5  Appendices

5.1 Identities

5.1.1 Trigonometric

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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