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Application of Integration
Business and Economics
Derivative as rate of change
• Measurements described as a rate of change is a derivative
• Key words:Cost PER unit produced, profit PER unit soldCost FOR EVERY YEAR, value PER yearMarginal cost: additional cost for producing one
extra item of a product. Additional cost per unit of production
Derivative as rate of change
Key words: Marginal revenue: additional revenue by
increasing product sales by 1 unit: extra revenue PER 1 unit increase in sales
Marginal profit: additional profit for each unit sold: MP=MR-MC.
Demand curve as a rate of change
Demand function: Quantity a consumer is willing to buy for each unit
increase in price
Supply curve as a rate of change
Supply functionThe quantity of goods a producer is willing to
produce for every unit increase in price
Integral as Anti derivative
dxxFxF '
axdxaC
adxdCadxdC
COST PER UNIT PRODUCED
Integral as an antiderivative
Example 1.1If cost of printing 1 book is 90 pesos, what is the
cost of printing 500 books?
xC
dxdCdxdC
90
9090
Php 000,4550090
500At
C
x
Integral as antiderivative
If cost of printing 1 book is 90 pesos, what is the cost of printing 500 books?
000,450000,45
90 90 5000
500
0
C
xdxC
Integral as antiderivative
Example 1.2If the cost of producing 1 book unit is 90+0.01x,
what is the cost of producing 500 books?
dxxdCxdxdC 01.09001.090
500
0
Php 250,46200
000,250000,452
90 5000
2
1001
xxC
Integral as antiderivative
Example 1.3In a certain factory the marginal cost of is 3(q-4)^2 when the level of production is q units.How much would the total cost increase if
production is increased from 6 to 10?
3
22
4
4343
qC
dqqdCqdqdC
Integral as antiderivative
Example 1.3Increase in total cost when production is
increased from 6 to 10.
dqqCC 210
643610
Php 208
8216264cost Total 33106
3
q
Definite Integral as area between curves
Example 1.4 (Area between curves)If the marginal revenue from selling books is
And the marginal cost is
xRdxdR 05.01'
22' xCdxdC
Definite integral as area between curves
Find the net profit when production is increased from 1 to 3?
Definite integral as area between curves
3
1P
dxxdxxP 23
1
3
12 05.01
Php 22
13
2
205.01
51
51
31
401
31
409
31
3312
401
23
1
xxx
dxxxP
Interpretation of graphs
Example 1.5 (Area between curves)Suppose that t years from now, investment A
will generate a profit per year according to
While investment B will generate a profit per year according to
21 5' ttP
ttP 520'2
Interpretation of graphs
a. What information is revealed by the two functions?
Graphs and definite integral
b. What is the accumulated profit of investment A after 6 years?
c. What is the accumulated profit of investment B after 6 years?
Definite integral and area under the curve
Php 1027230
55
1
60
3312
6
01
P
ttdttP
Definite integral and area under the curve
Php 21090120
20 520
2
60
225
6
02
P
ttdttP
Definite integral and area between curves
c. Identify the region in the graph which represents the excess profit earned by investment A over investment B over 6 years.
Definite integral and area between curves
108729090
15515
5 520
60
3312
252
6
0
26
0
6
0
n
n
n
P
tttdtttP
dttdttP
