Pure mathematics Year 2

1. ALGEBRAIC METHODS

1.1 Proof by contradiction

1.2 Algebraic fractions

1.3 Partial fractions

1.4 Repeated factors

1.5 Algebraic division

1.6 Mixed exercise 1

1.7 Review exercise for chapter 1

 

2. FUNCTIONS AND GRAPHS

2.1 The modulus function

2.2 Functions and mappings

2.3 Composite functions

2.4 Inverse functions

2.5 $ y = \vert \textrm f(x) \vert $ and $ y = \textrm f ( \vert x \vert ) $

2.6 Combining transformations

2.7 Solving modulus problems

2.8 Mixed exercise 2

2.9 Review exercise for chapter 2

 

3. SEQUENCES AND SERIES

3.1 Arithmetic sequences

3.2 Arithmetic series

3.3 Geometric sequences

3.4 Geometric series

3.5 Sum to infinity

3.6 Sigma notation

3.7 Recurrence relations

3.8 Modelling with series

3.9 Mixed exercise 3

3.10 Review exercise for chapter 3

 

4. BINOMIAL EXPANSION

4.1 Expanding $(1+x)^n$

4.2 Expanding $(a+bx)^n$

4.3 Using partial fractions

4.4 Mixed exercise 4

4.5 Review exercise for chapter 4

 

5. RADIANS

5.1 Radian measure

5.2 Arc length

5.3 Areas of sectors and segments

5.4 Solving trigonometric equations

5.5 Small angle approximations

5.6 Mixed exercise 5

5.7 Review exercise for chapter 5

 

6. TRIGONOMETRIC FUNCTIONS

6.1 Secant, cosecant and cotangent

6.2 Graphs of $\sec x$, $\csc x$ and $\cot x$

6.3 Using $\sec x$, $\csc x$ and $\cot x$

6.4 Trigonometric identities

6.5 Inverse trigonometric functions

6.6 Mixed exercise 6

6.7 Review exercise for chapter 6

 

7. TRIGONOMETRY AND MODELLING

7.1 Addition formulae

7.2 Using the angle addition formulae

7.3 Double-angle formulae

7.4 Solving trigonometric equations

7.5 Simplifying $a \cos x \pm b \sin x$ (Harmonic forms/identities)

7.6 Proving trigonometric identities

7.7 Modelling with trigonometric functions

7.8 Mixed exercise 7

7.9 Review exercise for chapter 7

 

8. PARAMETRIC EQUATIONS

8.1 Parametric equations

8.2 Using trigonometric identities

8.3 Curve sketching

8.4 Points of interection

8.5 Modelling with parametric equations

8.6 Mixed exercise 8

8.7 Review exercise for chapter 8

 

9. DIFFERENTIATION

9.1 Differentiating $\sin x$ and $\cos x$

9.2 Differentiating exponentials and logarithms

9.3 The chain rule

9.4 The product rule

9.5 The quotient rule

9.6 Differentiating trigonometric functions

9.7 Parametric differentiation

9.8 Implicit differentiation

9.9 Using second derivatives

9.10 Rates of change

9.11 Mixed exercise 9

9.12 Review exercise for chapter 9

 

10. NUMERICAL METHODS

10.1 Locating roots

10.2 Iteration 1 - Interval bisection

10.3 Iteration 2 - Fixed-point method

10.4 Iteration 3 - The Newton-Raphson method

10.5 Applications to modelling

10.6 Mixed exercise 10

10.7 Review exercise for chapter 10

 

11. INTEGATION

11.1 Integrating standard functions

11.2 Reverse chain rule 1: Integrating ${\rm f}(ax+b)$

11.3 Using trigonometric identities

11.4 Reverse chain rule 2: Integrating ${\rm f}'(x)/{\rm f}(x)$ and ${\rm f}'(x)[{\rm f}(x)]^n$

11.5 Integration by substitution (Reverse chain rule)

11.6 Integration by parts (Reverse product rule)

11.7 Partial fractions

11.8 Finding areas

11.9 The trapezium rule

11.10 Solving differential equations

11.11 Modelling with differential equations

11.12 Mixed exercise 11

11.13 Review exercise for chapter 11

 

12. VECTORS

12.1 3D coordinates

12.2 Vectors in 3D

12.3 Solving geometric problems

12.4 Application to mechanics

12.5 Mixed exercise 12

12.6 Review exercise for chapter 12

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