manyproofs and concepts are already exposed to

you. Should you be interested, please

research yourself for proofs.

- The use of the graphic calculator is notcovered in this note. It is assumed that you

have prior knowledge on its use.

- Assumptions are made to save time and cutback on redundancy. Strong O level concepts

greatly assist in reading this note.

- Included are some relevant concepts that arenot in the H2 Mathematics syllabus for

enthusiasts. (Denoted by *)

- Distribute only to students by email orthumbdrive. The usage of these notes by anyschool or tuition teacher is stricyl prohibited.

- This is meant for J1 students only. I stronglyrecommend all J2 students to practice on

problems instead of wasting time here.

- If you bought a copy of this, please ask for arefund. It is free!


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Contents Page

Basic O Level Revision 3l-5l

- Laws of Indices 3l- Laws of Logarithms 3l- Completing the Square 3r- Partial Fractions 3r- Graphs 4l- Trigonometry 4r-5lGraphing Techniques 5r-9l

- Rational Functions, Asymptotes 5r-6l- Conics 6l-7l- Parametric Equations 7l- Transformation of Graphs 7r-8l-

Special Graphs 8l-9l

Functions 9l-10r

- Definition / Set Notation 9l- One to One / Inverse Functions 9r- Composite Functions 10l- Piecewise Functions 10l- Hyperbolic Functions* 10rInequalities 11l-12l

-

Rational Functions 11l- Modulus Inequalities 11r- System of Linear Equations 12lDifferentiation 12r-16l

- Limits 12r- Differentiation by First Principle 13l- Techniques of Differentiation 13l- Implicit Differentiation 13r- Trigonometric Functions 13r- Inverse Trigonometric Functions 14l- Exponential / Logarithmic Function 14l- Parametric Equations 14r- Tangents and Normals 14r- Rates of Change 14r-15l- Stationary Points 15l-15r- Maxima / Minima problems 15r- Graph of derivatives 15r-16lIntegration 16l-20r

- Definitions 16l-16r- Integration of Standard Forms 16r-17r- Integration using Partial Fractions 17r-18l- Integration of Modulus Functions 18l

- Integration by Substitution 18l- Integration by Parts 18r- Special Types* 19l- Finding Area Under Graph 19l-19r- Integrating Parametric Equations 19r- Solid of Revolution 20l- Shell method* 20l- Approximation* 20r- Length of Curve* 20r- Surface of Revolution* 20rVectors

- Definitions 21l- Properties of Vectors 21l-21r- Ratio Theorem 22l- Scalar (dot) Product 22l-22r- Vector (cross) Product 22r- Projection Vectors 23l- Perpendicular Distance 23l- Area of Parallelogram / Triangle 23l- Straight Lines 23r- Interactions between Straight Lines 23r-24l- Planes 24r- Interactions (btw. Lines and Planes) 24r-25l- Interactions between Planes 25r-26r- Angle Bisectors 26r- Distance between Skew Lines 26rBinomial Expansion 27l-28l

- Pascals Triangle and Expansion 27l- The Binomial Series 27l-27r- Approximation 27r- Maclaurin/Taylor Series* 28lSequences and Series 28l-32l

- Arithmetic Progression (AP) 28l- Geometric Progression (GP) 28l-28r- Sequences / Recurrence Relation 28r- Sigma Notation and Properties 29l- Method of Differences 29l-29r- Method of Common Differences* 29r-30r- Mathematical Induction 30r- Convergence and Divergence* 31l-32lMiscellaneous 32l-32r

- Formulas for Shapes and 3D Shapes 32l-32rCredits 32r


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Chapter 1: Basic O Level Revision

Before we proceed, it is assumed that you have

got a moderate grasp of your O level syllabus. This

section only serves as a revision, not teaching

material. Hence, materials are mostly in brief.

If you wish to revise your O level syllabus, please

grab a copy of my notes catered for O levels.

