FP1

FP1

1. Complex Numbers

   • i= √(-1)
     • i^2= -1
     • i^odd= ± i
     • i^even= ± 1
   • Dividing
     • "realise" the denominator
       • if you have x/(a+bi)
         • times by (a-bi)/(a-bi)
           • sames as x1
           • complex conjugate
             • if z = a+ib
               • z*=a-ib
                 • zz*= real no.
                   • difference of 2 squares
       • make bottom a real no.
   • modulus and argument
     • argand diagram
       • vectors
         • magnitude
           • modulus, r or |z|
             • length
               • pythag
         • direction
           • argument, Arg(z)
             • angle between vector and x-axis
               • tanθ= y/x
     • modulus-argument form
       • if z=x+iy
         • by trig
           • x=rcosθ
           • y=rsinθ
         • z=r(cosθ+i sinθ)
   • in equations
     • equating real and imaginary
       • if a+ib=c+id
         • a=c and b=c
       • find √(15+8i)
         • let √(15+8i)= a+ib
           • 15+8i=(a+ib)^2
             • etc.
     • quadratic roots
       • if z is a root
         • so is z*

2. Co-ordinate Systems

   • Parabola
     • equation
       • cartesian
         • y^2=4ax
       • parametric
         • x=at^2
           • y=2at
   • Rectangular Hyperbola
     • equations
       • cartesian
         • xy=c^2
       • Parametric
         • x=ct
           • y=c/t
   • tangents and normals
     • m(t)=dy/dx
       • m(n)=-1/m(t)
         • m(n)x m(t)=-1
     • y-y1=m(x-x1)

3. Proof by Induction

   • method
     • prove true for n=1
       • assume true for n=k
         • show true for n=k+1
           • state true for all n≥1 where nϵ Ζ+
   • types
     • series
       • swap k for n
         • NOT r
     • divisibility
       • consider f(k+1)-mf(k)
         • show difference is divisible by a
           • therefore f(k+1) is also divisible by a
             • remember rules of indicies
               • a^n x a^m= a^n+m
               • (a^n)^m= a^nm
     • Matrices
       • Sub step 2 into step 3
         • M^k+1
           • same as M(M^k)
     • recurrence relationships
       • show true for
         • n=1 AND n=2
           • for step 3
             • use recurrence formula for U k+1
               • sub U k in
   • write your target!

4. Series

   • General formulae
     • r
     • r^2
     • r^3
     • 1
   • sum between 2 limits
     • sum up to top limit
       • minus sum up to bottom limit -1
   • to show a summation formula = ....
     • take out common factors

5. Numerical Methods

   • show root in interval [a,b]
     • find f(a) and f(b)
       • change in sign
         • root between a and b
   • Interval bisection
     • next estimate
       • midpoint
         • (a+b)/2
         • between a and b
           • where f(a) -ve and f(b) +ve
   • linear interpolation
     • if root lies in [a,b]
       • use similar triangles
       • or formula
         • not in data book
   • Newton Raphson
     • find f'(x)
       • write out f(a) and f'(a)
     • in formula book
       • doesn't work @ turning point
         • f'(x)=0

6. Matrices

   • multiplying
     • only multiply if
       • no. of columns of 1st matrix
         • same as no. of rows of 2nd
           • product dimensions
             • same no. rows as 1st matrix
             • Same no. of columns as 2nd
     • not comutative
       • AB≠BA
   • dimensions
     • (nxm)
       • n= rows
         • m=columns
           • simultaneous equations
   • Transformations
     • vectors
       • position vector
         • from origin
           • can be written (x,y)
             • translation vector
               • from given point
     • linear
       • linear expressions
         • point (0,0) unchanged
           • reflections
             • y-axis
               • x-axis
             • y=x
               • y=-x
           • enlargement
             • scale factor a
           • rotation
             • 180°
             • 90°
               • clockwise
               • anticlockwise
             • 45°
               • clockwise
               • anticlockwise
     • inverse matrix
       • transforms back to original
   • Inverse
     • A^-1
       • AA^-1=I
         • I= identity matrix
         • determinant
           • ad-bc
             • singular if =0
               • transform shape
                 • straight line
                   • 0 area

Bilder und Dateien der Mindmap