Learning Objectives for Section 14 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 Value of 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

Consumers’ Surplus 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 = CS x p 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

Example (continued) Step 2. Find the consumers’ surplus: Barnett/Ziegler/Byleen Business Calculus 11e

Producers’ Surplus 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 x p CS 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

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