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Further Pure 1 formulae

The 7 results the Further Pure 1 notes display, each with the conditions it needs to be true and a link to the lesson that states it. Of the 7, the formulae booklet prints 2 and 5 are yours to memorise.

Further Maths · 6 lessons with formulae.

A result the Edexcel formulae booklet prints is marked as such; one it does not print is marked to learn; and a result badged derived in the lesson is a working line the notes build in front of you rather than a standard result to memorise.

6 lessons

The t-formulae

sinθ=2t1+t2,cosθ=1-t21+t2,tanθ=2t1-t2\sin θ = \frac{2t}{1 + t^{2}}, \cos θ = \frac{1 - t^{2}}{1 + t^{2}}, \tan θ = \frac{2t}{1 - t^{2}}NOT IN THE BOOKLET: LEARN ITConditionst = tan(θ/2), so θ/2 must avoid odd multiples of 90°; the tan form additionally needs t² ≠ 1.

Taylor series

f(x)=f(a)+f′(a)(x-a)+f′′(a)2!(x-a)2+…\text{f}(x) = \text{f}(a) + \text{f}'(a)(x - a) + \frac{\text{f}''(a)}{2!}(x - a)^{2} + …NOT IN THE BOOKLET: LEARN ITConditionsf differentiable at a to the order used, and x inside the interval where the series converges.

Leibniz's theorem and the Weierstrass substitution

(fg)(n)=ΣnCrf(r)g(n-r)(\text{fg})^{(n)} = Σ \, ^{n}C_{r} \, \text{f}^{(r)} \text{g}^{(n-r)}NOT IN THE BOOKLET: LEARN ITConditionsf and g both n times differentiable.

Conic sections

x2a2+y2b2=1,x=acost,y=bsint\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, x = a \cos t, y = b \sin tIN THE FORMULAE BOOKLETConditionsa > 0 and b > 0; the parametrisation traces the whole ellipse once as t runs from 0 to 2π.

The vector product and the scalar triple product

a×b=(a2b3-a3b2,a3b1-a1b3,a1b2-a2b1)\text{a} × \text{b} = (a_{2}b_{3} - a_{3}b_{2}, \, a_{3}b_{1} - a_{1}b_{3}, \, a_{1}b_{2} - a_{2}b_{1})IN THE FORMULAE BOOKLETConditionsComponents taken in a right-handed set of axes.

Numerical methods for differential equations

dydx≈yn+1-ynh or yn+1-yn-12h\frac{dy}{dx} \approx \frac{y_{n+1} - y_{n}}{h} \text{ or } \frac{y_{n+1} - y_{n-1}}{2h}NOT IN THE BOOKLET: LEARN ITConditionsBoth improve as the step h shrinks, and the central form is the more accurate for the same h, but it needs a value on each side of the point.
∫abydx≈h3(y0+4y1+2y2+…+4yn-1+yn)\text{∫}_{a}^{b} y \, dx \approx \frac{h}{3}(y_{0} + 4y_{1} + 2y_{2} + … + 4y_{n-1} + y_{n})NOT IN THE BOOKLET: LEARN ITConditionsAn even number of strips of equal width h, so the strips pair off, with three ordinates to each pair. With an odd number of strips the rule does not apply.

Every formula in the library is indexed by unit on the equation index, and the Further Pure 1 notes are where these results are derived and used. The 5 the booklet does not print are on the list of every formula to memorise, alongside the rest of the course.

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