Unit FP2: Further Pure Mathematics 2

The manipulation and solution of The solution of inequalities such as algebraic inequalities and 1 x inequations, including those >, | x2 − 1 | > 2(x + 1). x − a x − b involving the modulus sign.

Summation of simple finite series n 1 ∑ using the method of differences.; Students should be able to sum series such as by r(r +1) r=1 1 1 1 using partial fractions such as = −. r(r +1) r r +1.

Euler’s relation eiθ = cos θ + i sin θ.; Students should be familiar with cos θ = (eiθ + e−iθ) and sin θ = (eiθ − e−iθ). 2i.

De Moivre’s theorem and its To include finding cos nθ and sin mθ in terms of powers of application to trigonometric sin θ and cos θ and also powers of sin θ and cos θ in terms identities and to roots of a complex of multiple angles.; Students should be able to prove number.; De Moivre’s theorem for any integer n.

Interpret and sketch complex-number loci and regions defined by modulus and argument conditions, including |z − a| = b, |z − a| = k|z − b| and arg(z − a) = β.

Elementary transformations from az + b Transformations such as w = z2 and w =, where the z-plane to the w-plane. cz + d a, b, c, d ∈, may be set. ℂ.

Further solution of first order The formation of the differential equation may be required. differential equations with Students will be expected to obtain particular solutions and separable variables. also sketch members of the family of solution curves.

First order linear differential The integrating factor e ∫Pdx may be quoted without proof. dy equations of the form + Py = Q dx where P and Q are functions of x.

Differential equations reducible to the above types by means of a given substitution.

The linear second order differential The auxiliary equation may have real distinct, equal or d2y dy complex roots. f(x) will have one of the forms equation a + b + cy = f(x) dx2 dx k epx, A + Bx, p + qx + cx2 or m cos ωx + n sin ωx. where a, b and c are real constants Students should be familiar with the terms ‘complementary and the particular integral can be function’ and ‘particular integral’. found by inspection or trial.; Students should be able to solve equations of the form d2y + 4y = sin 2x. dx2.

Differential equations reducible to the above types by means of a given substitution.

FP2.6.1Third and higher order derivatives

Third and higher order derivatives.

FP2.6.2Derivation

Derivation and use of Maclaurin The derivation of the series expansion of ex, sin x, cos x, series. ln (1 + x) and other simple functions may be required.

FP2.6.3Derivation

Derivation and use of Taylor series.; The derivation, for example, of the expansion of sin x in ascending powers of (x − π) up to and including the term in (x − π)3.

FP2.6.4Taylor series method for differential equations

Use of Taylor series method for Students may, for example, be required to find the solution series solutions of differential in powers of x as far as the term in x4,of the differential equations. equation d2y dy dy + x + y = 0, such that y = 1, = 0 at x = 0. dx2 dx dx.

Use polar coordinates (r, θ) with r ≥ 0 and sketch standard polar curves, including lines, circles, spirals, cardioids, limacons and lemniscates.

β The ability to find tangents parallel to, or at right angles to, Use of the formula 1 ∫ r2 dθ 2 the initial line is expected. α for area.