STEP formula Flashcards
What is the quadratic formula
(-b±Root(b^2-4ac))/2a
formula for coefficients of ax^2 + bx + c = 0 in terms of roots
α + β = −b/a
αβ = c/a
formula for coefficients of ax^3 + bx^2 + cx + d = 0
α + β + γ = −b/a,
αβ + βγ + γα = c/a,
αβγ = −d/a
4 indices laws
a^x*a^y=a^(x+y)
a^0=1
(a^x)^y=a^(xy)
a^x=e^(xlna)
4 log rules
x=a^n <=> n=loga(x)
loga(x)+loga(y) = loga(xy)
loga(x)-loga(y) = loga(x/y)
kloga(x)=loga(x^k)
nth term of arithmetic series
u(n) = a + (n − 1)d
a is initial n is number of terms, d is difference
nth term of geometric
ar^(n-1)
a is initial n is number of terms
sum of arithmetic
S(n) = 1/2(n{2a + (n − 1)d})
a is initial n is number of terms, d is difference
sum of geometric
(a(1-r^n))/(1-r)
limit of sum of geometric to infinity
a/(1-r)
mod(r) <1
nCr
n!/(r!(n-r)!)
(a+b)^n
sum (from r=0 to n) of nCr*a^(n-r)b^r
(1+x)^k
1+kx+k(k-1)/2! * x^2 + k(k-1)(k-2)/3! *x^3 …+ k(k-1)…(k-r+1)/r! x^r +…
mod(x) <1 to converge
sum of natural numbers sum of (from r=1 to n) of r
1/2*n(n+1)
maclaurin series
f(x) = sum of (from r=0 to infinity) 1/r! f^r(0)x^r
maclaurin e^x
= sum of (from r=0 to infinity) (x^r)/r!
maclaurin ln(1+x)
= sum of (from r=0 to infinity) (-1)^(r+1) * (x^r)/r
x -x^2/2 +x^3/3 …
maclaurin sinx
= sum of (from r=0 to infinity) (-1)^(r) * (x^(2r+1))/(2r+1)!
maclaruin cosx
= sum of (from r=0 to infinity) (-1)^(r) * (x^(2r))/(2r)!
which maclaurin converge
sinx,cosx e^x converge for all x
ln(1+x) converges for -1< (x) <=1
Straight line through point (x1,y1) and gradient m
y-y1 = m(x-x1)
perpendicular condition
m1m2 = -1
sine rule
a/sinA = b/sinB = c/sinC
cosine rule
a^2 = b^2 + c^2 − 2bc cos A
area of a triangle
1/2 ab sin C
trig pythag identity
cos^2 A + sin^2 A = 1
trig pythag tan
1 +tan^2 A = sec^2 A
trig pythag cot
cot^2 A +1 = cosec^2 A
sine double angle
sin A+-B = sin A cos B +_ sin B cos A
cosine double angle
cos A+-B = cos A cos B -+ sin A sin B
tan double angle
tan (A+B) = (tan A +- tan B)/(1 -+ tan A tan B)
small angle approximations
sin θ ≈ θ , cos θ ≈ 1 − 1/2 θ^2 , tan θ ≈ θ
θ in radians and small
sinhx
(e^x-e^-x)/2
coshx
(e^x+e^-x)/2
tanhx definition
= sinhx/coshx
hyperbolic trig pythag
cosh^2 A − sinh^2 A = 1
pythag sech A
1-tanh^2 A = sech^2 A
pythag cosech A
- coth^2 A +1 = -cosech^2 A
cosech^2A = coth^2 A -1
sinh double angle
sinh(A ± B) = sinh A cosh B ± cosh A sinh B
cosh double angle
cosh(A ± B) = cosh A cosh B ± sinh A sinh B
tanh double angle
tanh(A ± B) = (tanh A ± tanh B)/(1 ± tanh A tanh B)
d/dx sinx
cosx
d/dx cos x
-sinx
d/dx tanx
sec^2 x
d/dx cot x
-cosec^2 x
d/dx cosecx
-cosec x cotx
d/dx sec x
sec x tan x`
d/dx arcsin x
1/root(1-x^2)
d/dx arctan
1/(1+x^2)
why isnt arcos x included
bc arcos x = 1/2 π − arcsin x
d/dx sinhx
coshx
d/dx coshx
sinhx
d/dx tanhx
sech^2 x
d/dx cothx
- cosech^2x
d/dx sechx
sechx tanhx
d/dx arsinh x
1/(root(1+x^2)
d/dx tanhx
1/(1-x^2)
d/dx e^x
e^x
product rule
ab’+a’b
chain rule
d/dx f(g(x))= g’(x)f’(g(x))
quotient rule
d/dx u/v = (vu’-uv’)/v^2
integral x^-1
ln|x| +c
what do all integrals need but i cba to write
constants
integral x^n
1/(n+1) x^(n+1)
integral cos x
sin x
integral sinx
-cos x
integral sinhx
coshx
integral coshx
sinhx
integral 1/root(a^2-x^2)
arsin x/a
integral 1/(a^2+x^2)
1/a arctan x/a
