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Maths - Math Problem Example

Summary
As shown in the diagram above, producer surplus is the area above the supply function but below the equilibrium price, while consumer surplus is the area above optimum price but below the demand curve.
Consumer surplus is refers to the excess accruing to the customers when the…

Extract of sample "Maths"

Integration Question i) Given Let So that And dx = 2udu Substituting these values to the first term: Replacing u withThe definite integral is thus

ii) Given
Let y = x+2
And thus dy = dx
Substituting for y and dy
Then
Let u = ln y
Then
Substituting for u and du
Therefore
= 5(ln y)2 + c
= 5 [ln(x+2)]2
iii) Given
Let u =
Hence
Recall that u =
= (6.93 – 13.86) – (3.77 -11.31)
= -6.93 – 7.54
= 0.61
iv) Given
Expanding the denominator

Since the denominator is a quadratic equation, the integral becomes:
Question 2
Given
Inverse Supply function as P=6 + Q2/4 and inverse demand function P= 51-Q2
Graphically, consumer and producer surplus are indicated as below.
As shown in the diagram above, producer surplus is the area above the supply function but below the equilibrium price, while consumer surplus is the area above optimum price but below the demand curve.
At equilibrium
Supply = demand
Q* = 6
P* = 51-62 = 15
Producer surplus
=

90-18
= 72
Consumer surplus
=

234 -90
= 144
Consumer surplus is refers to the excess accruing to the customers when the optimal price (equilibrium price) is less than price customers are willing and able to pay (depicted by the demand curve). For instance, in the above case the market price is £15, for every single unit (1000 units), consumed. However, consumers are willing to pay higher prices if the quantity to be purchased is less than optimum. For example, if customers only demand (or can access) 5 units, they will be willing to pay 51- 52 = £26 (according to market demand, but according to the market supply and demand the market price remains at £15, hence consumers save (26-15 = £11), for the fifth unit bought. The £11 is thus the consumer surplus. It increases as the quantity demanded decreases.
Producer surplus on the other hand is the gain to producers for every unit of quantity supplied below the optimal quantity, due to the fact that the equilibrium price exceeds the price at which suppliers are willing and able to supply that quantity. According to the case above, producers are willing to supply 3 units at a price of £ 8.25 ( 6+ 32/4 ), but the market price gives them an advantage of (15-8.25 = £6.75) for the third unit sold. The excess is the producer surplus, which reduces as the quantity produces approaches equilibrium quantity.
Question 3
Given
For the first year t starts from 0 to 12: Therefore, the total is given by
Integrating by parts
Let u = 10t2 and dv =
u’
dv
10t2
e-0.4t
+
20t
-2.5 e-0.4t
-
20
6.25 e-0.4t
+
0
-15.625 e-0.4t
-
To obtain the indefinite integral, multiply the first u with the second dv and so on, and either and get the sum by making use of the signs in the last column.
=
44.64 - - 325
36
Part (ii)
Given that the rate of oil production r =
The maximum rate of production will be obtained when
Using product rule
Let u = -0.4t
Then the second term becomes
=eu
By chain rule derivative of the second term becomes;
At maximum rate
Implying that either
(20-4t) t = 0
Hence t=0, 20-4t= 0
4t= 20
t= 5
Hence the peak will be at the fifth month
Part iii
(-13.94 – 4.65 – 0.81) - (-84.58 -84.58 – 43.98)
(-19.14- - 213.14)
= 194
Between 15 and 25
(-0.71 -0.14 – 0.015) - (-13.94 – 4.65 – 0.81)
-0.865 - - 19.14
= 18.275
As observed in (ii) above the peak rate is at 5 from which the rate starts decreasing. Therefore as t approaches infinity (moves away from 5), then the rate approaches zero. This trend explains the large negative difference between the totals of the two periods.
Question 4
i. Given and as the demand and supply curves respectively,
The equilibrium will be attained at the point where supply = demand

Introducing ln
0.8Q ln( e) = ln 15 -0.2Qln(e)
Ln e =1
0.8Q + 0.2Q = ln15
Q* = ln 15 = 2.708 ≈ 3 units
ii.
Typically, consumer surplus is the area under the demand curve but above the equilibrium price. While producer surplus is the area above the supply curve limited by the horizontal line at the equilibrium price as shown in the diagram.

Consumer surplus
= 67.68-52.35
= 15.33
Producer surplus

52.35 – (27.55- 2.5)
52.35 – 25.05
27.3
iii. Note that in this case, producer surplus exceeds consumer surplus contrary to what was observed in question two. Probably, the producers are able to restrict supply and hence push price high (towards the demand curve) hence reducing the gain to consumers.
References
Childress, R. L. (2005). Calculus for business and economics (3rd ed.). Englewood Cliffs, NJ: Prentice-Hall. Read More
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