1. 6.4 Exponential Growth and Decay

2. What you’ll learn about Separable Differential Equations Law of Exponential Change Continuously Compounded Interest Modeling Growth with Other Bases Newton’s Law of Cooling … and why Understanding the differential equation gives us new insight into exponential growth and decay.

3. Separable Differential Equation

4. Example Solving by Separation of Variables

5. The Law of Exponential Change

6. Continuously Compounded Interest

7. Example Compounding Interest Continuously

8. Example Finding Half-Life

9. Half-life

10. Newton’s Law of Cooling

11. Example Using Newton’s Law of Cooling A temperature probe is removed from a cup of coffee and placed in water that has a temperature of T = 4.5 C. Temperature readings T, as recorded in the table below, are taken after 2 sec, 5 sec, and every 5 sec thereafter. Estimate (a)the coffee's temperature at the time the temperature probe was removed. (b)the time when the temperature probe reading will be 8 C. o S o

12. Example Using Newton’s Law of Cooling

13. is a model for the  t, T – T  = ( t,T  4.5) data. Thus,  4.5  61.66 0.9277 Example Using Newton’s Law of Cooling  Use exponential regression to find that According to Newton's Law of Cooling, T  T =  T – T  e  kt S O S where T = 4.5 and T is the temperature of the coffee at t  0. S O t S T  T  4.5 + 61.66 0.9277 is a model of the  t,T , data.  t (b) The figure below shows the graphs of y  8 and y  T  4.5 + 61.66  0.9277  t (a)At time t  0 the temperature was T  4.5 + 61.66  0.9277   66.16 C 