Question

Qn 1 a) what do you understand by Quantitative analysis and quantitative approach?

Qn 2. i)where do we apply differential calculus?

ii)Apply the knowledge of rules of differentiation on the following

a)use Product rule and differentiate the following,

Village fresh dairy farm faces the non linear demand schedule P= (500-0.2Q) ^1.5,what output should it sell to maximize its total revenue?

b)use quotient rule in the equation,derive marginal revenue ,Remember TR = P.Q

P   = 250 / (2+Q) ^0.5

Answer 1

#1. Quantitative analysis emphasises on statistical and mathematical tools which are collected through polls, large and small scale surveys, interviews, and uses computational techniques to quantify opinions, behaviours, and other variables. It uses data available and additional data from surveys to uncover patterns for research purposes.

#2. (i) Differential calculus has its applications in a wide range of optimization problems. Derivatives are used to find maximum and minimum values of cost, profit, loss, etc.

(ii) P = (500 - 0.2Q)^1.5

Total Revenue = P.Q = (500 - 0.2Q)^1.5.Q differentiating TR with respect to Q, we get

dTR/dQ = 1.5 (-0.2) (500 - 0.2Q)^0.5.Q + (500 - 0.2Q)^1.5.1 = 0 (product rule)

dividing throughout by (500 - 0.2Q)^0.5 , we get

=> - 0.3 Q + (500 - 0.2Q)¹ = 0 => 500 = 0.5Q => Q = 500/0.5 = 1000

The dairy should sell 1000 units to optimise total revenue

(iii) P   = 250 / (2+Q) ^0.5

Total Revenue = P.Q = 250 / (2+Q) ^0.5 .Q = 250.Q / (2+Q) ^0.5

differentiating TR with respect to Q to get MR, (quotient rule)

MR = [(2+Q) ^0.5 ] 250 - 250Q [0.5 ( 2+Q)^-0.5] / [ (2+Q) ^0.5 ]² = (2+Q) ^0.5 ] 250 - 125 (2+Q)^-0.5] / (2+Q) ¹

simplifying the numerator by dividing numerator and denominator by ( 2+Q)^-0.5 , we get

MR = ( 2+Q)¹ (250) - 125Q / (2+Q) ^1.5