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INFORMATIC PRINCIPLES Univariate Differential Calculus: Solutions to Review Questions
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INFORMATIC PRINCIPLES Univariate Differential Calculus: Solutions to Review Questions
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Univariate Differential Calculus: Solutions to Review Questions1-
Find the first derivative of the following functions
a) f ( x) 8 3 x 4 2 x 2 1
df ( x)
4(12 x 3 4 x)(3x 4 2 x 2 1) 1/ 2
dx
x2 1
f ( x)
b) 3x 3 2 x
df ( x) (2 x)(3x 3 2 x) ( x 2 1)(9 x 2 2)
dx (3x 3 2 x) 2
4 x)
f ( x) 4 xe ( x
2
c)
df ( x)
(4e ( x 4 x ) ) (4 x)(2 x 4)e ( x 4 x )
2 2
dx
2-
The total cost function for a manufacturer has the following form,
TC= 2Q3 - 45Q2 + 300Q where Q is the output level.
a)Find the level of output, which minimises the total cost,
b)Find the average cost at the production level, which minimises the cost.
a) FOC: MC 6Q 2 90Q 300 0 Q=10 , Q=5
SOC: MC ' 12Q 90
Q=10 MC’=30>0 Minimum, therefore Q=10 minimises the Total cost
Q=5 MC’=-30<0 Maximum
b) Q=10 TC = 500 AC=50
3-
The total revenue and marginal cost functions of a manufacturing firm have been defined as
TR = 5900Q-10Q2 MC = 6Q2 -8Q +140
where Q is the output level. If the firms fixed cost is 845, Find:
a)The level of output, which maximises the total revenue,
b)The maximum revenue
c)The level of output, which maximises the firm’s profit,
d)The total cost when profit is maximised,
e)The average cost when profit is maximised.
a) FOC: MR = 5900 – 20Q = 0 Q=295
SOC: MR’ = -20 <0 therefore Q=295 is the level which maximises TR
b) Q=295 TR= 870,250
120
, c) We know that TR TC 5900Q 10Q 2 TC
FOC:
d
dQ
0 TR TC 0 MR MC 0 5900Q 10Q 2 6Q 2 8Q 140 0
6Q 2 12Q 5760 0
Q=-32 not acceptable Q=30 acceptable
d 2 d 2
SOC: 12Q 12 Q=30 372 0
dQ 2 dQ 2
Q=30 maximises profit
d) We need to find the total cost when profit is maximised, but we know only MC and not TC i.e. we
know that
TC MC TC 6Q 2 8Q 140
We need to find the function that if we differentiate it will give us TC.
However basic differentiation rules state that: Q 1 , Q 2 2Q and, Q 3Q
3 2
Hence:
TC 6Q 2 8Q 140 3Q 2 3Q 2 4(2Q) 140(1)
And the above can be written as: TC Q3 Q3 4Q 2 140Q
Or: TC 2Q3 4Q 2 140Q
Which is TC 2Q3 4Q 2 140Q
TC 2 Q 3 4 Q 2 140Q
(Check: TC 2Q 4Q 140Q 6Q 8Q 140 )
3 2 2
The final total cost function includes also a fixed cost of 845, So:
TC 2Q3 4Q 2 140Q 845
And, for Q =30:
TC(30) = 55,445
e) AC = TC/Q = 55,445/30=1,848.167
4-
Given the following demand function,
Q=100-24P
a)Find the price elasticity of demand at P=1 and P=3,
b)Find the price at which the price elasticity demand is unity,
c)What is the relationship between the price level in part (b) and the supplier’s revenue?
a)
P=1 Q=76 , P=3 Q=28
121
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