1 / 12

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)

salene
Download Presentation

C4 REVISION

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. C4 REVISION

  2. 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!!

  3. 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

  4. 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

  5. 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 cos2θetc 1- tan2θ

  6. 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)

  7. 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!!!

  8. Integration by parts • Formula in booklet • ALWAYS use ln x = u and ex = dv/dx

  9. 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

  10. 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!!

  11. 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 μsatisfy all 3 equations If skew they only satisfy 2 equations If parallel there are no solutions for λ and μ

  12. 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.

More Related