1. Chapter 6 Integration Section 3 Differential Equations; Growth and Decay

2. 2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 6.3 Differential Equations, Growth, Decay ■ The student will be able to: ■ Identify differential equations. ■ Solve continuous compound interest problems using differential equations. ■ Solve problems involving exponential growth and decay using differential equations.

3. 3 Barnett/Ziegler/Byleen Business Calculus 12e Differential Equations A differential equation contains derivatives. First order differential equations contain only first derivatives Second order differential equations contain second derivatives (and possibly first derivatives) Examples:

4. 4 Steps for Solving Differential Equations 1.Separate the “like variables” on opposite sides of the equation. 2.Integrate both sides. 3.Solve for y Barnett/Ziegler/Byleen Business Calculus 12e Note that the constant C on the left side gets combined with the constant on the right side into a single constant.

5. 5 Example 1  Solve the differential equation using the steps mentioned. Barnett/Ziegler/Byleen Business Calculus 12e Rewrite the log equation as an exponential equation:

6. 6 Example 2  Solve for the particular equation that satisfies: Barnett/Ziegler/Byleen Business Calculus 12e Rewrite the log equation as an exponential equation: Solve for C:

7. 7 Continuous Compound Interest Revisited Barnett/Ziegler/Byleen Business Calculus 12e

8. 8 Application 1 - CCI Find the function A(t) from the information given. Find the balance in the account after 10 years. Barnett/Ziegler/Byleen Business Calculus 12e Change to exponential Solve for C A(t) Equation

9. 9 Application 1 Barnett/Ziegler/Byleen Business Calculus 12e After 10 years the account will have $6,412.36 Find the balance in the account after 10 years.

10. 10 Application 2 - Price-supply Barnett/Ziegler/Byleen Business Calculus 12e

11. 11 Application 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e

12. 12 Application 2 (continued)  First solve the differential equation: Barnett/Ziegler/Byleen Business Calculus 12e Rewrite the log equation as an exponential equation:

13. 13 Application 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e p(0)=$10 C = 10 p(50) = $12.84 Next, solve for C: Next, solve for r: Price-supply equation: Price based on 100 units:

14. 14 Homework Barnett/Ziegler/Byleen Business Calculus 12e

15. 15 Homework Barnett/Ziegler/Byleen Business Calculus 12e #6-3B Pg 379 (2, 8, 9, 12, 20, 22-24, 54, 56, 58)