1.1Laws of IndicesFor an index notation, we have the following:

As such, the law of indices states that:

1. 2. 3. ( )4. 5. 6. 7.

1.2Laws of LogarithmsRewriting the index notation, we obtain a

logarithmic expression as shown:

Similarly, the law of logarithms exists as such:

1. 2. 3.

4. 5. 6. (change of base)

Note:

1.3Completing the SquareThe objective of this technique is as follows:

Hence, starting from , This is exactly equal to:

Given , we observe the following:1. Minimum / maximum value = 2. Stationary point at 1.4Partial FractionsFor any function

,

We need to first perform long division. Otherwise,

the following sections detail the basic rules:

1.4.1 Linear Factors

1.4.2 Repeated Linear Factors

1.4.3 Irreducible Quadratic Factors

All methods can be used simultaneously.

Note: All 3 methods use substitution (of) orcomparing coefficients (of) to solve.


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1.5Graphs (Basic)

The red line is a linear graph with equation:

The blue line is a quadratic graph with equation:

To find the x-intercepts (or roots) are given by:

It is assumed the reader knows the proof, which

involves the use ofcompleting the square.

Using (discriminant), we know that:

Also, with the roots ( ) known, we deduce:

To plot a polynomial graph in general:

1.6TrigonometryConsider a right angle triangle as shown:

The 3 basic ratios are sine, consine, tangent

We have the following special values for special

angles for application to the 4 quadrants (ASTC):

Angle Sin Cos Tan

The 4 quadrants here represent which

trigonometric ratio will be positive. Angles are

calculated in an anti-clockwise manner.

O osite AdjacentHypotenuse


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Some basic identities include:

Advanced formulas include:

Furthermore 2 rules must be known for the

following triangle:

Chapter 2: Graphing Techniques

2.1Rational Functions, AsymptotesRational functions are defined as:

The 2 main types of asymptotes are as follows:

Vertical ( ) Horizontal ( ) Other asymptotes ( ) would depend onthe expression of

and

The following are some graphs with asymptotes:


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2.2ConicsGenerally, the equation of conics is formed by

intersecting a plane and a cone:

Source: http://mrhiggins.net/algebra2/?p=210

1.6.1 ParabolaThere are only 2 types of equations:

Properties follow either

that of a quadratic or

root graph.

1.6.2 Ellipse and Circle

Given the above diagram, an ellipse has a general

equation as follows:

To plot in your Graphic Calculator (GC) without

the Conics application, we rewrite it as follows:

Circles are formed when :

1.6.3 HyperbolaA hyperbola can be defined by 2 differentequations:

The vertices of this hyperbola is given by [i.e. the points A and B]The equation for asymptote (green) is given by:


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The vertices of this hyperbola is given by [i.e. the points A and B]The equation for asymptotes (green) is given by:

Note: It is highly unlikely your GC is of any use

here because neither the conics app nor normal

graph lets you find the asymptotes and vertices.

2.3Parametric Equations

Parametric equations occur when x and y are

expressed as a function of a third variable, i.e.

To plot the graph in GC, press mode. Then, select

Par (for parametric) before keying the equations.

Sometimes, x and y can be expressed in Cartesian

form. If possible, just substitute x = y to eliminate

the 3rd

variable. 2 standard instances are:

Do take note of the restrictions / domain for the

3rd

variable, e.g.

The limit can be set in your GC by accessing

window and changing the value of Tmin and Tmax.

2.4Transformation of Graphs2.4.1 Translation / Reflecting GraphsThe graph below shows the translation of

:

Up/Down Left/Right The graph below shows the scaling of

Scale parallel to y-axis Scale parallel to x-axis

(by factor of1/a)

The graph below shows the reflection of

Reflect in y-axis Reflect in x-axis


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The sequence of transformations is as follows:

Note: Always apply to the variable x or y only [not

the parameter inside the function].

