calculus

Calculus notations

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Calculus

$\displaystyle \max_{x \in [a,b]} f(x)$			The maximum of the continuous function in the range to .
$\displaystyle \min_{x \in [a,b]} f(x)$			The minimum of the continuous function in the range to .
$\displaystyle\sup_{x \in [a,b]} f(x)$			The supremum of the function over the interval , i.e. the smallest real number that is greater than or equal to every value of with $x \in [a,b]$ .
$\displaystyle \inf_{x \in [a,b]} f(x)$			The infimum of the function over the interval , i.e. the biggest real number that is smaller than or equal to every value of with $x \in [a,b]$ .
$\displaystyle (x_n)_{n \geq 0}$			This denotes an infinite sequence of real numbers , , , ... that are given through some particular formula depending on the index of each term. For example the sequence $(\sin(n \alpha))_{n \geq 0}$ is the sequence of terms $\sin(0 \cdot \alpha),\, \sin(1 \cdot \alpha),\, \sin(2 \cdot \alpha),\, \sin(3 \cdot \alpha),\, \ldots$ (2)
$\displaystyle \lim_{n \rightarrow \infty} x_n$			This denotes the limit of the sequence $(x_n)_{n \geq 0}$ , whenever it exists. Intuitively this is the value that the term approaches as the index gets closer and closer to infinity. It can be shown for example that $\lim_{n \rightarrow \infty} \frac{\sin n + \cos n}{n} = 0.$ (3)
$\displaystyle \sum_{i=0}^{\infty} x_i$			This is called an infinite series of the sequence $(x_n)_{n \geq 0}$ and it is defined through a sequence of partial sums $(S_n)_{n \geq 0}$ whose terms are given by the formula $S_k = \sum_{i=0}^k x_i \qquad \forall k \geq 0.$ (4) Whenever it exists, the limit of the sequence $(S_n)_{n \geq 0}$ is called the value of the infinite series and thus $\sum_{i=0}^{\infty} x_i = \lim_{n \rightarrow \infty} S_n.$ (5)
$\displaystyle \prod_{i=0}^{\infty} x_i$			This is called an infinite product of the sequence $(x_n)_{n \geq 0}$ and it is defined through a sequence of partial products $(P_n)_{n \geq 0}$ whose terms are given by the formula $P_k = \prod_{i=0}^k x_i \qquad \forall k \geq 0.$ (6) Whenever it exists, the limit of the sequence $(P_n)_{n \geq 0}$ is called the value of the infinite product and thus $\prod_{i=0}^{\infty} x_i = \lim_{n \rightarrow \infty} P_n.$ (7)
$\displaystyle \lim_{x \rightarrow a} f(x)$			This denotes the limit of the function at point , whenever it exists. Intuitively this is the value that the function approaches as the argument gets closer and closer to . It can be shown for example that $\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1.$ (8)
$\displaystyle f`(\alpha), \quad \frac{df}{dx}(\alpha)$			This denotes the first derivative of the function $f:(a,b) \rightarrow \mathbb{R}$ at point $\alpha \in (a,b)$ , whenever it exists. It can be defined through the following formula using limits $\frac{df}{dx}(\alpha) = \lim_{h \rightarrow 0} \frac{f(\alpha+h) - f(\alpha)}{h}.$ (9)
$\displaystyle f^{(n)}(\alpha), \quad \frac{d^n f}{dx^n}(\alpha)$			This denotes the -th derivative of the function $f:(a,b) \rightarrow \mathbb{R}$ at point $\alpha \in (a,b)$ , whenever it exists. It can be defined recurrently through the following formulae $\frac{d^n f}{dx^n}(\alpha) = \lim_{h \rightarrow 0} \frac{1}{h} \left(\frac{d^{n-1}f}{dx^{n-1}}(\alpha+h) - \frac{d^{n-1}f}{dx^{n-1}}(\alpha)\right), \qquad \frac{d^1 f}{dx^1}(\alpha) = \frac{df}{dx}(\alpha).$ (10)
$\displaystyle \frac{\partial}{\partial x_k} f(a)$			The partial derivative of a function $f:U \rightarrow \mathbb{R}$ , $U \subseteq \mathbb{R}^n$ with respect to the -th variable, at a point $a = (a_1, a_2, \ldots, a_n) \in \mathbb{R}^n$ . This is defined through the following formula $\frac{\partial}{\partial x_k} f(a) = \lim_{h \rightarrow 0} \frac{f(a_1, a_2, \ldots, a_{k-1}, a_k + h, a_{k+1}, \ldots, a_n) - f(a_1, a_2, \ldots, a_n)}{h}.$ (11) Basically this evaluates the first derivative of the function $g_k:I \rightarrow \mathbb{R}$ , $I \subseteq \mathbb{R}$ defined by $g_k(x) = f(a_1, a_2, \ldots, a_{k-1}, x, a_{k+1}, \ldots, a_n).$ (12) Therefore it will also be valid to write $\frac{\partial}{\partial x_k} f(a) = \lim_{h \rightarrow 0} \frac{g_k(a_k + h) - g_k(a_k)}{h} = g_k`(a_k).$ (13) As an example consider the function $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ given by $f(x, y, z) = \sin x + \cos y + \tan z$ . Using the basic differentiation rules it is easy to see that $\frac{\partial f}{\partial x} = \cos x \qquad \frac{\partial f}{\partial y} = -\sin y \qquad \frac{\partial f}{\partial z} = 1 + (\tan z)^2.$ (14)
$\displaystyle \frac{\partial^n f}{\partial x_k^n}$			If $f: U \subseteq \mathbb{R}^p \rightarrow \mathbb{R}$ is a function then the -th order partial derivative of with respect to the -th variable is defined through the following recurrence relation $\frac{\partial^n f}{\partial x_k^n} = \frac{\partial}{\partial x_k} \left( \frac{\partial^{n-1}}{\partial x_k^{n-1}} f \right), \qquad \frac{\partial^1}{\partial x_k^1} f = \frac{\partial f}{\partial x_k}.$ (15) For example if $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is defined by , then $\frac{\partial f}{\partial x} = 3x^2 + y^3 \qquad \frac{\partial^2 f}{\partial x^2} = 6x \qquad \frac{\partial^3 f}{\partial x^3} = 6 \qquad \frac{\partial^4 f}{\partial x^4} = 0$ (16) while $\frac{\partial f}{\partial y} = 3xy^2 \qquad \frac{\partial^2 f}{\partial y^2} = 6xy \qquad \frac{\partial^3 f}{\partial y^3} = 6x \qquad \frac{\partial^4 f}{\partial y^4} = 0.$ (17)
$\displaystyle \int_a^b f(x) \,dx$			The Riemann integral of the non-negative real-valued function on the interval . This basically gives the area below the graph of calculated from point to point .
$\displaystyle \int_a^{\infty} f(x) \,dx$			The improper integral of the non-negative real-valued function defined through $\int_a^{\infty} f(x) \,dx = \lim_{b \rightarrow \infty} \int_a^b f(x) \,dx.$ (18) This gives the area below the graph of calculated from point to infinity.
$\displaystyle \int_{-\infty}^b f(x) \,dx$			The improper integral of the non-negative real-valued function defined through $\int_{-\infty}^b f(x) \,dx = \lim_{a \rightarrow -\infty} \int_a^b f(x) \,dx.$ (19) This gives the area below the graph of calculated from minus infinity to point .