6.4 Exponential Growth and Decay

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.

Separable Differential Equation

Example Solving by Separation of Variables

The Law of Exponential Change

Continuously Compounded Interest

Example Compounding Interest Continuously

Example Finding Half-Life

Half-life

Newton’s Law of Cooling

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

Example Using Newton’s Law of Cooling

is a model for the  t, T – T  = ( t,T  4.5) data. Thus,  4.5  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  is a model of the  t,T , data.  t (b) The figure below shows the graphs of y  8 and y  T    t (a)At time t  0 the temperature was T     C 