F Maths Pure Flashcards
Series
1 = n
r = 0.5n(n+1)
Roots of Polynomials
Sum of alpha = -b/a Sum of alpha x beta = c/a Sum of alpha x beta x gamma = -d/a Sum of alpha x beta x gamma x delta = e/a
Sums of Squares
Alpha^2 + Beta^2 etc = (Sum of alpha )^2 - 2 x Sum of alpha x beta
Sum of Cubes
Alpha^2 + Beta^2 etc = (a + b) ^2 - 3 x (sum of alpha x beta) x sum of alpha + 3 x (sum of alpha x beta x gamma )
Volumes of revolution
X axis : Etc y^2
Y axis : etc x^2
Parametric: (y^2 dx/dt )dt vice versa
Cylinder
Pi r^2 h
Cone
1/3 pi r^2 h
Determinate of Matrix
ad - bc
Angle between lines
Cos 0 = (a•b) / |a||b|
Angles between planes and line
sin 0 = (a•b) / |a||b|
Angles between plane and plane
Cos 0 = (n•n) / |n||n|
Shortest Distance from point to plane
Point (alpha, beta, gamma)
Plane ( ax + by + cz = d)
|a x alpha + b x beta + c x gamma - d | / root (a^2 + b^2 + c^2)
Inverse of 3x3 matrix
Find det Do matrix of minors Do matrix of cofactors Transpose Multiply by 1/det
Skew
Not parallel and does not intersect
Un
Sn - Sn-1
Differential equations (second order homogenous) (complementary function)
b^2 > 4ac
y=Ae^(ax) + Be^(bx)
b^2 = 4ac
y=(A + Bx)^(ax)
b^2 < 4ac
y=e^px(A(cosqx)+ B(sinqx))
Differential equations (second order homogenous) (particular integral)
p to a p + qx to (a + bx) p + qx + rx^2 to (a + bx + cx^2) pcos(wx) + qsin(wx) to acos(wx) + bsin(wx) pe^(kx) to ae^(kx)
Methods of Differences
f(n) - f(n+1)
Hyperbolic Functions
sinhx = 0.5(e^x - e^-x)
coshx = 0.5(e^x + e^-x)
tanhx = (e^2x -1)/(e^2x + 1)
Osborne’s Rule
Sinhx x sinhx = -
Eulers Relation
z = re^(i0) = r(cos0 + isin0)
De Moivres Theorem
(r(cos0 + isin0))^n = r^n(cos(n0) + isin(n0))
Sin0 and cos0 in terms of z
Z + 1/z = 2c
Z- 1/z = 2s
Sum of z series
W + wz + wz^2 … wz^(n-1)
= (w(z^n - 1))/(z-1)