integral e^x
e^x
d/dx arcosh x
1/root(x^2-1)
integral 1/root(x^2-a^2)
arcosh x/a
integral 1/(a^2-x^2)
1/a artanh x/a
intergral 1/(x^2-a^2)
either partial fractions or
1/2a ln| (x-a)/(x+a) |
integration by parts
integral of uv’ = uv - integral of vu’
first principles derivatives
lim (h–> infinty) = (f(x+h)-f(x))/h
parametric derivatives
dy/dx = dy/dt / dx/dt
volume of rev about x axis
π integral y^2 dx
volume of revolution about y axis
π integral x^2 dy
trapezium rule
(1/2 h)(y0 + yn + h(y1 + y2 + ··· + yn−1))
h = (b − a)/n , yr = y(a + rh)
shm equation and solution
x¨ = −ω^2 x ⇒ x = R sin(ωt + α)
or x = R cos(ωt + β) or x = A cos ωt + B sin ωt
arc length of circle
rθ
area of a circle radians
1/2 r^2 θ
eulers identity
e^(iθ) = cos θ + i sin θ
de moivres theorem
z = r(cos θ + i sin θ) ⇒ z^n = r^n(cos nθ + i sin nθ)
roots of unity
z^n = 1 has roots z = e^(2πki/n)
half line
arg(z − a) = θ
complex circle locus
|z − a| = r
the magnitude of a vector
|xi + yj + zk| =
root (x^2 + y^2 + z^2)
dot product
a.b = a1b1 + a2b2 + a3b3 = |a| |b| cos θ
vector product
a × b = (a2b3 − a3b2)i + (a3b1 − a1b3)j + (a1b2 − a2b1)k = |a| |b||sin θ|n̂
euqation of line vectors
r = a + kb
equation of plane
(r − a).n = 0
or r.n = d
det of 2x2
det A = ad − bc
det of a 3x3
a(minor) -b(minor) +c(minor
inverse of a 2x2
1/det (a) * (d -b)
(-c a)
(AB)^-1
B^-1 * A^-1
reflection matrix in line y=+-x
(0 +-1)
+-1 0
rotation in matrix 2x2
(cos θ -sin θ)
(sin θ cos θ)
anticlockwise about origin
rotation about x axis 3x3
(1 0 0 )
(0 cos θ -sin θ)
(0 sin θ cos θ)
rotation about y axis 3d
(cos θ 0 sin θ )
(0 1 0 )
(-sin θ 0 cos θ)
roation about z axis
(cos θ -sin θ 0)
(sin θ cos θ 0)
( 0 0 1 )
Reflection in place z=0
(1 0 0)
(0 1 0)
(0 0 -1)
Perpendicular distance from a point to a plane
|n1α +n2β + n3γ + d|/root(n1^2 + n2^2 + n3^2)
polar coordinates area of a secor
1/2 intergral of r^2 dθ
Sum of square numbers
1/6 n (n+1)(2n+1)
Sum of cube numbers
1/4n^2(n+1)^2
arsinh in ln
ln(x +root(x^2 + 1))
Arcosh in ln
ln(x±root(x^2 - 1))
artanh in ln
1/2 ln( (1+x)/(1-x) )
change of base log formula
loga x = logb x / logb a
newton raphson formula
x{n+1}=x{n}-f(x{n})/{f’(x_{n})
Probability addition rule
P(A∪B)=P(A)+P(B)−P(A∩B)
Probability multiplication rule
P(A∩B)=P(B)P(A|B)
Bayes rule
P(B|A) = (P(B)P(A|B))/P(A)
nPr
n!/(n-r)!
SHM
x’’ = -ω^2 x
t=
tan(x/2)
t formula sin
sinθ = 2t/(1+t^2)
t formula cos
cosθ = (1-t^2)/(1+t^2)
t formula tan
tanθ = 2t/(1-t^2)
dx = (for t-formula)
dx= (2dt)/(1+t^2)
Variance
E(X^2) - (E(X))^2
Expection
E(X) = sum xi * P(X=xi) E(X) = integral xf(x) dx
E(X^2)
E(X^2) = sum (xi)^2 * P(X=xi) E(X) = integral x^2 f(x) dx
E(aX + bY +c )
aE(X) + bE(Y) +c
Var(aX+b)
a^2Var(x)
Var(aX+bY+c)
If independent
a^2Var(x)+ b^2Var(x)
Binomial
(n/x)p^x(1-p)^x
E(x) = np
Var(x) = np(1-p)
Uniform distribution discrete
1/n
E(X) = 1/2 (n+1)
Poisson
lambda^x e^-x /x!
E(X) = Var(X) = lambda
Continuous uniform
1/b-a
E(x) = 1/2 (a+b)
Normal
E(x) = mu Var(x) = sigma^2
Independent random variables
P(X=x, Y=y) = P(X=x)P(Y=y)
Discrete random variables
P(X=x, Y=y)
=> P(X=x) = sum of y from 1 to n of f(x,y)
Mutually exclusive
P(AUB) = P(A) + P(B) P(AnB) = 0
Independent
P(AnB) = P(A)P(B)
P(AUB) =
P(A) + P(B) + P(AnB)
P(A|B)
P(AnB)/P(B)
