Things Flashcards
Centre of mass of a semicircle
4r/3π
Centre of mass of solid cone
1/4 h
Centre of mass of hollow cone
1/3 h
Centre of mass of a solid hemisphere
3/8 r
Centre of mass of a hollow hemisphere
1/2 r
Even function
f(x)=f(-x)
Odd function
f(-x)=-f(x)
I.e. rotational symmetry of 2
d/dx(tan x)
sec^2(x)
d/dx(cosec x)
-cosec x cot x
d/dx(sec x)
sec x tan x
d/dx(cot x)
-cosec^2(x)
S tan x dx
ln(sec x) + c
S cosec x dx
-ln(cosec x + cot x) + c
S sec x dx
ln(sec x + tan x) + c
S cot x dx
ln(sin x) + c
SinCos -+ identities
cos(π-α) = -cos(α) cos(α-π) = -cos(α) cos(-α) = cos(α)
sin(π-α) = sin(α) sin(α-π) = -sin(α) sin(-α) = -sin(α)
Double angle formulae
sin(2A)=2sinA cosA
cos(2A)=cos^2(A)-sin^2(A)
=2cos^2(A) - 1
=1 - 2sin^2(A)
Half angle formulae
Use double angle formulae and replace A with θ/2
PF denominator with linear factor
(ax+b) -> A/(ax+b)
PF denominator with irreducible quadratic factors
(ax2 + bx + c) -> (Ax + B)/(ax2+bx+c)
PF denominator with a repeated factor
(ax+b)^2 -> A/(ax+b) + B/(ax+b)^2
PF improper
When degree of numerator is greater than or equal to degree of numerator, quotient is polynomial of order top - bottom
Order4/order3 -> Ax + B + other fractions from other rules
Integration by parts
S uv’ = uv - S u’v
Integrating odd powers of sinx and cosx
Split sin^3(x) into sinx and sin^2(x), then convert the latter into 1-cos^2(x). Expand brackets and separate integrals.
Integrating even powers of sinx and cosx
Use double angle formula of cos to turn them into cos(2x)
Even powers of secx and cosecx
Use tan sec and cot cosec identities and differentials. Split into sec^2(x)sec^2(x)
Odd powers of secx and cosecx
Write sec^(n-1)xsecx and then integrate by parts
Dividing polynomials
To divide say x2 + 4x - 5 by (x+3),
Set it identically equivalent to (x+3)(ax+b)+r
Where ax+b is the quotient and r is the remainder
Remainder theorem
When f(x) is divided by (αx-β) the remainder is f(β/α)
Special sums
Σr = n(n+1)/2 Σr^2 = n(n+1)(2n+1)/6 Σr^3 = n^2(n+1)^2/4
Newton Raphson iteration
x(n+1) = xn - f(xn)/f’(xn)
Limitations include a slow convergence speed when the tangent is shallow near the root, and the fact that some starting points fail to find the nearby root, and find other roots instead.
A divide by zero error can occur, which causes the method to fail.
Binomial expansion
(1+x)^n=1+nx+n(n-1)/2!x^2…
Valid for |x|<1
Maclaurin expansion
f(x)=f(0) + f’(0)x + f’’(0)x^2/2! +…
How to determine if g(x) converges
If 1
Plotting a complex conjugate
Reflects in real axis
Modulus and argument of z^n
arg(z^n)=narg(z)
|z^n|=|z|^n
Modulus and argument of wz
|wz|=|w||z|
arg(wz)=arg(w)+arg(z)
Rotation with matrix
Clockwise
(cosθ sinθ)
(-sinθ cosθ)
Inverse matrix
1/(ad-bc) (d -b)
(-c a)
(AB)^-1
B^-1 * A^-1
