FP1 Flashcards
Sum of roots
alpha + beta
-b/a
Product of roots
Alpha X beta
c/a
For new roots α^2 and β^2, what are a b and c
α^2 + β^2 = (α+β)^2 - 2αβ = (-b/a)^2 -2(c/a)
α^2β^2 = (αβ)^2 = (c/a)^2
New roots mα and mβ. What is equation?
Replace x in original equation by y/m and simplify
Σr from 1 to n
1/2n(n+1)
Σr^2 from 1 to n
1/6n(n+1)(2n+1)
Σr^3 from 1 to n
1/4n^2(n+1)^2
Identity matrix
(1 0)
0 1
How to transform point (x,y) by matrix (ab|cd)
M(x|y) = (ab|cd)(x|y)
Reflection matrix in y=X
(0 1)
1 0
Reflection in y=tanθx
(cos2θ sin2θ)
sin2θ -cos2θ
Rotation anti-clockwise matrix
(cosθ -sinθ)
sinθ cosθ
How to find vertical asymptote of rational function
What makes denominator 0
How to find horizontal asymptote of rational function
cancel constants and see what happens as x tends to infinity
If denominator of a quadratic rational function doesn’t have real roots
The curve will not have a vertical asymptote
How to find stationary points of rational function
Make curve = k and solve, putting in form b^2-4ac =0
General formula for a parabola
y^2=4ax
General formula for a ellipse
x^2/a^2 +y^2/b^2 =1
General formula for a hyperbola
x^2/a^2 - y^2/b^2 =1
What is a rectangular hyperbola and general formula
Asymptotes at 90°
xy=c^2
i^2
-1
Complex conjugated
z=x+iy
z*=x-iy
Differentiating from first principles
Using chord AP with A(x1,y1) and P(x+h,y2) as h tends to 0
Two things which make an integral improper
- infinity or negative infinity in limits
- integral undefined at a limit or between them IE. Crosses an asymptote
General solution for cosθ
θ= 360n° +/- α θ= 2nπ +/- α
General solution for sinθ
θ= 180n° +(-1)^n α θ= nπ +(-1)^n α
General solution for tanθ
θ= 180n° + α θ= nπ + α
Interval bisection
If a root lies between f(a) and f(b), try x=a+b/2. Sign determines which side root is closer to. Repeat until desired accuracy
Linear interpolation
Using similar triangles to compare x/y ratios and find estimation for the root
Newton raphson method
If α is an approximation of a root of f(x)=0, hen a better approximation is given using α-f(α)/f’(α)
Euler formula for step by step solution to differential equations
Yn+1 ~Yn + hf(xn)
How to prove something has a linear correlation
Get in the form
logy = loga +nlogx
