Chapter 7
Additional
Integration
Topics
Section 2
Applications in
Business and
Economics
Learning Objectives for Section 7.2
Applications in Business/Economics
The student will be able to:
1. Construct and interpret probability
density functions.
2. Evaluate a continuous income
stream.
3. Evaluate the future value of a
continuous income stream.
4. Evaluate consumers’ and producers’
surplus.
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Continuous Income Stream
 In the previous lesson, we learned how to calculate the
total income produced from a continuous income stream
as:
𝑏
𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑐𝑜𝑚𝑒 =
𝑓 𝑡 𝑑𝑡
𝑎
 What if the money produced by this continuous income
stream is invested as soon as it is received? In other words,
what if the money earns interest?
 We can use calculus to determine the future value when
the income stream is compounded continuously.
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Future Value
of a Continuous Income Stream
If f (t) is the rate of flow of a continuous income stream,
0  t  T, and if the income is continuously invested at a rate r
compounded continuously, then the future value FV at the end
of T years is given by:
𝑇
𝑓(𝑡)𝑒 𝑟(𝑇−𝑡) 𝑑𝑡
𝐹𝑉 =
0
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Comparison
𝑇
𝐴 = 𝑃𝑒 𝑟𝑡
𝑓(𝑡)𝑒 𝑟(𝑇−𝑡) 𝑑𝑡
𝐹𝑉 =
0
 In both cases, the interest is compounded continuously.
 With FV, deposits are made in a continuous flow.
 With 𝐴 = 𝑃𝑒 𝑟𝑡 , a single deposit is made.
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Definitions
 Total income produced from a continuous income stream:
• For example, a total of your monthly allowance over a
period of time.
 Future value of a continuous income stream compounded
continuously:
• The total of your allowance plus the interest it has
earned over a period of time.
 Interest earned = Future value – Total income
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Example 1
A. Find the total income produced in the first 5 years by
the continuous income stream if the rate of flow is:
f (t) = 5000 e0.04 t (use graphing calculator)
B. Find the future value at a rate of 9% interest
compounded continuously for 5 years of the same
rate of flow. (use graphing calculator)
C. Find the interest earned for the 5 years.
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Example 1
A. Find the total income produced in the first 5 years by
the continuous income stream if the rate of flow is:
f (t) = 5000 e0.04 t (use graphing calculator)
5
5000𝑒 .04𝑡 𝑑𝑡
𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑐𝑜𝑚𝑒 =
0
≈ $27,675.35
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Example 1
A. Find the future value at a rate of 9% interest
compounded continuously for 5 years of the same
rate of flow. (use graphing calculator)
r = 0.09, T = 5, f (t) = 5000 e 0.04t
𝑇
𝑓 𝑡 𝑒 𝑟(𝑇−𝑡) 𝑑𝑡
𝐹𝑉 =
0
5
5000𝑒 .04𝑡 𝑒 .09(5−𝑡) 𝑑𝑡
=
0
≈ $34,690.94
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Example 1
C. Find the interest earned for the 5 years.
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝐹𝑢𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒 − 𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑐𝑜𝑚𝑒
= 34,690.94 − 27,675.35
= 7015.59
The interest earned over 5 years is $7,015.59
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Review: Supply and Demand
 p = D(x) is the price-demand equation for a product, where
x is the number of units of the product that consumers will
purchase at a price of $p per unit.
 p = S(x) is the supply-demand equation for a product, where
x is the number of units of the product that producers will
supply at a price of $p per unit.
𝑝 = 𝐷(𝑥)
𝑝 = 𝑆(𝑥)
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Review: Supply and Demand
 According to economic theory, in a free market, the
demand for a product decreases as the price increases.
 And the supply of a product increases as the price
increases.
 The intersection of the graphs is
called the equilibrium point. This
is where supply and demand are
equal. (P*, Q*)
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Consumers’ Surplus
𝑃𝑟𝑖𝑐𝑒
 In the Price-demand graph below:
• Let 𝑝 be the current price for a product.