2.4.2 Applying the ModulusFor , we apply the modulus function in 2ways to transform the graph:

Using the graph

Reflect graph below y-

axisupwards

Reflect graph where

x>0 in y-axis

2.5Special GraphsTo deduce from ,

y-coordinate

Positive + +

Negative - -Gradient Vertical

Asymptote

xintercept atx-intercept Vertical

Asymptote Horizontal

Asymptote

Horizontal

Asymptote

Oblique

asymptote

Approach Stationary

Points

Maximum Minimum Minimum Minimum

To deduce from ,

Take only the positive part of the graph!

Following which, reflect it in the x-axis.

Note: for

, take the negative portion

of the graph! (So that is positive)


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Defined0 or 1

y-intercept

x-intercept Vertical

asymptote

Horizontal

Asymptote

Stationary

Points

Maximum Maximum Minimum

Minimum

Minimum

MaximumChapter 3: Functions

3.1Definition / Set NotationFor a typical set defined as follows,

Note that [] stands for closed interval (inclusive)

and () stands for open interval (exclusive).

Furthermore, other typical sets include:

By definition, a function (f) is a rule assigning x

(

) to y (

) (i.e. mapping x to y). Hence, X

is the domain () and Y is the range ().

3.2One to One / Inverse FunctionsTo verify a function, we use the vertical line test,

ensuring that maps only 1 y-value for everyx-value. (For instance, circles fail the test)

If a function is one to one, we use the horizontal

line test to verify its existence. (The blue function

is not one to one, unlike the red)

Note: The domain also determines if a function is

one to one. For instance, if is the domain,then the blue function is a one to one function too.

Hence, for all one to one functions, given that:

For instance, to find from for thegraph below, we make x the subject.

We see that graphically, is a reflection inthe line of.


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3.3Composite FunctionsFor a composite function, we have a function in

another. As an example,

()

Hence, in order for to exist, we need toensure the following holds:

This means that the range of g is a subset of or is

equal to the domain of f.

.

Here are some properties of composite functions:

3.4Piecewise FunctionsPiecewise functions use different rules for

different parts of the domain. For instance,

We can obtain repeated patterns from piecewise

functions. As an example, we have:

3.5Hyperbolic Functions*The diagram below shows how forms a circle and how form theright half of an equilateral hyperbola.

Source: http://en.wikipedia.org/wiki/File:HyperbolicAnimation.gif

Hence, the hyperbolic sine and cosine functions

are defined by:

Using Eulers Formula ( ),

Hence, we observe that trigonometric and

hyperbolic functions share the relation:

Thus, they share many similar properties with

trigonometric ratios, (e.g. addition theorems,

double argument formulas)

Lastly, here are some graphs for and: (Do note the range)


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Chapter 4: Inequalities

4.1Rational FunctionsTo solve for all inequalities with rational functions,

1) Bring all terms to one side. (DO NOT cancel)2) Remove all factors that are always positive

(proven via complete the square)

3) Plot the graph with the roots.4) Determine interval of graph that satisfies the

inequality (i.e. your solution)

For all factorisable or ,

To summarize the above, an example is provided:

Note: Sometimes the GC is required to solve

inequalities, done via finding intersection points.

E.g.:

4.2Modulus InequalitiesFirst, we note that for ,

Then, we observe useful results for inequalities:

Most of the time, we can just plot the graph using

GC. However, to get values in exact form, we

must solve them.

For instance,

After plotting the graphs using GC, we know that:

Also note that by letting , we can solveother inequalities if a substitution can occur. This

is illustrated in the following example:

x-1 1 2


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4.3System of Linear EquationsA system of linear equations is a set of2 or more

equations with 2 or more variables. Its solution is

a set of values satisfying all equations in the

system (not all systems have unique solutions).

To do it manually*, we use the Gauss-Jordan

Elimination technique, leveraging on the

elementary row operations (ERO) on matrices.

For any Matrix [A][B], we wish to get [I][S], where

A and B represent the LHS and RHS of the

equations, I represents the identity matrix and S

represents the solution. For example,

However, please use the Graphic Calculator

(Plysmlt2) to solve any system. You will only be

tested on your ability to formulate the equations,

not solve them manually.

