C4 REVISION

C4 REVISION

Partial Fractions

• Set up as 3 separate fractions – usually one its own, one squared and one of each next to other

• Sub in values of x to determine A, B and C (first 2 picks should make a bracket worth 0)

• If you have to integrate at the end don’t forget to look for logs and to put +C on the end!!

Binomial

• Formula in booklet – no need to learn• Don’t forget (-3x)² = + 9x²• To get range of valid values, reciprocate coefficient of x.

e.g. (1 + 4x)½ is |x| < ¼

R - α

• If they don’t give you an identity to use then pick Rcos(θ – α)

• Quick trick – square, add and root coefficients to get R and use

tan -1 sin coefficient to get α cos coefficient

Double angle formulae• You MUST learn the following:• sin 2θ = 2sinθcosθ Generally, pick the formula that will eventually• cos 2θ = cos2θ – sin2θ set up a quadratic (or something that

factorises).• cos2θ = 1 – 2sin2θ Sometimes, these will also have to be

rearranged - • cos2θ = 2cos2θ – 1 we cant ∫ sin2 or cos2 so rearrange to get• tan2θ = 2tanθ ½ + ½ cos2θ etc 1- tan2θ

Parametric Equations

• Find dx/dt and dy/dt then calculate dy/dt x dt/dx to get dy/dx

• This gives us the gradient of the Tangent. For a ‘normal’ reciprocate and negate.

• You will probably then have to sub in some value of p to create a given expression. If this is a cubic that needs to be solved, do this by trial and improvement (its usually either 1, -1, 2 or -2)

Volume of Solids of Revolution• For y = ƒ(x) then vol = π ∫ y²dx.• The limits are the coords that cut the x axis.• Look out for Trig!! Remember we cant ∫ sin² or cos² so swap it

using the ‘double angle formulae’• If the function is a y = 3+x² type, when you square it don’t

forget to write as (3+x²)(3+x²) to get 9+6x²+x4 NOT just 9+x4

• Leave in terms of π unless otherwise stated.• Look out for top being differential of bottom – Logs!!!

Integration by parts

• Formula in booklet• ALWAYS use ln x = u and ex = dv/dx

Integration by Substitution

• LEARN THIS ROUTINE :• Step 1 – find du/dx and rearrange so we can get rid of the

dx• Step 2 – find the new limits by substituting the original

limits into the u = equation• Step 3 – re-write the whole integral with new limits, new u

and new du• Step 4 – integrate this and finish

Differential Equations

• If a simple direct proportion, write as dp/dt = kp then separate and ∫ to get ∫ 1/p dp = ∫ k dt which gives ln |p| = ½kt

• Now take “e’s” to get p = Aekt (not always a simple direct proportion though!!)

• They’ll give you an initial value (i.e. t = 0) which you sub in to get A then another value to work out k.

• Check your final answer make sense in context!!

Vectors• Square, add, root to find magnitude of vector |a|• AB = b – a• Parallel vectors are multiples of each other e.g. 3i + 2j + 5k and 9i + 6j +

15k are parallel• Dot product a.b = aibi + ajbj + akbk and = 0 if PERPENDICULAR• Angle between vectors cos θ = a.b |a||b|• Intersecting lines: set up simultaneous equations to work out λ and μ : If they intersect λ and μ satisfy all 3 equations If skew they only satisfy 2 equations If parallel there are no solutions for λ and μ

Proof by contradiction

• Learn √2 proof from notes. √3 and √5 proofs are the same except swap all 2’s for 3’s/5’s.

• Other proofs usually involve getting a quadratic and showing that they have no roots.