Mathematics: Further Pure 2

Mathematics: Further Pure 2

1. Rational Functions and Graphs

   • Relationship between y=f(x) and y-squared =f(x)
   • Features
     • Asymptotes
     • Restrictions
     • Turning points
     • Points of intersection
   • Partial fractions
     • Includes top-heavy fractions and quadratics in the denominator which can't be factorised eg. (\(x^2\)+\(a^2\))

2. Polar Coordinates

   • Identify features of polar curves
     • Symmetry
     • Max/min r values
     • Tangents at pole
   • Sketch polar curves
   • Relationship between polar and cartesian
   • Integrate to find area of a sector
     • \(\frac{1}{2}\)\(\int\)\(r^2\)d\(\theta\)

3. Hyperbolic Functions

   • Derive and use ientities
     • \(cosh^2\)x-\(sinh^2\)x = 1 sinh2x=2sinhxcoshx
   • Sketch graphs of hyperbolic functions
   • Inverse hyperbolics
     • Derive and use expressions in terms of logarithm
   • Define hyperbolic functions in terms of exponentials

4. Differentiation and Integration

   • Use Maclaurin series of e^x, sinx, cosx and ln(1+x)
   • Derive and use the derivatives of hyperbolics
   • Derive and use derivatives of inverse trig
   • Derive and use the first few terms of the Maclaurin series of simple functions
   • Integrate given expressions and use trip or hyperbolic substitutions to integrate
     • Integrate: \(\frac{1}{\sqrt(a^2-x^2)}\)\(\frac{1}{\sqrt(x^2-a^2)}\)\(\frac{1}{\sqrt(x^2+a^2)}\)\(\frac{1}{(a^2+x^2)}\)
   • Derive and use reduction formulae to integrate
   • Approximate area under a curve using rectangles and use to set bounds for the area

5. Numerical Methods

   • Convergence (and failure) of iterative formulae
     • Staircase
     • Cobweb
   • Errors
     • The ratio of two errors is approximately F'(X)
     • The subsequent error is proportional to the previous error squared if F'(X) = 0
   • Newton-Raphson
     • When does it fail?
   • Derive and use Newton-Raphson iterations