Chapter 5: Differentiation

Legend: 5.1Limits5.1.1 Introduction

The above expression means that as x

approaches , approaches L. Also, andrepresent approaching from the right and leftof respectively (one-sided limit), possiblydifferent for discontinuous functions.

For rational functions, where

is

denoted by ,

() () () () () ()

5.1.2 l'Hpital's Rule

5.1.3 Squeeze Theorem


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5.2Differentiation by First Principle

For the graph , we take 2 points, and .

5.3Techniques of DifferentiationAssuming you do not already know this, brief

proofs will be given throughout this section:

5.3.1 Polynomials

5.3.2 Chain Rule

5.3.3 Product / Quotient Rule

5.4Implicit Differentiation() ()

As an example offinding a differential,

Hence, differentiate y w.r.t. y, then times dy/dx.

5.5Trigonometric FunctionsAssuming you dont know the proof, we start with:

Trigonometric derivatives are summarized below:


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5.6Inverse Trigonometric Functions

To summarize,

5.7Exponential / Logarithmic Functions

To summarize the derivatives,

5.8Parametric Equations

5.9Tangents and NormalsAt a point along a curve ,

Note: for parametric equations, substitute: Also remember to expressin terms of.5.10 Rates of Change

This is used to prove if a curve is strictly

increasing/decreasing. For instance,


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The chain rule is also used to solve problems

involving connected rates of change. For instance,

A spherical balloon is deflated. When its radius is

3m, its Surface Area is decreasing at 2m2s

-1. Find

the rate of decrease ofradius and volume at the

same instant.

5.11 Stationary Points

The red graph shows the global and local

stationary points (maximum and minimum). The

blue graph shows a non-stationary point of

inflexion, and the greenshowing a stationary

point of inflexion (POI).

5.11.1 Second Derivative TestCondition Conclusion

Maximum at

Minimum at

Possiblestationary POI. Possible non-stationary POI.

5.11.2 First Derivative Test (left) (right)

Maximum + 0 -

Minimum - 0 +POI Same Sign 0 / Not 0

(stationary

or not)

Same Sign

5.12 Maxima / Minima problemsWith the above knowledge, we can now solve

maxima/minima problems. As an example:

An 8cm wire is cut into 2 wires. The first of

is

bent into a circle of circumference . Thesecond is bent to form a square of perimeter . Prove that the sum of areas of the squareand circle is a minimum when the radius of the

circle is

5.13 Graph of derivatives

When we plot the graph of derivatives (blue), we

look at the value of and .


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The relevant changes are as follows:

Above x-axis

Below x-axis

Vertical asymptote at Vertical asymptote at Any other asymptote Non-vertical asymptoteAs with regards to the concavity of the curve,

Hence, a point of inflexion is the point wherechanges in sign.Chapter 6: Integration

6.1DefinitionsIntegration was discovered by Newton and

Leibniz in the 17th

century. It is generally regarded

as the anti-derivative, i.e.

2 properties are observed for indefinite integrals:

The definite integral introduces 2 limits, the

lower limit a and upper limit b:

Important properties for definite integrals are:

6.2Integration of Standard FormsUsing anti-derivatives, we immediately observe

the following standard forms:

6.2.1 Polynomials

6.2.2 Some Fractions

6.2.3 Exponential Functions

Sometimes, we need to multiply the numerator

and denominator by the same factor to

differentiate. E.g.:

6.2.4 Trigonometric Functions


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The following is the full prooffor integrating :

Hence to summarize the above,

Using double angle, factor formulas and

trigonometric identities, we have the following:

6.3Integration using Complete the Square

We first express it in this form:

Hence, the first part is immediately solvable:

For the second part, we complete the square:

Then, we finalize our result with the following:

Form Result

The proofs can be derived using implicit

differentiation and partial fractions.


