5  Appendices

5.1 Identities

5.1.1 Trigonometric

[Trig Identities]

5.1.2 Antiderivatives

Simple functions, nb relationship to gradient and derivative is zero at max/min of the function.

Table 5.1: Antiderivatives of Common Functions

(a) Common Functions
\(\int f(x)\, dx\) \(f(x)\)
\(x^n\) \(nx^{n-1}\)
\(\exp{g(x)}\) \(g'(x) \exp{g(x)}\)
\(\log g(x)\) \({g'(x)}/{g(x)}\)
\(\tan^{-1}(x)\) \(\frac{1}{1+x^2}\)
\(\sin^{-1}(x)\) \(\frac{1}{\sqrt{1-x^2}}\)
(b) Trigonometric
\(\int f(x)\, dx\) \(f(x)\)
\(\sin (x)\) \(\cos (x)\)
\(\cos (x)\) \(-\sin (x)\)
\(\sec^2 (x)\) \(\tan (x)\)
\(\sec(x)\tan(x)\) \(\sec(x)\)
\(\csc^2(x)\) \(-\cot(x)\)
\(\csc(x)\cot(x)\) \(-\csc(x)\)

5.2 Simple Formulae

  • arithmetic series
    if \(u_n = a + (n-1)d\) then \(S_n = \frac{1}{2}n[2a+(n-1)d]\) (cf Gauss)
  • geometric series
    if \(u_n = ar^{n-1}\) then \(S_n = \frac{a}{1-r} \quad |r|<1\) (cf \(S-xS=a\))
  • binomial expansion
    \((a+b)^n = \binom{n}{1} a^n + \binom{n}{2} a^{n-1}b + \dots\) with
    \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
  • quadratic formula (cf Vieta’s formulae)
    \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
  • Newton-Raphson
    \(x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}\)

5.2.1 Bounds

  • Markov Inequality
  • AM/GM Inequality
  • Cauchy-Schwartz Inequality
  • Triangle Inequality
  • Cramer-Rao Lower Bound