Core Flashcards
Method of differences
If Uₙ = f(n) - f(n+1)
then sum to n from r=1 = f(1) + f(n+1)
Sums of cubes for quadratic and cubic eq
a^3+b^3 = (a+b)^3-3ab(a+b)
a^3+b^3+y^3
= Σa^3 - 3(Σa)(Σab)+Σaby
Conditions for a transformation to be linear
- Maps origin onto itself
- Can be represented by a matrix
- |ax+by|
|cx+dy|
Properties of linear transformations
T|kx| kT|x|
|ky| = |y|
T((x+x0)(y+y0) = T(x,y) + T(x0,y0)
Properties of stretches (linear transformations)
|a 0|
|0 b| stretch sf a parallel to x axis and b to y axis
stretch parallel to x axis - points on y axis (x=0) invariant n vice versa
reflections self inverse P^2 = I
detM = area scale factor - negative if shape is reflected, 1 if shape is rotated
Reflections about x,ynz axes
x |1 0 0| y |c 0 s|
|0 c -s| |0 1 0|
|0 c s| |-s 0 c|
z |c -s 0|
|s c 0 |
|0 0 1| s=sin etc angle is measured in anticlockwise direction from positive axis
Order of applying multiple transformations
AB (x) = B then A
f(n) = 3^4n + 2^(4n+2)
show f(k+1)-f(k) div by 15
then prove f(n) is div by 5
Write out subtraction, factor 3 n 2 terms, note that 3 coefficient (80) is 5x16, now there’s a factor of 3^k x 5, which will always be a multiple of 15 :)
Assume that f(k) is divisible by 5.
It would follow that f(k)=5m = where
m is an integer
f(k+1) = f(k) +5(16)(3^4k) + 60(2^4k)
take out factor of 1/3
15(16)(3^4k-1) + 60(2^4k)
also non negative integers means prove for 0 azwel
1/i
-i (i/i^2)
argand straight line
Re(z) = k
meth odf differences simplifications for sum of f(r)-f(r+1) n f(r+2)
sum (fr - fr+1 = f(1)-f(n+1)
fr - fr+2 = f1 + f2 - f(n+1) - f(n+2)
What makes a function improper
+ convergent/divergent
If one of the limits is infinite, or if the function is undefined at one of the limits, or in between the interval [a,b]
Convergent - if improper integral exists
Divergent - if integral doesn’t exist
Dont write integral from -inf to inf as integral from -t to t, instead from -t to 0 then 0 to t
both must converge to say function converges
When to STOP using D,I meth
When there’s a 0 in the d row, or where you can integrate product of a row
Mean value transformations
f(x)+k — f+k
kf(x) —– kf
-f(x) —— -f*
mean of f(x) + g(x) = sum of means on interval
where f* is the mean value
Geometric reasoning to explain why mean value of sinx^5 is 0 on interval [0,2pi]
Integral on interval [0,π] is negative of integral on interval [π,2π], so summing them gives 0
Definition of mean value of a function
Average value of a function over a given domain
Don’t forget to square in volumes of rev
Yes king
Polar integration tip
Integrating in 3rd n 4th quadrants doesn’t give negatives
Find values that give beginning and end of loop by solving r=0 e.g. r=asin4θ (0 n π/4)
general solution to an equation with auxiliary solution sqrtk(k^2-n^2) where n>k and both are positive
Acos(sqrtn^2-k^2)…
as roots = isqrt(n^2-k^2) since n>k inside will be neg
Invariant point Vs invariant line Vs line of invariant points
Point - point remains in same spot after matrix transformation (0,0) a solution for all linear transformations
Line - points that start on the line remain on the line (not necessarily remaining fixed)
Line of points - points that remain fixed, on a line.
Centre of a polygon definition
The centre of a circle passing through all of the polygon’s vertices
Conjugate of a complex number don’t get caught on lack
a-bi - IMAGINARY PART NEGATED
Finding eq of tangent polar edition
E.g parallel to og line, calculate theta then sub in r=y/sintheta to find eq in terms of y, then after that sub back into
Conditions on certain integrals
For arcsin(x/a) mod a needs to be less than x (cant arcsin smth more than one)
For arctan a needs to be greater than 0 and above
Integrating smth to minus one
For LN, MAKE SURE U CHECK IF X COEFFICIENT IS NEG - THEN DIVIDE BY -1 FOR REVERSE CHEN RUL MAYOWA PLEASE
Damped oscillations
discriminant>0 two roots heavy damping. No oscillations as resistive force large compared to restoring force.
Discriminant = 0 critical damping no oscillations
Discriminant<0 light damping oscillations decrease in magnitude exponentially
Heavy and critical motion determined from initial conditions. Light from period of oscillations
Particular integral
Equation which satisfies differential eqs
constant a
linear ax+b
quad ax^2…
pe^kx = ae^kx
pcoswx+qsinwx = acoswx + bsinwx
Ways of shifting summations
Sum from 0 to n of f(k) = Sum from 1 to n+1 of f(k-1)
When one of the auxilliary roots is zero
Multiply PI by x
Same as cf and both contain a constant term
Matrix properties
Non commutative - AB not BA
If AB exists doesn’t mean BA exists
Associative - (BC)A = B(CA)
Distributive A(B+C)=AB+AC
Orientation reversed after linear t if
Determinant less than one
Period of trig function
2pi/coefficient
for tan piover
Damped vs forced harmonic motion
Damped - homogeneous, forced is non homogeneous (=f(t))
Arcosh
positive root, valid for x>/1
x+sqrt(x^2-1), multiply by conjugate for diff of squares, rearrange for one then ln both sides - lnu = -ln(1/u)
each solution represents each x value that can produce y value
Convex vs dimple polar curves
If a curve isn’t convex there will be more than two tangents perpendicular to the initial line
Hyp functions
Sinh looks like a cubic, cosh a quadratic, tanh like arctan
Sinh odd, cosh even, tanh odd
Parametric integration
y(t)dx/dt dt
for vs of rev, piy^2dx/dt dt
area of rhombus + parallelogram
Rhombus - 1/2 d1d2
Parallelogram base x PERP height
