\(TC=20+5{ Q }^{ 2 }+{ 3Q }^{ 3 }\)
\(MC=\frac { d(20) }{ dQ } +\frac { d({ 5Q }^{ 2 }) }{ dQ } +\frac { d(3Q^{ 3 }) }{ dQ } \)
\(=0+5\times 2\quad { Q }^{ 2-1 }+3(3)\quad { Q }^{ 3-1 }\)
\(=0+10Q+9{ Q }^{ 2 }\)
\(MC=10Q+{ 9Q }^{ 2 }\)

\(TC=15+3{ Q }^{ 2 }+{ 7Q }^{ 3 }\)
\(MC=\frac { d(15) }{ dQ } +\frac { d({ 3Q }^{ 2 }) }{ dQ } +\frac { d(7Q^{ 3 }) }{ dQ } \)
\(=0+3(2)\quad { Q }^{ 2-1 }+7(3)\quad { Q }^{ 3-1 }\)
\(MC=6Q+{ 21Q }^{ 3 }\)

y = 3x² + 16x³
Differentiating,
\(\frac{dy}{dx}=3(2)x^{2-1}+16(3)x^{3-1}\)
= 6x¹ + 48x²
\(\frac{dy}{dx}=6x+48x^{2}\)

Given
Q_d = 50 - 5P
5P = 50 - Q_d
5P = 50 when Q_d is zero.
\(p=\frac { 50 }{ 5 } \)
P = 10
When P = 10, Demand is 0
Hence P = 10, which is the maximum price beyond which nobody will demand the commodity.

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:
\({ P }_{ s }={ P }_{ 0 }{ x }_{ 0 }-\int _{ 0 }^{ { x }_{ 0 } }{ g(x)dx } \)
\(=8-\int _{ 0 }^{ 4 }{ \frac { x }{ 2 } } dx\)
\(=8-\frac { 1 }{ 2 } \int _{ 0 }^{ 4 }{ xdx } \)
\(=8-\frac { 1 }{ 2 } { { { { { \left[ \frac { { x }^{ 2 } }{ 2 } \right] }^{ 4 } }_{ 0 } } } }\)
\(=8-\frac { 1 }{ 4 } [{ \left( 4 \right) }^{ 2 }-(0)]\)
\(=8-\frac { 1 }{ 4 } [16]=8-4\)
\(\therefore { P }_{ s }=4\quad units \)

MS office Excel is used in data analysis by using formula. A spread sheet is a large sheet of paper which contains rows and columns. The intersection of rows and columns is termed as 'cell'. MS Excel 2007 version supports up to 1 million rows and 16 thousand columns per worksheet.
Start:
You can start Excel from various options.
Click Start → Program → Microsoft Excel
Double click the MS Excel Icon from the Desktop.
Worksheet:
A worksheet is a table like document containing rows and columns with data and formula. There are four kinds of calculation operators. They are arithmetic, comparison, text concatenation (link together) and reference. MS Excel helps to do data analysis and data presentation in the form of graphs, diagrams, area chart, line chart.

\(\mathrm{TC} =2.5 \mathrm{q}^{3}-13 \mathrm{q}^{2}+50 \mathrm{q}+12  \)
\(\mathrm{MC} =\frac{\mathrm{d}(\mathrm{TC})}{\mathrm{dq}}  \)
\(=2.5\left(3 \mathrm{q}^{2}\right)-13(2 \mathrm{q})+50(1)+0  \)
\(=7.5 \mathrm{q}^{2}-26 \mathrm{q}+50  \)
\(\mathrm{AC} =\frac{2.5 \mathrm{q}^{3}-13 \mathrm{q}^{2}+50 \mathrm{q}+12}{\mathrm{q}}  \)
\(=2.5 q^{2}-13 q+50+\frac{12}{q}\)
\(\text {Ans: } \mathrm{MC} =7.5 \mathrm{q}^{2}-26 \mathrm{q}+50  \)
\(\mathrm{AC} =2.5 \mathrm{q}^{2}-13 \mathrm{q}+50+\frac{12}{\mathrm{q}}\)