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.

Given \(TC(Q)=2Q^{ 3 }+{ 8Q }^{ 2 }+12Q+20\)
To find MC differentiate the function with respect to Q.
\(MC(Q)=\frac { d(TC) }{ d(Q) } =2(3){ Q }^{ 3-1 }+8(2){ \quad Q }^{ 2-1 }+12(1)\quad { Q }^{ 1-1 }+0\)
\(={ 6Q }^{ 2 }+16Q^{ 1 }+12\)
\(When\quad Q=4\)
\(MC(Q)=6{ (4) }^{ 2 }+16(4)+12\)
\(=96+64+12\)
\(MC(Q)=172\)
To find AVC
\(AVC(Q)=TVC(Q)/Q\)
\(AVC(Q)=TC(Q)-TFC(Q)\)
In  TC(Q)-Constant Value is TFC(Q) [\(\therefore \) Value of constant =20]
\(\therefore TFC(Q)={ 2 }Q^{ 3 }+{ 8Q }^{ 2 }+12Q\)
\(AVC(Q)=\frac { TVC(Q) }{ Q } =\frac { { 2Q }^{ 3 }+{ 8Q }^{ 2 }+12Q }{ Q } \)
\(={ 2Q }^{ 2 }+8Q+12\)
\(when \ Q=4\)
\(AVC(Q)=2({ 4) }^{ 2 }+8(4)+12\)
\(=2(16)+8(4)+12=32+32+12\)
\(AVC(Q)=76\)
\(marginal\ cost=172;AV\quad cost=76\)

Microsoft Power Point :
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 Powerpoint - Click.
(iv) New PowerPoint 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 classroom teaching uses the PPT to deliver the information in an effective way to enhance the quality of teaching.

This theory was developed by the Alfred Marshall. The demand function P(x) reveals the relationship between the quantities that the people would buy at given price. It can be expressed as P = f (x) Consumer surplus is the difference between the price one is willing to pay and the price that is actually paid. It is represented in the following diagram.

Mathematically, the consumer's surplus (CS) can be defined as CS = (Area under the demand curve from x =0 to x = x0) - (Area of the rectangle OX0BP0)
CS = \(\left[ \int _{ x }^{ x0 }{ p(x)dx } \right] -{ x }_{ 0 }{ p }_{ 0 }\)

1. \(\frac{dc}{dx}\) = 0 Where C is a constant (Read differentiation of 'C' with respect to 'x' is)
2. \(\frac { d\left( x^{ n } \right) }{ dx } =nx^{ n-1 }\)
3. \(\frac { d\left( x \right) }{ dx } =1x^{ 1-1 }=1x^{ 0 }=1\)
4. \(\frac { d\left( u+v \right) }{ dx } =\frac { du }{ dx } +\frac { dv }{ dx } \)
5. \(\frac { d\left( u-v \right) }{ dx } =\frac { du }{ dx } -\frac { dv }{ dx } \)