1. AP Calculus BC Tuesday, 02 February 2016 OBJECTIVE TSW solve exponential growth and decay problems. ASSIGNMENTS DUE FRIDAY –WS Bases Other Than e  given Monday, 02/01/16 –WS Differential Equations: Growth and Decay  given today, Tuesday, 02/02/16 –WS Newton’s Law of Cooling  given Wed/Thur, 02/03-04/16 QUIZ: Other Bases; Growth and Decay will be on Friday, 05 February 2016.

2. Differential Equations: Growth and Decay

3. In this section, we’ll use differential equations to solve application problems. Ex:Solve

4. Differential Equations: Growth and Decay In many applications, the rate of change of a variable y is proportional to the value of y. If y is a function of time t, the proportion can be written as Rate of change of y is proportional to y. k Constant of proportionality

5. Differential Equations: Growth and Decay

6. If y is a differentiable function of t such that for some constant k, then Exponential growth k > 0 exponential decay k 0 and exponential decay occurs when k < 0. T HEOREM : Exponential Growth & Decay Model

7. Differential Equations: Growth and Decay Ex:The rate of change of y is proportional to y. When t = 0, y = 2. When t = 2, y = 4. What is y when t = 3? When t = 0, y = 2:When t = 2, y = 4: Store into calculator.

8. Differential Equations: Growth and Decay Ex:The rate of change of y is proportional to y. When t = 0, y = 2. When t = 2, y = 4. What is y when t = 3? When t = 3, y = ?: REMEMBER: Unless told otherwise, 3 decimal places.

9. Differential Equations: Growth and Decay Ex:10 g of an isotope is released. It has a ½- life of 24,360 years. How long will it take the isotope to decay to 1 g? If labels are given, use them ! ! !

10. Differential Equations: Growth and Decay Ex:A fruit fly population grows according to the law of exponential growth. On the 2 nd day, there were 100 flies; on the 4 th day, there were 300 flies. Approximately how many flies were in the original population?