AQA Maths Core 2

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Indices

1. am x an = am+n

 

2. am / an = am-n

 

3. (am)n = amn

 

4. a-n = 1/an

 

5. a0 = 1

 

6. a1/n = nÖa

 

7. am/n = nÖam

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Futher Differentiation

If y=xn then dy/dx  = nxn-1

 If y=cxn then dy/dx  = cnxn-1

 If y=f(x)+g then dy/dx = f’(x)+g’(x)

 Tangent to (x1,y1) is given at y-y1 = m(x-x1) when m=gradient at the point

 Normal to a point has gradient of 1/m where m = gradient of tangent

 Stationary point ® second derivative: negative = maximum , positive = minumun

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Futher intergration & trapezium rule

If dy/dx=xn then y=(1/n+1)xn+1 +c

 

In definite integral ® no limits ® need a +c

 

Definite integral has limits ® then do F(b)-F(a)

 

Trapezium rule:  (http://images-mediawiki-sites.thefullwiki.org/09/4/0/7/4213441325076158.png)      where h=(b - a)/n

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Basic Trigonometry

y=sinθ crosses at 0, 180, 360

 

y=cosθ crosses at 90, 270

 

y=tanθ crosses at 0, 180, 360 heading up

 

Area = ½ ab sinC

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Simple transformation of graphs

A translation of         transforms the graph of y=f(x) to y=f(x)+b

A translation of         transforms the graph of y=f(x) to y=f(x-a)

A translation of         transforms the graph of y=f(x) to y=f(x-a)+b

A reflection in y=0 transforms the graph of y=f(x) to y=-f(x)

A reflection in x=0 transforms the graph of y=f(x) to y=f(-x)

A stretch of scale factor d in the y direction transforms the graph of y=f(x) to y=df(x)

A stretch of scale factor c in the x direction transforms the graph of y=f(x) to y=f(x/c)

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Solving Triginmetrical equations

To solve sin bx=k substitute u for bx to simplify, find interval of u, solve sin u=k, substitute back in

cos2θ + sin2θ = 1

tanθ=sinθ/cosθ

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Factorials and Binomial expansions

n!= n x (n-1) x (n-2) ... x 2 x 1

n!= n x (n-1)!

0!=1

nCr is the number of ways of taking r objects out of n objects

Pascal’s Triangle adds the two numbers above to get the number below

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Sequences and Series

Inductive relates one to the previous         eg un=un-1+4n3

 

Arithmetic have first term a, common difference d , last term l 

 eg a + (a+d) + (a+2d)...

A series is an added sequence, sigma notation means series - used as short hand

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Radian Measures

1° = 2p/360 C

 

l=rθ

 

A= 0.5r2θ

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Futher trigonmetry with radians

30=p/6

 

45=p/4

 

60=p/3

 

90=p/2

 

180=p

 

360=2p

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Exponentials and logarithms

The curve y=ax will always pass through (0,1) when x is positive

 

Logarithm of y to base a is written as logay

y=ax  ® x=logay

 

logaa=1    loga1=0

 

logax + logay = logaxy

 

logax - logay = loga(x/y)

 

k logax = loga(xk)

 

ax=b   ®   x= logb/loga          ß Take logs of both sides, rearrange and solve by calculator

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Geometrical series

a+ar+ar2+ar3+ar4….  First term a, common ratio r

 

S¥=a/(1-r)

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