(i) Marginal product of a factor of production refers to addition to total product due to the use of an additional unit of a factor.
(ii) In symbols\(​​​​MP=\frac {d(TP) }{ dQ } =\frac { \triangle (TP) }{ \triangle Q } \)

(i) Matrix is rectangular array of numbers systematically arranged in rows and columns within brackets.
(ii) In a matrix, if the number of rows and columns are equal, it is called a 'square matrix'.

Q_s = f (P_x) Where 'Q_s' stands for Quantity supplied of a commodity and P_x is the price of that commodity:

MS word is a word processor, which helps to create, edit, print and save documents for future retrieval and reference.

The evolution of ICT has five phases. They are:
(a) Computer
(b) PC
(c) Microprocessor
(d) Internet and
(e) Wireless links

\(Y={ 5x }^{ 4 }+{ 3x }^{ 3 }\)
Differentiating,
\(\frac { dx }{ dy } =5(4){ x }^{ 4-1 }+3(3){ x }^{ 3-1 }=20{ x }^{ 3 }+{ 9x }^{ 2 }\)
\(20{ x }^{ 3 }+{ 9x }^{ 2 }\)

Given function \(y=3x^{ 3 }\)
slope=\(\frac { dy }{ dx } \)
\(\frac { dy }{ dx } \)=3(3) \({ x }^{ 3-1 }={ 9x }^{ 2 }\)
When x=5, then slope=\({ 9(5) }^{ 2 }\)=\(9\times 5\)=225
slope=225

A worksheet is a table like document containing rows and columns with data and formula. There are Four kinds of calculation operators. They are
i) arithmetic
ii) comparison
iii) text concatenation(link together) and
iv) reference

\(I=\int { \left( { 4x }^{ 5 }+6{ x }^{ 3 }+\frac { 2 }{ 3 } x \right) } dx\)
\(=4\left[ \frac { { x }^{ 6 } }{ 6 } \right] +6\left[ \frac { { x }^{ 4 } }{ 4 } \right] +\frac { 2 }{ 3 } \left[ \frac { { x }^{ 2 } }{ 2 } \right] +c\)
\(I=\left( \frac { { 2x }^{ 5 } }{ 3 } +\frac { { 3x }^{ 4 } }{ 2 } +\frac { { x }^{ 2 } }{ 3 } \right) +c\)

For Market equilibrium
Quantity demanded = Quantity supplied
\(\frac { 16 }{ x+4 } \frac { x }{ 2 } \)
\(x(x+4)=16\times 2\)
\({ x }^{ 2 }+4x=32\)
\({ x }^{ 2 }+4x-32=0\)
\((x-4)(x-8)=0\)
\(x=4;x=-8.\)
\(but\quad x=-8\quad is\text { in admissible.}\)
\(\therefore x=4(i.e.,){ x }_{ 0 }=4\)
\( { p }_{ 0 }{ x }_{ 0 }=\left[ \frac { 16 }{ 4+4 } \right] \times 4\)
\(=\frac { 16 }{ 8 } \times 4\)
\({ p }_{ 0 }{ x }_{ 0 }=8\)
Consumer Surplus,
\(CS=\int _{ 0 }^{ { x }_{ 0 } }{ f(x)dx-{ p }_{ 0 } } { x }_{ 0 }\)
\(=\int _{ 0 }^{ 4 }{ \frac { 16 }{ (x+4) } dx-8 } \)
\(=16[\log { (4+4)-\log { (0+4)]-8 } } \)
\(=16\left[ \log { \left( \frac { 8 }{ 4 } \right) } \right] -8\)
\({ C }_{ s }=16\log { 2-8\quad units } \)
Producer's Surplus:
\({ P }_{ s }={ P }_{ 0 }{ x }_{ 0 }-\int _{ 0 }^{ { x }_{ 0 } }{ g(x)dx } \)
\(=8-\int _{ 0 }^{ 4 }{ \frac { x }{ 2 } } dx\)
\(=8-\frac { 1 }{ 2 } \int _{ 0 }^{ 4 }{ xdx } \)
\(=8-\frac { 1 }{ 2 } { { { { { \left[ \frac { { x }^{ 2 } }{ 2 } \right] }^{ 4 } }_{ 0 } } } }\)
\(=8-\frac { 1 }{ 4 } [{ \left( 4 \right) }^{ 2 }-(0)]\)
\(=8-\frac { 1 }{ 4 } [16]=8-4\)
\(\therefore { P }_{ s }=4\quad units \)

It is a software used to perform computer based presentation. Steps involved in making presentation:
(i) Click Start Menu
(ii) Click Program
(iii) Select Microsoft Power Point - Click.
(iv) New Power Point file will open, and then type the title and subtitle if wanted.
(v) A new slide can be inserted by 'click' on icon 'new slide' or using short key 'Ctrl + M'.
(vi) We can type the content, insert the table, pictures, movies, sounds, etc., with the content.
(vii) Tab 'Design' helps to design the slides (can select common design for all slides or separate slide for each slide).
(viii) Click icon slide show, one can run slide show either starting from the first slide or starting from the current slide.
The Power Point Presentation (PPT) facilitates the key points to be kept in memory and understand the particular topic. Recently, the smart class room teaching uses the PPT to deliver the information in an effective way to enhance the quality of teaching.

Differential calculus measures the rate of change of functions. In Economics it is also necessary to reverse the process of differentiation and find the function F(x) whose rate of change has been given. This is called integration. The function F(x) is termed an integral or anti-derivative of the function f(x). The integral of a function f(x) is expressed mathematically as \(\int { f\left( x \right) dx=F\left( x \right) +C } \)
Here the left hand side of the equation is read "the integral of f(x) with respect to x", The symbol J is an integral sign, f(x) is integrand, C is the constant of integration, and F(x) + c is an indefinite integral. It is so called because, as a function of X, which is here unspecified, it can assume many values.

The coefficient and the constant terms are given below for the equations
\(\triangle =\begin{vmatrix} 7 & 5 \\ 5 & -6 \end{vmatrix}=(-42)-(25)=-67\)
\( { \triangle x }_{ 1 }=\begin{vmatrix} 89 & 5 \\ -13 & -6 \end{vmatrix}=-534+65=-469\)
\({ \triangle x }_{ 2 }=\begin{vmatrix} 7 & 89 \\ 5 & -13 \end{vmatrix}=-91-445=-536\)
\(\therefore \quad { x }_{ 1 }=\frac { \triangle { x }_{ 1 } }{ \triangle } =\frac { -469 }{ -67 } =7\)
\(\therefore { \quad x }_{ 2 }=\frac { { \triangle x }_{ 2 } }{ \triangle } =\frac { -536 }{ -67 } =8\)
\(\therefore \quad { x }_{ 1 }=7,\quad { x }_{ 2 }=8\)