$\inline&space;\displaystyle&space;\max_{x&space;\in&space;[a,b]}&space;f(x)$ The maximum of the continuous function $\inline&space;f$ in the range $\inline&space;a$ to $\inline&space;b$. $\inline&space;\displaystyle&space;\min_{x&space;\in&space;[a,b]}&space;f(x)$ The minimum of the continuous function $\inline&space;f$ in the range $\inline&space;a$ to $\inline&space;b$. $\inline&space;\displaystyle\sup_{x&space;\in&space;[a,b]}&space;f(x)$ The supremum of the function $\inline&space;f$ over the interval $\inline&space;[a,b]$, i.e. the smallest real number that is greater than or equal to every value of $\inline&space;f(x)$ with $\inline&space;x&space;\in&space;[a,b]$. $\inline&space;\displaystyle&space;\inf_{x&space;\in&space;[a,b]}&space;f(x)$ The infimum of the function $\inline&space;f$ over the interval $\inline&space;[a,b]$, i.e. the biggest real number that is smaller than or equal to every value of $\inline&space;f(x)$ with $\inline&space;x&space;\in&space;[a,b]$. $\inline&space;&space;\displaystyle&space;(x_n)_{n&space;\geq&space;0}$ This denotes an infinite sequence of real numbers $\inline&space;x_0$, $\inline&space;x_1$, $\inline&space;x_2$, ... that are given through some particular formula depending on the index of each term. For example the sequence $\inline&space;(\sin(n&space;\alpha))_{n&space;\geq&space;0}$ is the sequence of terms $\sin(0&space;\cdot&space;\alpha),\,&space;\sin(1&space;\cdot&space;\alpha),\,&space;\sin(2&space;\cdot&space;\alpha),\,&space;\sin(3&space;\cdot&space;\alpha),\,&space;\ldots$(1) $\inline&space;&space;\displaystyle&space;\lim_{n&space;\rightarrow&space;\infty}&space;x_n$ This denotes the limit of the sequence $\inline&space;(x_n)_{n&space;\geq&space;0}$, whenever it exists. Intuitively this is the value that the term $\inline&space;x_n$ approaches as the index $\inline&space;n$ gets closer and closer to infinity. It can be shown for example that $\lim_{n&space;\rightarrow&space;\infty}&space;\frac{\sin&space;n&space;+&space;\cos&space;n}{n}&space;=&space;0.$(2) $\inline&space;&space;\displaystyle&space;\sum_{i=0}^{\infty}&space;x_i$ This is called an infinite series of the sequence $\inline&space;(x_n)_{n&space;\geq&space;0}$ and it is defined through a sequence of partial sums $\inline&space;(S_n)_{n&space;\geq&space;0}$ whose terms are given by the formula $S_k&space;=&space;\sum_{i=0}^k&space;x_i&space;\qquad&space;\forall&space;k&space;\geq&space;0.$(3) Whenever it exists, the limit of the sequence $\inline&space;(S_n)_{n&space;\geq&space;0}$ is called the value of the infinite series and thus $\sum_{i=0}^{\infty}&space;x_i&space;=&space;\lim_{n&space;\rightarrow&space;\infty}&space;S_n.$(4) $\inline&space;&space;\displaystyle&space;\prod_{i=0}^{\infty}&space;x_i$ This is called an infinite product of the sequence $\inline&space;(x_n)_{n&space;\geq&space;0}$ and it is defined through a sequence of partial products $\inline&space;(P_n)_{n&space;\geq&space;0}$ whose terms are given by the formula $P_k&space;=&space;\prod_{i=0}^k&space;x_i&space;\qquad&space;\forall&space;k&space;\geq&space;0.