barnett/ziegler/byleen business calculus 11e1 learning objectives for section 14.2 applications in...

Barnett/Ziegler/Byleen Business Calculus 11e 1

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.


Probability Density Functions

A probability density function must satisfy:

1. f (x) 0 for all x

2. The area under the graph of f (x) is 1

3. If [c, d] is a subinterval then

Probability (c x d) = d

cdxxf )(


Probability Density Functions(continued)

Sample probability density function


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.

0)(Otherwise.0if15.)( 15. xfxexf x

23.0

15.)6 (3y Probabilit

450906

3

15.

15.06

3

|

.-.–x

x

e - ee

dxex


Insight

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

015.)3 (3y Probabilit 15.03

3 dxex x

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.


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

b

adttf )( income Total

a Total Income b


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


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

275,1$

100010

00010

600 income Total

120

20

060

2

0

06.0

) – (e,

e,

dte

.

t .

t


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

T trTrT tTr dtetfedtetfFV00

)( )()(


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%.


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%.

1,408.59 (1.92209) (1.22140) (600)

600

600

)(

2

0

04.02.0

2

0

10.006.0)2(10.0

0

dtee

dteee

dtetfeFV

t

tt

T trTr

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


If 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 .

Consumers’ Surplus

p

p

CS

xp

( , )x p

0

[ ( ) ]x

CS D x p dx px

p


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

p = D (x) = 200 – 0.02x

Example

120p


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

p = D (x) = 200 – 0.02x

Example

120p

x

Step 1. Find the demand when the price is

120p

000,4

02.0200120

02.0200

x

x

xp


Example (continued)

Step 2. Find the consumers’ surplus:

000160$000160000320

01080

)02.080(

)12002.0200(

)(

|4000

0

2

4000

0

4000

0

0

, , – ,

x.x –

dxx

dxx

dxpxDCSx


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 .

If 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

0

[ ( )]x

p PS p S x dx

Producers’ Surplus

p

p

p

x

p

CS

( , )x p

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

x x

p

p


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

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

Example

$ p 55


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:

Example

$ p 55

100x

x

401000300

0030101555

00301015

2

2

2

x.x.

x.x.

x.x.p

$ p 55

x


Example (continued)

Step 2. Find the producers’ surplus:

500200015000004

001005040

)003.01.040(

])003.01.015(55[

)(

100

|100

0

32

100

0

2

100

0

2

0

, $ , – – ,

x. – x.x –

dxxx

dxxx

dxxSpPS

xx


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.

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