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Learning Objectives for Section 14.2 Applications in Business/Economics

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Learning Objectives for Section 14.2 Applications in Business/Economics

1. The student will be able to construct and interpret probability density functions.

2, The student will be able to evaluate a continuous income stream.

3. The student will be able to evaluate consumers’ and producers’ surplus.

Barnett/Ziegler/Byleen Business Calculus 11e

Probability Density Functions

- A probability density function must satisfy:
- f (x) 0 for all x
- The area under the graph of f (x) is 1
- If [c, d] is a subinterval then Probability (c x d) =

Barnett/Ziegler/Byleen Business Calculus 11e

Probability Density Functions(continued)

Sample probability density function

Barnett/Ziegler/Byleen Business Calculus 11e

Example

In a certain city, the daily use of water in hundreds of gallons per household is a continuous random variable with probability density function

Find the probability that a household chosen at random will use between 300 and 600 gallons.

Barnett/Ziegler/Byleen Business Calculus 11e

Insight

The probability that a household in the previous example uses exactly 300 gallons is given by:

In fact, for any continuous random variable x with probability density function f (x), the probability that x is exactly equal to a constant c is equal to 0.

Barnett/Ziegler/Byleen Business Calculus 11e

Continuous Income Stream

Total Income for a Continuous Income Stream:

If f (t) is the rate of flow of a continuous income stream, the total income produced during the time period from t = a to t = b is

a Total Income b

Barnett/Ziegler/Byleen Business Calculus 11e

Example

Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is

f (t) = 600 e 0.06t

Barnett/Ziegler/Byleen Business Calculus 11e

Example

Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is

f (t) = 600 e 0.06t

Barnett/Ziegler/Byleen Business Calculus 11e

Future Valueof a Continuous Income Stream

From previous work we are familiar with the continuous compound interest formula

A = Pert.

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, the the future value FV at the end of T years is given by

Barnett/Ziegler/Byleen Business Calculus 11e

Example

Let’s continue the previous example where

f (t) = 600 e0.06 t

Find the future value in 2 years at a rate of 10%.

Barnett/Ziegler/Byleen Business Calculus 11e

Example

Let’s continue the previous example where

f (t) = 600 e0.06 t

Find the future value in 2 years at a rate of 10%.

r = 0.10, T = 2, f (t) = 600 e 0.06t

Barnett/Ziegler/Byleen Business Calculus 11e

x

p

Consumers’ SurplusIf is a point on the graph of the price-demand equation

P = D(x), the consumers’ surplus CS at a price level of is

which is the area between p =

and p = D(x) from x = 0 to x =

The consumers’ surplus represents the total savings to consumers who are willing to pay more than for the product but are still able to buy the product for .

Barnett/Ziegler/Byleen Business Calculus 11e

Example

Find the consumers’ surplus at a price level of for the price-demand equation

p = D (x) = 200 – 0.02x

Barnett/Ziegler/Byleen Business Calculus 11e

Example

Find the consumers’ surplus at a price level of for the price-demand equation

p = D (x) = 200 – 0.02x

Step 1. Find the demand when the price is

Barnett/Ziegler/Byleen Business Calculus 11e

x

p

Producers’ SurplusIf is a point on the graph of the price-supply equation p = S(x), then the producers’ surplus PS at a price level of is

which is the area between and p = S(x) from x = 0 to

The producers’ surplus represents the total gain to producers who are willing to supply units at a lower price than but are able to sell them at price .

Barnett/Ziegler/Byleen Business Calculus 11e

Example

Find the producers’ surplus at a price level of for the price-supply equation

p = S(x) = 15 + 0.1x + 0.003 2

Barnett/Ziegler/Byleen Business Calculus 11e

Example

Find the producers’ surplus at a price level of for the price-supply equation

p = S(x) = 15 + 0.1x + 0.003x2

Step 1. Find , the supply when the price is

Solving for using a graphing utility:

Barnett/Ziegler/Byleen Business Calculus 11e

Summary

- We learned how to use a probability density function.
- We defined and used a continuous income stream.
- We found the future value of a continuous income stream.
- We defined and calculated a consumer’s surplus.
- We defined and calculated a producer’s surplus.

Barnett/Ziegler/Byleen Business Calculus 11e

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