\(\int { 4{x }^{ 3 } } dx=4\int { { x }^{ 3 } } dx\)
\(=4\frac { { x }^{ 3+1 } }{ 3+1 } +c=4\frac { { x }^{ 4 } }{ 4 } +c={ x }^{ 4 }+c\)

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\(\mathrm{AC}=\frac{\mathrm{TC}}{x} \)
\(\mathrm{TC}=60+10 x+15 x^{2} \)
\(\mathrm{AC}=\frac{60}{x}+10+15 x\)

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.
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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.

(i) Mathematical methods help to present the economic problems in a more precise form.
(ii) It helps to explain economic concepts.
(iii) It helps to use a large number of variables in economic analysis.
(iv) To quantify the effect of any economic activity implemented by government.

Given that TC(Q) = 128 + 60Q + 8Q²
We know TC = Fixed cost + variable cost
MC (Q) =\(\frac { d(TC) }{ dQ } \)
= 0 + 60(1)Q^1-1 + 8(2)Q^2-1
= 0 + 60Q⁰ + 16Q¹(Since, Q⁰= 1)
MC = 60 + 16Q
Average Cost =\(\frac { TC }{ dQ } \)
=\(\frac { 128+60Q+8Q^{ 2 } }{ Q } \)
AC = \(\frac { 128 }{ Q } \) + 60 + 8Q
Constant value is known as fixed cost
Fixed cost = 128
FC = 128
Average Fixed cost = \(\frac { 128 }{ Q } \)
AFC = \(\frac { 128 }{ Q } \)
Average Variable cost = 60 + 8Q (total variable cost divided by Q)
∴ AVC = 60 + 8Q

For equilibrium P_d = P_s
\(1600-x^{2} =2 x^{2}+400  \)
\(1600-400 =2 x^{2}+x^{2}  \)
\(3 x^{2} =1200  \)
\(x^{2} =400  \)
\(x =\pm 20  \)
\(P_{\mathrm{d}} =1600-(20)^{2} \)
=1600-400 
=1200 
\(\mathrm{P}_{\mathrm{s}} =2(20)^{2}+400 \)
= 2(400) + 400 
= 800 + 400 
= 1200
Consumer's surplus:
\(\mathrm{C}_{\mathrm{s}}=\int_{0}^{20}\left(1600-x^{2}\right) \mathrm{d} x-(20 \times 1200) \)
\(=\left[1600 x-\frac{x^{3}}{3}\right]_{0}^{20}-(24000)=\left[1600(20)-\frac{20^{3}}{3}\right]-24000 \)
\(=32000-\frac{8000}{3}-24000=5,333.34\)
Producer's surplus:
\(\mathrm{P}_{\mathrm{s}} =(20 \times 1200)-\int_{0}^{20}\left(2 x^{2} +400\right) \mathrm{d} x  \)
\(=24000-\left[\frac{2 x^{3}}{3}+400 x\right]_{0}^{20}   \)
\(=24000-\left[\frac{2(20)^{3}}{3}+400(20)\right]=24000-\left[\frac{2 \times 8000}{3}+8000\right]\)
= 24000-5333.33 - 8000 = 10,666.67