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We always need to complete the square. Only

then can we apply the following:

The proof is done by implicit differentiation.

6.4Integration of Modulus FunctionsFor the portion of the graph that is negative, we

integrate the negative of it. This suggests we

must find the x-intercepts of that graph. E.g.:

[ ] [ ]

6.5Integration by SubstitutionTo solve an integral given the following:

The following are common substitutions*:

Given Use Result -

6.6Integration by PartsGiven the product rule, we derive the following:

Hence, we see that we need integrate v and

differentiate u. We choose u like this:

L I A T E

Log / Ln Inv. Trig Algebraic Trig Exp.

The following 2 examples illustrate the above:

Integrating by parts a second time:


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6.7Special Types*

6.8Finding Area Under GraphFor H2 mathematics, we explore the use of the

Riemann Integral in solving area under graph.

Regardless of the way we place the rectangles of

width (right, minimum, maximum or left), weobtain the following:

As approximation is more accurate as ,

The following is useful for :

From the above diagram, it is obvious that the

area between the graphs is the red portion minus

the green portion:

Using the example, the integral for intersecting

graphs is:

Also, the area between the curve and the y-axis

for some interval :

6.9Integrating Parametric EquationsSometimes curves are expressed in parametric

equations. To solve them, we observe that:


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6.10 Solid of RevolutionThe solid of revolution is obtained by rotating a

curve about a straight line.

Rotated about the x-axis, we see that this solid

comprises many disks. Hence, the Disk method to

calculate its volume is used as follows:

To express rotation about the y-axis, rewrite y,

making x the subject.

6.11 Shell method*To save yourself some trouble, we use a new

method for finding the volume when isrotated about the y-axis.

From this view, we observed that instead of

adding up disks, we add cylindrical shells (left).

Each shell (right) when unfolded gives a volume:

Hence, to find the volume of the solid,

Note:

is rotated about the y-axis.

6.12 Approximation*Instead of using rectangles, your graphic

calculator uses the trapezium rule to approximate

your definite integrals.

( ) ( ) ( )

( )

6.13 Length of Curve*

Hence, length of curve from to is:

6.14 Surface of Revolution*For interest, the surface area is given by:

This is given by Pappuss centroid theorem (first),

where the surface area is equal to the product of

the length of curve and distance travelled by the

geometric centroid. i.e.:


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Chapter 7: Vectors

7.1DefinitionsScalars have a magnitude.

Vectors however, have magnitude and direction.

For this notes, vectors are represented as:

Furthermore, this note tries to use minimal

diagrams because anything beyond 3D is not

really visual anymore. This also forces you to use

spatial imagination.

7.2Properties of Vectors

From the above, we observe that:

The parallelogram law of addition (left)

demonstrates how to sum 2 vectors, and the

polygon law of addition shows how to sum all

vectors. Hence, we conclude:

Other useful properties include:

Vectors (3D) can be defined by using the co-ordinates:

Source:http://www.technology2skill.com/science_mathematics/vect

or_analysis/vector_picture/position_vector_xyz.png

Hence, expressing in column notation:

Hence, the magnitude is of is therefore:

Unit vectors (denoted by the ^ above it)

[magnitude = 1] of any vector is thus:

Position vectors originate from the origin O.

Displacement vectors can be any other vector.

They can be equal in magnitude and direction as

a position vector.

Lastly, for collinear points, we observe that:


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7.3Ratio Theorem

Given that p divides AB is the ratio ,

Conditions for using ratio theorem:

- a, b and p must all point inwards or outwards- there must be a common pointMy personal opinion is that it is only a shortcut;

otherwise it is more or less redundant.

7.4Scalar (dot) ProductFor any two vectors and :

Here is the acute/obtuse angle formed whenboth vectors point towards/away from a point.

The result is a scalar.

