01. Differentiation 1
- Gradient of a curve
- First principles
- The dy/dx notation
- Tangents and Normals
- Velocity and Acceleration
02. Differentiation 2
- d/dx and f'(x) notation
- Greatest and Least values
- Maxima and Minima
- Curve sketching
- Second derivative
- Small changes
03. Integration
- Reverse of differentiation
- Velocity and Acceleration
- Area under a curve
- Definite integrals
- Area cut off by a line
- Area enclosed by two curves
- Solids of revolution
- Center of gravity
04. Further Differentiation
- The Chain rule
- Rates of change
- Products and Quotients
- Implicit functions
- Parameters
- Second derivative
05. Algebra 1
- Rules of Surds
- Rules of Logarithm
- Quadratic equations
- Completing the square
- The quadratic formula
- The discriminant b²-4ac
- Mini and maxi values
- Sum and Product of roots
- The Remainder and Factor theorems
06. Binomial theorem
- The Pascal’s triangle
- Factorial notation
- Combinations
- The Binomial theorem
- The term independent of x
- Binomial theorem for any index
07. Algebra 2
- Inequalities
- Linear inequalities
- Quadratic inequalities
- Rational fraction inequalities
- Further equation methods
- Square root equations
- Equations reducing to quadratics
- Simultaneous equations in 3-unknowns
- Substitution and elimination
08. Series
- Arithmetical progression (AP)
- General term of an AP
- Sum of an AP
- Geometrical progression (GP)
- General term of a GP
- Sum of a GP
- Sum to infinity of a GP
- Proof by induction
- Further series
09. Trigonometry 1
- Trigonometric ratios of angles
- Ratios of 30°, 45°, and 60°
- Secant, cosecant and cotangent
- Trigonometric equations
- Pythagoras theorem
- Compound angle formula
- Double angle formula
- The t– formula
- The form acos(x) + bsin(x)
- The factor formula
- Cosine and Sine rule
- Area of a triangle
10. Trigonometry 2
- Radian measure
- Length of an arc and area of a sector
- Differentiation of trig functions
- Small angles
- Integration of trig functions
- Reverse differentiation
- Even and odd powers
- Using factor formula
- Definite integrals
- Differentiation of inverse trig
- General solutions
11. Further integration 1
- Recognizing a function and its derivative
- Change of variable
- Changing limits
- Integration using inverse trig functions
- Using sin⁻¹(x)
- Using tan⁻¹(x)
- Quadratic denominators
- Definite integrals involving trig functions
12. Exponential and Log functions
- The Exponential function
- Deriving y = eˣ
- The natural logarithm function
- Recognizing a function and its derivative
- Logarithm of negative limits
- d/dx(aˣ) and ∫(aˣ) dx
- Logarithmic differentiation
13. Partial fractions
- CASE I: Denominator with only linear factors
- Cover-up method
- CASE II: Denominator with a quadratic factor
- CASE III: Denominator having repeated roots
- CASE IV: Improper fractions
- Differentiation after Partialization
- Integration after Partialization
- Binomial expansion after Partialization
14. Further integration 2
- Integration by parts
- Integration by parts with limits
- Taking dv/dx as 1
- Integration by parts more than once
- Integration by parts where the original integral appears again
- Change of variable t = tan(ˣ⁄₂)
- Change of variable t = tan(x)
- Splitting the numerator
15. Coordinate geometry 1
- The Circle
- Cartesian equation
- Parametric equation
- Parametric equation
- Circle from 3 points
- Circle from diameter ends
- Intersection of circle and line
- Gradient at a point
- Tangent from a point
- Intersection of 2 circles
- Orthogonal circles
- Conics 1 [The Parabola]
- The Parabola (e = 1)
- Form (y – k)² = 4a(x – h)
- Tangents / Normals
- Parametric equation
- Chord to parabola
18. Differential equations
- Solving differential equations
- By separating variables
- Inserting boundary conditions
- Exact differential equations
- Integrating factor method
- Homogenous equations
- Other useful substitutions
- Differential equations problems
- Forming differential equations
19. Complex numbers
- Algebra of complex numbers
- Conjugate complex
- Square root a complex number
- Further complex roots
- The Argand diagram
- Modulus and Argument
- Laws of Modulus
- Rules of Arguments
- Form r(cosθ + i sinθ)
- De Moivre’s theorem
- Identities in form cosⁿθ or sinⁿθ
- Identities in form cosⁿθ or sinⁿθ
- The nᵗʰ root of a complex number
- Complex loci
- Complex loci with inequalities
20. Vectors in 3D
- The Unit Vector
- Products of vectors
- Scalar product (Dot product)
- Vector product (Cross product)
- Vector equation of a line
- Intersection of 2 lines
- Angle between 2 lines
- Shortest distance of a point from a line
- The plane
- Area of a Parallelogram
- Area of a Triangle
- Intersection of planes
- Angle between planes
- Intersection of line and plane
- Shortest distance of a plane from a point