# Mathematics: Further Pure 2

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## A-level Maths (FP2) Mind Map on Mathematics: Further Pure 2, created by declanlarkins on 01/24/2014.

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Mathematics: Further Pure 2
1 Rational Functions and Graphs
1.1 Relationship between y=f(x) and y-squared =f(x)
1.2 Features
1.2.1 Asymptotes
1.2.2 Restrictions
1.2.3 Turning points
1.2.4 Points of intersection
1.3 Partial fractions

Annotations:

• Includes top-heavy fractions and quadratics in the denominator which can't be factorised eg. ($$x^2$$+$$a^2$$)
2 Polar Coordinates
2.1 Identify features of polar curves
2.1.1 Symmetry
2.1.2 Max/min r values
2.1.3 Tangents at pole
2.2 Sketch polar curves
2.3 Relationship between polar and cartesian
2.4 Integrate to find area of a sector

Annotations:

• $$\frac{1}{2}$$$$\int$$$$r^2$$d$$\theta$$
3 Hyperbolic Functions
3.1 Derive and use ientities

Annotations:

• $$cosh^2$$x-$$sinh^2$$x = 1 sinh2x=2sinhxcoshx
3.2 Sketch graphs of hyperbolic functions
3.3 Inverse hyperbolics
3.3.1 Derive and use expressions in terms of logarithm
3.4 Define hyperbolic functions in terms of exponentials
4 Differentiation and Integration
4.1 Use Maclaurin series of e^x, sinx, cosx and ln(1+x)
4.2 Derive and use the derivatives of hyperbolics
4.3 Derive and use derivatives of inverse trig
4.4 Derive and use the first few terms of the Maclaurin series of simple functions
4.5 Integrate given expressions and use trip or hyperbolic substitutions to integrate

Annotations:

• Integrate: $$\frac{1}{\sqrt(a^2-x^2)}$$$$\frac{1}{\sqrt(x^2-a^2)}$$$$\frac{1}{\sqrt(x^2+a^2)}$$$$\frac{1}{(a^2+x^2)}$$
4.6 Derive and use reduction formulae to integrate
4.7 Approximate area under a curve using rectangles and use to set bounds for the area
5 Numerical Methods
5.1 Convergence (and failure) of iterative formulae
5.1.1 Staircase
5.1.2 Cobweb
5.2 Errors
5.2.1 The ratio of two errors is approximately F'(X)
5.2.2 The subsequent error is proportional to the previous error squared if F'(X) = 0
5.3 Newton-Raphson
5.3.1 When does it fail?
5.4 Derive and use Newton-Raphson iterations