Alternatively, we also know that:

Given this, we can find the angle between the 2

vectors. For instance,

Important results from dot product:

7.5Vector (cross) Product

From the above diagram, we observe that:

Note that n is the perpendicular vector to both a

and b. Alternatively, it is computed as follows:

Basically, we ignore the first row for the first row

of our product, second row for the second and

third row for the third.

Important results from cross product:

The vector product can give us a vector that is

perpendicular to both a and b. For instance:


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7.6Projection Vectors

From the diagram, the projection vector ofa on b

is ON.

| | | | | | || | |

( )

7.7Perpendicular Distance

By Pythagoras theorem, we know that:

| | Alternatively, we know that since:

| | || | |

7.8Area of Parallelogram / TriangleGiven the above, we know that:

7.9Straight Lines

Source: http://www.netcomuk.co.uk/~jenolive/vecline.gif

We see that a line as such would take the form:

In this case, a is the position vector and b is the

displacement vector.

For any line AB, to get the displacement vector,

When expressed in column notation,

This is the Cartesian equation of the line.

7.10 Interactions between Straight Lines7.10.1 Intersecting, Parallel and Skew Lines

The above represents the 3 possible interactions

involving straight lines. Hence, given 2 straight

lines:

First, we test for parallel lines:


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Otherwise, equate both lines, solving :For instance,

Equation both lines:

Use the GC (to save time) to solve (using PlySmlt2).

7.10.2 Angles between intersecting/skew LinesLike what we do when finding angle between

vectors, we just need the displacement vectors

this time round:

7.10.3 Foot of Perpendicular (Point to Line)Like what we do when finding projection vectors,

we just need a point on the line:

( ) | |

7.10.4 Reflection in line for a pointTo do this we use the ratio theorem ( ):

7.11 Planes

A plane is defined by a point A and 2 directional

vectors that must be parallel to the plane. In the

parametric form:

Alternatively, knowing that:

This is the scalar product form of a plane. Also,

The Cartesian form of a plane is:

7.12 Interactions between Lines and Planes7.12.1 Intersecting Lines and Planes

7.12.2 Angle between Lines and PlanesFirst, we check ifline is parallel to plane:


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If line is not parallel to plane,

Otherwise,

7.12.3 Angle between Line and PlaneTo find the angle between lines and planes,

7.12.4 Distance between Parallel Line and Plane

To put it in short, we just wish to find.For vectors, do create points when you need

them. For instance:

| |

Note: For distance from point to plane, A is given!

7.13 Interactions between Planes7.13.1 Intersection of 2 Planes

One could equate both equations of the planes

but I prefer substitution:

To do it by equating both planes:

Note:


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7.13.2 Intersection of 3 Planes

Source: http://www.vitutor.com/geometry/space/three_planes.html

By no intersection we mean that there is no

common point/line of intersection for all 3 planes.

The fastest way to solve this is just to solve a

system of equations in 3 variables (i.e. PlySmlt2).

The 2 examples show the results:

7.13.3 Angle between Planes

We see that the angle between the planes is the

angle between their normal vectors. Hence,

7.13.4 Distance between 2 Planes

Very obviously, we once again let A and B be

points in the respective planes:

| | | |

| |7.14 Angle Bisectors

To find the angle bisector, we realize that the

resultant vector must be as such:

Hence, in general, the displacement vector for

any angle bisector is: The point of intersection (x) (if lines are given)

can be solved by using your GC.

(

)

8.1Distance between Skew Lines

No IntersectionAt a Point

No IntersectionAt a LineNo Intersection


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Chapter 8: Binomial Expansion

8.1Pascals Triangle and ExpansionPascals triangle (below) shows the binomial

coefficient of each term for

:

If for , then we know that:

If you encounter something like this:

We either group the terms together before use

binomial expansion or we can use the

multinomial theorem*.

