1. Aim: Growth & Decay Course: Calculus Do Now: Aim: How do we solve differential equations dealing with Growth and Decay Find

2. Aim: Growth & Decay Course: Calculus Separation of Variables – Liebniz notation Solve the differential equation General Solution

3. Aim: Growth & Decay Course: Calculus Separation of Variables Solve the differential equation General Solution

4. Aim: Growth & Decay Course: Calculus Growth & Decay Models The rate of change of a variable is often proportional to the value of the variable. Rate of change of y proportional to y is

5. Aim: Growth & Decay Course: Calculus Growth & Decay Models e n  n   Exponential growth Continuous compounding Continuous growth/decay k is a constant (±) y is differentiable function of t such that y > 0 and y’ = ky, for some constant k. C is the initial value of y, and k is the proportional constant. Exponential growth occurs when k > 0 and exponential decay occurs when k < 0. The rate of change of a variable is often proportional to the value of the variable

6. Aim: Growth & Decay Course: Calculus Model Problem The rate of change of y is proportional to y. When t = 0, y = 2. When t = 2, y = 4. What is the value of y when t = 3? y’ = kyFind value of C; find value of k

7. Aim: Growth & Decay Course: Calculus Model Problem Suppose 10 grams of the plutonium isotope Pu-239 (half-life 24,360 years) was released in the Chernobyl nuclear accident. How long will it take for the 10 gram to decay to 1 gram? y represents mass of plutonium Find value of C; find value of k

8. Aim: Growth & Decay Course: Calculus Model Problem Suppose an exponential population of fruit flies increases according to the law of exponential growth. There were 100 fruit flies after the second day of the experiment and 300 flies after the fourth day. Approximately how many flies were in the original population? y represents # of flies Find value of C; find value of k

9. Aim: Growth & Decay Course: Calculus Model Problem Suppose an exponential population of fruit flies increases according to the law of exponential growth. there were 100 fruit flies after the second day of the experiment and 300 flies after the fourth day. Approximately how many flies were in the original population?

10. Aim: Growth & Decay Course: Calculus Model Problem Newton’s Law of Cooling state that the rate of change in the temperature of an object is proportional to the difference between the object’s temperature (y) and the temperature of the surrounding medium. Let y represent the temperature (in 0 F) of an object in a room whose temperature is kept at a constant 60 0. If the object cools from 100 0 to 90 0 in 10 minutes, how much longer will it take for its temperature to decrease to 80 0 ? y’ = k(y – 60), 80 < y < 100 Rate of change of y proportional to y is

11. Aim: Growth & Decay Course: Calculus Model Problem Let y represent the temperature (in 0 F) of an object in a room whose temperature is kept at a constant 60 0. If the object cools from 100 0 to 90 0 in 10 minutes, how much longer will it take for its temperature to decrease to 80 0 ? y’ = k(y – 60) differential equation separate variables integrate both sides find antiderivatives

12. Aim: Growth & Decay Course: Calculus Model Problem Let y represent the temperature (in 0 F) of an object in a room whose temperature is kept at a constant 60 0. If the object cools from 100 0 to 90 0 in 10 minutes, how much longer will it take for its temperature to decrease to 80 0 ?

13. Aim: Growth & Decay Course: Calculus Model Problem Let y represent the temperature (in 0 F) of an object in a room whose temperature is kept at a constant 60 0. If the object cools from 100 0 to 90 0 in 10 minutes, how much longer will it take for its temperature to decrease to 80 0 ? t  24.09 minutes It will require 14.09 more minutes to cool.

14. Aim: Growth & Decay Course: Calculus The Product Rule