Chapter 7 Useful Math Facts
7.1 Exponents and Logarithms
for positive constants \(a\) and \(b\):
\(a^0 = 1\)
\(a^{-n}=\frac{1}{a^n}\) for positive integer \(n\)
\(a^{x+y}=a^x a^y\)
\(a^{x-y}=\frac{a^x}{a^y}\)
\((a^x)^y=a^{xy}\)
\((ab)^x=a^xb^x\)
\(\log_a(x) = y\) implies \(a^{y} = x\)
If you’re working in statistics it’s usually a safe bet that \(\log(x)\) means \(\log_e(x)\) not \(\log_{10}(x)\).
\(\log(1) = 0\)
\(\log(e) = 1\)
\(\log(xy)=\log(x) + \log(y)\)
\(\log(x/y)=\log(x)-\log(y)\)
7.2 Sequences and Series
7.2.1 Monotonicity
A sequence of numbers \(\{x_1,x_2,\ldots\}\) is monotone increasing if \(x_n \le x_{n+1}\) for all \(n\). If \(x_n \ge x_{n+1}\) for all \(n\), the sequence is monotone decreasing
7.3 Calculus - Derivatives
\(f(x)=c \;(\text{constant}) \qquad f'(x)=0\)
\(f(x) = x^{r},\, r \in \mathbb{R} \qquad f'(x) = rx^{r-1}\) (Power Rule)
\(f(x) = e^x \qquad f'(x) = e^x\)
\(f(x)=\log(x) \qquad f'(x) =\frac{1}{x}\)
\(f(x)=g(x)\pm h(x) \qquad f'(x)=g'(x) \pm h'(x)\)
\(f(x)=g(x)h(x) \qquad f'(x)= g(x)h'(x)+g'(x)h(x)\) (Product Rule)
\(f(x) =\frac{g(x)}{h(x)} \qquad f'(x)=\frac{g'(x)h(x)-g(x)h'(x)}{[h(x)]^{2}}\,\) (Quotient Rule)
\(f(x)=g \circ h(x) = g(h(x)) \qquad f'(x) = g'(h(x))h'(x)\) (Chain Rule)
7.4 Calculus - Integrals
\(f(x) = c \qquad \int f(x)\,dx = cx\)
\(f(x) = x^r \qquad \int f(x)\,dx = \frac{x^{r+1}}{r+1},\,r \ne -1\)
\(f(x) = \frac{1}{x} \qquad \int f(x)\,dx = \log(x)\)
\(f(x) = e^x \qquad \int f(x)\,dx = e^x\)
\(f(x) = cg(x) \qquad \int f(x)\,dx = c \int g(x)\, dx\)
\(f(x) = g(x)\pm h(x) \qquad \int f(x)\,dx = \int g(x)\, dx \pm \int h(x)\, dx\)