8.2The Binomial SeriesHowever, if nis a negative integer or fraction,then we use:

This only applies if .Hence, we can also extend it to the following:

Very useful shortcuts (it is quite obvious actually):

For any rational function, we should try to express

in partial fractions first:

( )

Note: to find coefficient for term :Use if r is positive when even and negativewhen odd, otherwise use or 8.3Approximation using Binomial ExpansionBeing an infinite series, we can use this series to

approximate our solutions. For instance,


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8.4Maclaurin / Taylor Series*For any real and complex function that isinfinite differentiable in a neighbourhood ofreal

or complex

can be expressed as:

Hence, when , we get the Maclaurin Series:

This is useful in helping us get the derivatives at

the point .Chapter 9: Sequences and Series

9.1Arithmetic Progression (AP)For an arithmetic progression, the n

thterm is

defined as the following:

The sum of the first n terms is given by:

Here are some useful properties of GP:

9.2Geometric Progression (GP)For an arithmetic progression, the n

thterm is

defined as the following:

The sum of the first n terms is given by:

Here are some useful properties of GP: For an infinite GP, the sum

only exists when

Additionally, we have the Arithmetic Mean (AM)

and Geometric Mean (GM)

9.3Sequences / Recurrence RelationA sequence can be defined by expressing the n

th

term in terms of n:

A sequence can also be defined using arecurrence relation, expressing a known term in

terms of its relationship with consecutive terms.

For instance, the Fibonacci sequence is:

A sequence may/may not converge:


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9.4Sigma Notation and PropertiesAs seen sometimes previously in this note, the

sigma notation is used for sums:

Properties ofsigma notation:

9.5Method of DifferencesIf the general term can be expressed as 2 or more

terms, then we can use the method of differences:

Questions usually provide For instance,

( )

9.6Method of Common Differences*We use the method ofcommon differences to try

and deduce the polynomial that generated the

sequence.

The main idea behind is that:

9.6.1 Sequences generated using Exponentials

We observe that the sequence repeats in the

differences. For instance,

Here we see that

repeats itself every

time we take differences.


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9.6.2 Sequences generated using Polynomials

To prove the following used in Chap 6 (integration)

We generate the first few and takedifferences as shown below:

When we observe a common difference, we can

stop. Since we have taken differences 3 times toget a constant, we can assume that:

The 3! appears because we differentiated the

function 3 times. Then we proceed to generate

the same triangle for :

After repeating as shown below,

We finally obtain:

9.6.3 Others

Fibonacci sequences have the following pattern:

Generally, any sequence that adds the previous

terms see that their differences go in the reverse

order.

9.7Mathematical InductionA method for proving, it is generalized as follows:

1) Write the proposition2) Verify proposition with smallest value ofn3) Let P(k) be true for some positive integer k4) Use P(k) to prove that P(k+1) is true5) State Conclusion.The following shows 2 examples of proving:

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9.8Convergence and Divergence*The following applies to all sequences that are

eventually non-negative.

9.8.1 Comparison Test

An example is provided as follows:

9.8.2 Integral Test

Source: http://en.wikipedia.org/wiki/Integral_test_for_convergence

Note that:

From the above, we observe that:

Hence, for any operating on interval :

An example is provided as follows:

{

9.8.3 Ratio Test


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The following 2 examples demonstrate its use:

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Chapter 10: Miscellaneous

10.1 Formulas for Shapes and 3D Shapes

Shape Area

Square Circle Triangle Sector Trapezium Ellipse Parallelogram

3D Shape Surface Area Volume

Regular

Pyramid

Cube Cone Cylinder Sphere Rectangular

Prism

Credits

This set of math notes is done by Ang Ray Yan,

Hwa Chong Institution 11S7B.

The following people deserve their due

recognition in the making of this set of notes:

- Mr Yee, my math tutor, for his wonderfulapplets that helped in my illustration for many

points, especially integration.

- Sim Hui Min for correcting many of mycalculation mistakes.

- Phang Zheng Xun for correcting my formattingand introducing other ways of presenting

certain oncepts.

- Yuan Yu Chuan for improving my command ofEnglish that is deemed as powderful.

Math j1 v1.0 Final

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