Chapter 7 Additional Integration Topics Section 2 Applications in Business and Economics

2 Learning Objectives for Section 7.2 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 the future value of a continuous income stream. 4. The student will be able to evaluate consumers’ and producers’ surplus.

3 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) over the interval (–∞, ∞) is 1 3.If [c, d] is a subinterval of (–∞, ∞) then Probability (c ≤ x ≤ d) =

4 Probability Density Functions (continued) Sample probability density function

5 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.

6 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.

7 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

8 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

9 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

10 Future Value of a Continuous Income Stream From previous work we are familiar with the continuous compound interest formula A = Pe rt. 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

11 Example Let’s continue the previous example where f (t) = 600 e 0.06 t Find the future value in 2 years at a rate of 10%.

12 Example Let’s continue the previous example where f (t) = 600 e 0.06 t Find the future value in 2 years at a rate of 10%. r = 0.10, T = 2, f (t) = 600 e 0.06t

13 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 CS x p

14 Find the consumers’ surplus at a price level of for the price-demand equation p = D (x) = 200 – 0.02x Example

15 Find the consumers’ surplus at a price level of for the price-demand equation p = D (x) = 200 – 0.02x Example Step 1. Find the demand when the price is

16 Example (continued) Step 2. Find the consumers’ surplus:

17 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 Producers’ Surplus x p CS which is the area between and p = S(x) from x = 0 to

18 Find the producers’ surplus at a price level of for the price-supply equation p = S(x) = x Example

19 Find the producers’ surplus at a price level of for the price-supply equation p = S(x) = x x 2 Step 1. Find, the supply when the price is Solving for using a graphing utility: Example

20 Example (continued) Step 2. Find the producers’ surplus:

21 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.