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Notes MathematicsPure Mathematics 3

Mathematics 9709

Pure Mathematics 3 — Revision Notes

Hover a chapter to open it, then pick a topic to start reading.

1.1The modulus function
1.2Graphs of y = |f(x)|
1.3Solving modulus inequalities
1.4Division of polynomials
1.5The factor theorem
1.6The remainder theorem

2.1Logs and the laws: the toolkit you apply everywhere
2.2Solving ln / log equations (combine, then undo with e)
2.3Unknown stuck in the power: a^x = b and the hidden quadratic
2.4e^x and ln x: inverse twins, graphs, and the e/ln cancel
2.5Straightening curves: turn y = kxⁿ or y = Ae^(bx) into Y = mX + c

3.1Cosecant, secant and cotangent
3.2Compound angle formulae
3.3Double angle formulae
3.4Further trigonometric identities
3.5Expressing a sin θ + b cos θ as R sin(θ ± α) or R cos(θ ∓ α)

4.1Product and Quotient Rule
4.2Differentiating e^x, a^x and ln x
4.3Differentiating sin, cos and tan
4.4Implicit Differentiation
4.5Parametric Differentiation

5.1Integration of exponential functions
5.2Integration of 1/(ax + b)
5.3Integration of sin, cos and sec² of (ax + b)
5.4Further integration of trigonometric functions
5.5Definite integrals and the area under a curve

6.1Locating a Root: Sign Change and Sketching a Pair of Graphs
6.2Rearranging f(x)=0 to x=F(x), and Proving It Converges to the Root
6.3Running the Iteration to a Required Accuracy (and Confirming the Root)

7.1Improper algebraic fractions
7.2Partial fractions
7.3Binomial expansion for rational n
7.4Binomial expansion of (a + x)ⁿ
7.5Partial fractions with binomial expansions

8.1Integrating to ln: when the top is the derivative of the bottom
8.2The arctan integral: 1/(x²+a²) and splitting (linear)/(x²+a²)
8.3Integration by substitution (when they give you the substitution)
8.4Integrating rational functions using partial fractions
8.5Integration by parts: integrating a product

9.1Displacement vectors
9.2Position vectors
9.3Unit vectors
9.4The scalar (dot) product
9.5The vector equation of a line
9.6The angle between two lines
9.7Intersection of two lines
9.8Foot of perpendicular & shortest distance

10.1What a differential equation is, and how to solve one
10.2Solve a given DE and rearrange to the requested form
10.3When the integral needs partial fractions
10.4Forming a DE from a rate statement, and interpreting the solution

11.1Imaginary numbers
11.2Complex arithmetic and the conjugate
11.3The Argand diagram: modulus and argument
11.4Polar and exponential form; multiply and divide
11.5Solving equations with complex roots
11.6Loci in the complex plane

Mathematics Pure Mathematics 3 — Revision Notes · Conquer Your A Levels