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