Further Pure 1

What further pure 1 covers

One of the eight optional papers of 9FM0, sat as Paper 3A, so revise it only if it is on your timetable. It is also the gateway to Further Pure 2, which may be taken only as a matching pair with it. The content is broad: half-angle substitutions, series, conics, cross products and the limits ordinary calculus does not reach.

The main ideas

• The t-formulae with t equal to the tangent of half the angle, used to prove identities and to turn a cos x + b sin x = c into a quadratic.
• Taylor series about a general point, assembled from a table of derivatives at the anchor, and the Taylor method for series solutions of differential equations.
• Limits of indeterminate forms, by series expansion or by L'Hospital's rule, with products, differences and power forms forced into a quotient first.
• Leibnitz's theorem for the nth derivative of a product, and the Weierstrass substitution for rational trigonometric integrals.
• The four standard conics in cartesian and parametric form, with eccentricity, foci, directrices and asymptotes, then their tangents, normals and loci.
• The vector product as an area and the scalar triple product as a volume, with the coplanarity test and direction cosines.
• Forward and central difference approximations, Simpson's rule, and rational and modulus inequalities solved from a sign diagram.

The results it turns on

with t = tan(θ/2): sin θ = 2t/(1 + t²), cos θ = (1 − t²)/(1 + t²)

the t-formulae, which do not reach theta = pi

f(x) = f(a) + (x − a)f'(a) + (x − a)²f''(a)/2! + …

the Taylor series about x = a

the limit of f/g becomes the limit of f'/g'

L'Hospital's rule, while the form stays indeterminate

parabola (at², 2at); ellipse (a cos t, b sin t); hyperbola (a sec t, b tan t); rectangular hyperbola (ct, c/t)

the four conics and their parametrisations

|a × b| = |a||b| sin θ, and |a·(b × c)| is a parallelepiped volume

the vector product as an area, the triple product as a volume

(h/3)[y₀ + 4y₁ + 2y₂ + … + 4y_n−1 + y_n], with n even

Simpson's rule, on an even number of strips

Where it usually goes wrong

• The half-angle substitution cannot produce the angle pi, so that value is tested by hand whenever the interval contains it.
• L'Hospital's rule differentiates the top and the bottom separately. Reaching for the quotient rule here computes something else entirely.
• The eccentricity relations differ in sign between the ellipse and the hyperbola. Read them across the booklet's conics table rather than trusting memory.
• A tetrahedron takes one sixth of the parallelepiped built on the same three vectors, not one third, and the triple product needs its modulus taken before it is called a volume.

Where to start

The t-formulae and the Weierstrass substitution belong together, since both run on the same substitution. Taylor series before both limits and series solutions. Conics is the longest block and deserves its own sessions, with tangents and normals after the standard forms. Vector products, numerical methods and inequalities are independent of everything else here.