MathsGeeks. Everything You Need to Know A Level Edexcel C4. March 2014 MathsGeeks Copyright 2014 Elite Learning Limited

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1 Everything You Need to Know A Level Edexcel C4 March 4 Copyright 4 Elite Learning Limited

2 Page of 4 Further Binomial Expansion: Make sure it starts with a e.g. for ( x) ( x ) then use ( + x) n + nx + n(n )x!! Parametric Equations: to find + n(n )(n )x3. 3! e.g. x 7cost cos7t, 7sint + 7sin7t and 7cost 7cos7t so any point is and the gradient of the normal. 7cost 7cos7t 7sint+7sin7t Partial Fractions: There are three methods only ) Linear expand as e.g. A(x 3)+B(x ) (x )(x 3) y 7sint sin7t then. Recall that the gradient at x A + B (x )(x 3) x x 3 and compare coefficients x, x and numbers of top line the top line e.g. A + B and 3A B then solve simultaneously. ) Repeated factor in the denominator e.g. x x (x+)(x ) solve as follows A + B + x+ x C (x ) and solve as above i.e. x x A(x ) + B(x + )(x ) + C(x + ). 3) When the top is bigger than the bottom x +8x+7 divide the denominator into the numerator by long division and you are left with. bottom and solve as method ). Further Differentiation x +5x+6 x+5 x +5x+6. Factorise the Implicit Differentiation: Recognise these as questions that ask for from functions of x AND y e.g. () Find for x3 + xy y 3 5 e.g. () Find for y xey. Differentiate each term in turn e.g. () d (x3 ) + d (xy ) + d ( y3 ) d (5). Don t forget to differentiate any products using the product rule and remember that d (f(y)) d f(y). So e.g.() is 3x + y + x. y 3y. Then rearrange to make the subject so 3x y yx 3y. e.g. () d (y) d (xey ), ey y + xe and rearrange ey xey. (Remember this can be used to find a gradient of a tangent/normal etc.). Separating variables: Recognise these questions as having x, y and e.g. () x y cos x, e.g. (). Separate so that all the y s are on the side and all the x s are x + y on the side and integrate both sides. e.g. () x therefore ln y y x + ln x + + A. e.g.() y cos x therefore y sinx + A. Rate Questions: the word rate means differentiate with respect to time t. e.g. the rate of change of volume would be dv. Naturally occurring differential equations occur for exponential growth or decay e.g. growth of yeast cells given by dn kn and solving by separating variables (above) gives n n e kt. Common question type e.g. A right cylinder is expanding as heated. After t seconds the radius is x cm and the length is 5x cm. The cross-

3 sectional area of the cylinder is increasing at the constant rate of.3 cm. a) Find when the radius of the rod is cm. b) Find the rate of increase of the volume when x. a) In the question we are given da.3 therefore da and we need a relationship da between x and A. But A is the cross sectional area of a cylinder so A πx so differentiating gives da πx. Therefore.3.5 πx cms. b) The rate of increase in volume is dv dv and so using part a) we get dv and we need a relationship between V and x which for a cylinder V πx (5x) 5πx 3 differentiating to give dv 5πx and therefore dv.5 5π 4.48 cm3 s. Vectors Equation of a line is given by r x i + y j + z k + t(x i + y j + z k) where x i + y j + z k is any point on the line, x i + y j + z k is the direction of the line and t is a variable that has specific value for each point on the line. e.g. find a vector equation for the line going through A and B where A i + 6j k and B 3i + 4j + k then AB b a (3 )i + (4 6)j + ( )k i j k and therefore a vector equation of the line through AB can be written as r (i + 6j k) + t(i j + k). Intersection of lines: given two vector equations of lines and asked to show they meet (or not) you must resolve the parts i, j and k independently because for intersection the i, j and k values must ALL be the equal. e.g. if r ( 9i + k) + s(i + j k) and r (3i + j + 7k) + t(3i j + 5k) then 9 + s 3 + 3t and s t solve these two simultaneously and show these values also work for k i.e. s 7 + 5t. Length of a line the distance between two points is the length of the vector so for AB above the length of AB is AB is given by + ( ) + ( ). Angles between lines: if asked to find the angle between two lines then do cosθ r.r r r where r and r are the direction vectors of the lines e.g for A and B above cosθ (i+6j k).(3i+4j+k) and solve. Integration Techniques: recall the trapezium rule from C and apply to functions including e x and recall exact integration techniques from C. Other new techniques are summarised on the next page. Recall area under the curve is given by Area π y. Learn that the volume of revolution around the x-axis is Volπ y and around y-axis is Volπ x. Learn to use formulas in the book and don t forget to use the differentiation formulas in reverse. Page of 4

4 What it looks like How to do it Example Recognition: Use when the function has almost the differential included. e.g.() sinx cos 3 x e.g. () xe x Substitution. e.g. () x (x 3 + ) e.g. () π/ cos xsin 3 x (note they usually suggest the substitution) Partial fractions e.g. (x )(x 3) (with a quadratic in the denominator) Look at it backwards: differentiate the approximate answer. () d (cos4 x) 4( sinx)cos 3 x () d (ex ) xe x e.g. () Substitute is u x 3 + Then differentiate both sides du..3x Change the limits When x then u When x then u e.g.() Substitute is u sinx du.. cos x When x π/ then u. When x then u. Separate into partial fraction as shown earlier. (x )(x 3) A(x 3) + B(x ) (x )(x 3) Page 3 of 4 And then rearrange to give: () sinx cos 3 x 4 cos4 x + K () xe x ex + K e.g. () x (x 3 + ) 3 u du 3 [ 3 u3 ] π e.g.() cos xsin 3 x ( ) 9 u 3 du [ u4 4 ] 4 (x ) + (x 3) x + x 3 ln x + ln x 3 + K (then simplify this answer!) ce ax+b c a eax+b + K e.g. 3e x+4 3 ex+4 + K ln x Use product rule with u ln x ( du ) and x dv (v x). ln x xlnx x x xlnx x + K ax + b a ln ax + b + K e.g. ln 4 3x + K 4 3x 3 (ax + b) n (ax + e.g. b)n+ (x + 3) 5 (x + 3)6 + K a(n + ) Convert to ( + x) ( + x) ( + x) + K Integration by parts: u dv uv v du xe x xex ex (This is in the formula book) e.g. xe x Let u x and dv ex then du xex 4 ex + K and v ex. Trigonometry c cos (ax + b) c a sin(ax + b) + K e.g. 5 cos(4 3x) 5 3 sin(4 3x) + K (note the a and b are the other way around!)

5 c sin(ax + b) c a cos(ax + b) + K 7 sin(8x + 9) 7 cos(8x + 9) 8 + K sin x or cos x e.g. cos 5 x or any odd power of cos or sin. e.g. sin 3 xcos x or any multiple powers of sin and cos. Convert to cosx using cos x sin x cosx and cos x + sin x. So sin x ( cosx) or cos x ( + cosx) Convert using cos x sin x cos 5 x ( sin x) cosx Convert using cos x + sin x to give a single cos or sin. e.g. sin x ( cosx) x sinx + K. 4 (note: this comes up on almost every paper!) Expand to get ( sin x + sin 4 x) cos x sinx 3 sin3 x + 5 sin5 x + K e. g. sinx( cos x)cos x sinxcos x sinxcos 4 x cos3 x 3 + cos5 x 5 + K e.g. tan 5 x or any power of tan x Use tan x sec x e.g. tan 3 x(sec x ) sec xtan 3 x tanx(sec ) 4 tan4 x tan x + ln sec x + K ***END*** Page 4 of 4

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