• Let 𝑥 be the demand (number of units that consumers will
buy at that price)
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦
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Consumers’ Surplus
 Price-demand graph:
• If the price was p, the demand would be x. This would
represent consumers who are willing to pay a higher price.
• Consumers who are willing to pay more than 𝑝, but who
end up buying the product at 𝑝, have saved money.
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Consumers’ Surplus
 The amount consumers save on an item when they are able
to pay less than they were planning to pay is called
consumers' surplus. The total amount all of the consumers
saved is represented by the area between the price-demand
curve and the price-level equation.
𝑃𝑟𝑖𝑐𝑒
𝑥
𝐶𝑆 =
𝐷 𝑥 − 𝑝 𝑑𝑥
0
𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦
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Example 2
Find the consumers’ surplus at a price level (𝑝) of $120
for the price-demand equation
p =D (x) = 200 – 0.02x
𝑥
𝐶𝑆 =
𝐷 𝑥 − 𝑝 𝑑𝑥
0
Step 1. Find the demand 𝑥 when the price 𝑝 is $120.
𝑝 = 200 – 0.02𝑥
120 = 200 − 0.02𝑥
−80 = −0.02𝑥
4000 = 𝑥
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Example 2 (continued)
Step 2. Find the consumers’ surplus:
4000
𝑥
𝐷 𝑥 − 𝑝 𝑑𝑥 =
𝐶𝑆 =
200 − 0.02𝑥 − 120 𝑑𝑥
0
4000
0
=
80 − 0.02𝑥 𝑑𝑥
0
= 80𝑥
− 0.01𝑥 2
4000
0
= 80 4000 − 0.01 4000
2
−0
= 160,000
The total savings to consumers who were willing to pay a
higher price than $120 is $160,000.
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Producers’ Surplus
 Similarly, if p = S(x) is the price-supply equation for a
product, 𝑝 is the current price, and 𝑥 is the current supply.
 The shaded region represents suppliers who could supply
some units at a lower price than 𝑝.
 The additional money that suppliers gain from charging more
when they could’ve charged less is called producers’ surplus.
𝑥
𝑃𝑆 =
𝑝 − 𝑆(𝑥) 𝑑𝑥
0
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Example 3
Find the producers’ surplus at a price level of p  $55
for the price-supply equation
p = S(x) = 15 + 0.1x + 0.003x 2
𝑥
𝑃𝑆 =
𝑝 − 𝑆(𝑥) 𝑑𝑥
0
Step 1. Find 𝑥 , the supply when the price is $55.
𝑝 = 15 + 0.1𝑥 + 0.003𝑥 2
55 = 15 + .1𝑥 + 0.003𝑥 2
0 = −40 + .1𝑥 + 0.003𝑥 2
𝑥 = 100 Use Option 2 to find the
x-intercept
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Example 3 (continued)
Step 2. Find the producers’ surplus:
100
𝑥
55 − (15 + 0.1𝑥 + 0.003𝑥 2 ) 𝑑𝑥
𝑝 − 𝑆(𝑥) 𝑑𝑥 =
𝑃𝑆 =
0
100
0
40 − 0.1𝑥 − 0.003𝑥 2 ) 𝑑𝑥
=
0
0.1𝑥 2 0.003𝑥 3 100
= 40𝑥 −
−
2
3
0
100
2
3
= 40𝑥−.05𝑥 −.001𝑥
0
= 4000 − 500 − 1000
= 2500 The producers’ surplus is
$2,500
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Lesson 7-2 Summary
■ We learned how to calculate probability from a probability
density function.
■ We found the total income from a continuous income
stream.
■ We found the future value of a continuous income stream.
■ We found the amount of interest earned from a continuous
income stream.
■ We defined and calculated consumer’s surplus.
■ We defined and calculated producer’s surplus.
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Homework
#7-2B
Pg 430
(11, 14, 22, 29,
31, 33, 43-51 odd)
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