Induction Proofs

Proving Results Involving Series

Steps

  1. Prove Sn is true for n = 1
  1. Assume Sn is true for n = k
  1. Try to prove Sn is true for n = k + 1
  1. Hence, Sn is true for all positive integers n

Proving Divisibility

To be divisible
integer needs to be a factor of that number

Proving Results Involving Inequalities

Move entire equation to one side and leave the other side as zero

Vectors

Introduction to Vectors

Vector Quantity

Magnitude

Direction

Represent with


Horizontal
Vertical

Positive = right and up
Negative = left and down

Vector Arithmetic

Adding and Subtracting Vectors

Add vertical and horizontal components

Multiplying by a Scalar

Multiply each component by separate non-vector integer

Vectors in the Cartesian Plane

Magnitude of Vectors

square root of
x-value squared + y-value squared

Unit Vector

Magnitude of 1

Form

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Further Operations with Vectors

Dot Products and the Angle between Two Vectors

Geometrical Applications

Properties of Vectors

Solving Motion Problems Using Vectors

Geometric Results and Proofs Using Vectors

Dot product / Scalar product

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Angle between Two Vectors

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image magnitude

Found by squaring i and j value
Adding
Square rooting

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Lie on the same line

Same vectors

Same Direction

Different magnitude

Parallel Vectors

Unit Vectors

Inverse of magnitude

Rationalise result

Vector Projection

Scalar projection of a onto b

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Vector Projection

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a perpendicular to b

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velocity

Speed + Direction

Trigonometric Equations

The Auxiliary Angle Method

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Steps

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Equals to co-efficient in actual equation

same for Rsin

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R = co-efficient squared and then square rooted

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Results for Sum & Differences of Angles

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Double Angle Results

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The T Formula

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Solving Trig Equations Using the T Method

Tan

Negative

Final Check

Sub x = pi into original equation

Quadrants 2 & 4

Positive

Quadrants 1 & 3

Number not easily inversed

Change calc to radians

  • pi

Solving Trigonometric Equations

Calculus

Further Calculus Skills

Integration by Substitution

Steps

Let u = equation

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make dx subject

Integrate like normal

Sub original equation back into u

Solve

Limits

Have to be changed

Original limits are subbed into x in u

Limits are changed with results

Trigonometry Differentiation

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Log and Exponential Functions

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Rules

Product Rule

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Quotient Rule

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Power Rule

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Chain Rule

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