$(5) Whenever it exists, the limit of the sequence $\inline&space;(P_n)_{n&space;\geq&space;0}$ is called the value of the infinite product and thus $\prod_{i=0}^{\infty}&space;x_i&space;=&space;\lim_{n&space;\rightarrow&space;\infty}&space;P_n.$(6) $\inline&space;&space;\displaystyle&space;\lim_{x&space;\rightarrow&space;a}&space;f(x)$ This denotes the limit of the function $\inline&space;f(x)$ at point $\inline&space;a$, whenever it exists. Intuitively this is the value that the function $\inline&space;f$ approaches as the argument $\inline&space;x$ gets closer and closer to $\inline&space;a$. It can be shown for example that $\lim_{x&space;\rightarrow&space;0}&space;\frac{\sin&space;x}{x}&space;=&space;1.$(7) $\inline&space;&space;\displaystyle&space;f(\alpha),&space;\quad&space;\frac{df}{dx}(\alpha)$ This denotes the first derivative of the function $\inline&space;f:(a,b)&space;\rightarrow&space;\mathbb{R}$ at point $\inline&space;\alpha&space;\in&space;(a,b)$, whenever it exists. It can be defined through the following formula using limits $\frac{df}{dx}(\alpha)&space;=&space;\lim_{h&space;\rightarrow&space;0}&space;\frac{f(\alpha+h)&space;-&space;f(\alpha)}{h}.$(8) $\inline&space;&space;\displaystyle&space;f^{(n)}(\alpha),&space;\quad&space;\frac{d^n&space;f}{dx^n}(\alpha)$ This denotes the $\inline&space;n$-th derivative of the function $\inline&space;f:(a,b)&space;\rightarrow&space;\mathbb{R}$ at point $\inline&space;\alpha&space;\in&space;(a,b)$, whenever it exists. It can be defined recurrently through the following formulae $\frac{d^n&space;f}{dx^n}(\alpha)&space;=&space;\lim_{h&space;\rightarrow&space;0}&space;\frac{1}{h}&space;\left(\frac{d^{n-1}f}{dx^{n-1}}(\alpha+h)&space;-&space;\frac{d^{n-1}f}{dx^{n-1}}(\alpha)\right),&space;\qquad&space;&space;\frac{d^1&space;f}{dx^1}(\alpha)&space;=&space;\frac{df}{dx}(\alpha).$(9) $\inline&space;&space;\displaystyle&space;\frac{\partial}{\partial&space;x_k}&space;f(a)$ The partial derivative of a function $\inline&space;f:U&space;\rightarrow&space;\mathbb{R}$, $\inline&space;U&space;\subseteq&space;\mathbb{R}^n$ with respect to the $\inline&space;k$-th variable, at a point $\inline&space;a&space;=&space;(a_1,&space;a_2,&space;\ldots,&space;a_n)&space;\in&space;\mathbb{R}^n$. This is defined through the following formula $\frac{\partial}{\partial&space;x_k}&space;f(a)&space;=&space;&space;\lim_{h&space;\rightarrow&space;0}&space;\frac{f(a_1,&space;a_2,&space;\ldots,&space;a_{k-1},&space;a_k&space;+&space;h,&space;a_{k+1},&space;\ldots,&space;a_n)&space;-&space;f(a_1,&space;a_2,&space;\ldots,&space;a_n)}{h}.$(10) Basically this evaluates the first derivative of the function $\inline&space;g_k:I&space;\rightarrow&space;\mathbb{R}$, $\inline&space;I&space;\subseteq&space;\mathbb{R}$ defined by $g_k(x)&space;=&space;f(a_1,&space;a_2,&space;\ldots,&space;a_{k-1},&space;x,&space;a_{k+1},&space;\ldots,&space;a_n).$(11) Therefore it will also be valid to write $\frac{\partial}{\partial&space;x_k}&space;f(a)&space;=&space;&space;\lim_{h&space;\rightarrow&space;0}&space;\frac{g_k(a_k&space;+&space;h)&space;-&space;g_k(a_k)}{h}&space;=&space;g_k(a_k).