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

2 Barnett/Ziegler/Byleen Business Calculus 12e 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.

3 Continuous Income Stream Barnett/Ziegler/Byleen Business Calculus 12e

4 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:

5 Comparison Barnett/Ziegler/Byleen Business Calculus 12e

6 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 Barnett/Ziegler/Byleen Business Calculus 12e

7 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 e 0.04 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.

8 Barnett/Ziegler/Byleen Business Calculus 12e 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 e 0.04 t (use graphing calculator)

9 Barnett/Ziegler/Byleen Business Calculus 12e 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

10 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 C.Find the interest earned for the 5 years. The interest earned over 5 years is $7,015.59

11 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. Barnett/Ziegler/Byleen Business Calculus 12e

12 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*) Barnett/Ziegler/Byleen Business Calculus 12e

13 Consumers’ Surplus Barnett/Ziegler/Byleen Business Calculus 12e

14 Consumers’ Surplus Barnett/Ziegler/Byleen Business Calculus 12e

15 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. Barnett/Ziegler/Byleen Business Calculus 12e

16 Barnett/Ziegler/Byleen Business Calculus 12e Example 2

17 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (continued) Step 2. Find the consumers’ surplus: The total savings to consumers who were willing to pay a higher price than $120 is $160,000.

18 Producers’ Surplus Barnett/Ziegler/Byleen Business Calculus 12e

19 Barnett/Ziegler/Byleen Business Calculus 12e Find the producers’ surplus at a price level of for the price-supply equation p = S(x) = x x 2 Example 3 Use Option 2 to find the x-intercept

20 Barnett/Ziegler/Byleen Business Calculus 12e Example 3 (continued) Step 2. Find the producers’ surplus: The producers’ surplus is $2,500

21 Barnett/Ziegler/Byleen Business Calculus 12e 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.

22 Homework #7-2B Pg 430 (11, 14, 22, 29, 31, 33, odd) Barnett/Ziegler/Byleen Business Calculus 12e