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
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You MUST learn the following:
sin 2θ = 2sinθcosθ
cos 2θ = cos2θ – sin2θ
Generally, pick the formula that will eventually
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
1- tan2θ
½ + ½ cos2θ etc
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
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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.