$(12) As an example consider the function $\inline&space;f:\mathbb{R}^3&space;\rightarrow&space;\mathbb{R}$ given by $\inline&space;f(x,&space;y,&space;z)&space;=&space;\sin&space;x&space;+&space;\cos&space;y&space;+&space;\tan&space;z$. Using the basic differentiation rules it is easy to see that $\frac{\partial&space;f}{\partial&space;x}&space;=&space;\cos&space;x&space;\qquad&space;&space;\frac{\partial&space;f}{\partial&space;y}&space;=&space;-\sin&space;y&space;\qquad&space;&space;\frac{\partial&space;f}{\partial&space;z}&space;=&space;1&space;+&space;(\tan&space;z)^2.$(13) $\inline&space;&space;\displaystyle&space;\frac{\partial^n&space;f}{\partial&space;x_k^n}$ If $\inline&space;f:&space;U&space;\subseteq&space;\mathbb{R}^p&space;\rightarrow&space;\mathbb{R}$ is a function then the $\inline&space;n$-th order partial derivative of $\inline&space;f$ with respect to the $\inline&space;k$-th variable is defined through the following recurrence relation $\frac{\partial^n&space;f}{\partial&space;x_k^n}&space;=&space;\frac{\partial}{\partial&space;x_k}&space;\left(&space;\frac{\partial^{n-1}}{\partial&space;x_k^{n-1}}&space;f&space;\right),&space;\qquad&space;&space;\frac{\partial^1}{\partial&space;x_k^1}&space;f&space;=&space;\frac{\partial&space;f}{\partial&space;x_k}.$(14) For example if $\inline&space;f:\mathbb{R}^2&space;\rightarrow&space;\mathbb{R}$ is defined by $\inline&space;f(x,y)&space;=&space;x^3&space;+&space;xy^3$, then $\frac{\partial&space;f}{\partial&space;x}&space;=&space;3x^2&space;+&space;y^3&space;\qquad&space;&space;\frac{\partial^2&space;f}{\partial&space;x^2}&space;=&space;6x&space;\qquad&space;&space;\frac{\partial^3&space;f}{\partial&space;x^3}&space;=&space;6&space;\qquad&space;&space;\frac{\partial^4&space;f}{\partial&space;x^4}&space;=&space;0$(15) while $\frac{\partial&space;f}{\partial&space;y}&space;=&space;3xy^2&space;\qquad&space;&space;\frac{\partial^2&space;f}{\partial&space;y^2}&space;=&space;6xy&space;\qquad&space;&space;\frac{\partial^3&space;f}{\partial&space;y^3}&space;=&space;6x&space;\qquad&space;&space;\frac{\partial^4&space;f}{\partial&space;y^4}&space;=&space;0.$(16) $\inline&space;&space;\displaystyle&space;\int_a^b&space;f(x)&space;\,dx$ The Riemann integral of the non-negative real-valued function $\inline&space;f$ on the interval $\inline&space;[a,b]$. This basically gives the area below the graph of $\inline&space;f$ calculated from point $\inline&space;a$ to point $\inline&space;b$. $\inline&space;&space;\displaystyle&space;\int_a^{\infty}&space;f(x)&space;\,dx$ The improper integral of the non-negative real-valued function $\inline&space;f$ defined through $\int_a^{\infty}&space;f(x)&space;\,dx&space;=&space;\lim_{b&space;\rightarrow&space;\infty}&space;\int_a^b&space;f(x)&space;\,dx.$(17) This gives the area below the graph of $\inline&space;f$ calculated from point $\inline&space;a$ to infinity. $\inline&space;&space;\displaystyle&space;\int_{-\infty}^b&space;f(x)&space;\,dx$ The improper integral of the non-negative real-valued function $\inline&space;f$ defined through $\int_{-\infty}^b&space;f(x)&space;\,dx&space;=&space;\lim_{a&space;\rightarrow&space;-\infty}&space;\int_a^b&space;f(x)&space;\,dx.$(18) This gives the area below the graph of $\inline&space;f$ calculated from minus infinity to point $\inline&space;b$.