Advanced Placement Calculus
          Differential Equations and Mathematical Modeling
                Chapter 7       Section 4
                 Exponential Growth and Decay
       Essential Question:  Explain how the laws of exponential change are used in the 
                    applications of compound interest, radioactivity, Netwon’s Law 
                    of Cooling, and resistance proportional to velocity?
       Student Objectives: The students will be able to solve problems involving exponential 
                    growth and decay in a variety of applications.
       Vocabulary:
          Compound interest
          Compounded continuously
          Continuous interest rate
          Half-life
          Newton’s Law of Cooling
          Radioactive decay
          Radioactive growth
          Rate constant
          Resistance
          Velocity
              Key Ideas:
                     Exponential Growth
                     Exponential Decay
                     Half-life
                     Newton’s Law of Cooling
                     Resistance proportional to velocity
              Mathematical Formulas:
                   1.   Exponential growth and decay formula:
                            y(t) = Cekt, where
                           C = the initial amount
                            t = time
                            k = the growth constant (k > 0 is growth and k < 0 is decay)
                   2.   Compound Interest
                                          nt
                                   ⎛   r ⎞
                            A(t)= A 1+
                                   ⎜     ⎟
                                   ⎝   n⎠
                            A(t) =  Ending amount
                           A = Initial amount
                            r =  interest rate
                            n = number of times the interest is collected in a year
                            t = number of years 
                   3.   Newton’s Law of Cooling
                            T −T = T −T ekt
                                s  ( 0   s )
                            T = the temperature at an given time
                            T= the stabilizing temperature
                            s
                            T = the initial temperature
                             0
                            k = the cooling rate (k < 0)
                            t = the time period being measured
                    4.   Resistance proportional to velocity
                                             −kt
                                    v(t)= v e m
                                           0
                                                  v m⎛     −kt⎞
                                    s(t) = ∫v(t)= 0   ⎜1−e m ⎟
                                                   k ⎝        ⎠
                                    Stopping distance = lims(t)
                                                      t→∞
                                    v = the velocity at any given time
                                    v0 = the initial velocity
                                    m = the mass of the moving object
                                    k = the constant resisting force
                                    t = the time period being measured
               Sample Questions:
                 1.   The half-life of a certain substance is 5.8 hours.  What percent of the substance if left 
                      after 2 days?
                   2.   $500.00 is can be placed into an account that will earn 3.75% compound quarterly for 5 
                        years or placed into an account that would earn 3.8% compounded continuously for 5 
                        years.  Which account would earn the most amount of money?  How much interest did 
                        this account earn?
                   3.   Coffee that measures 120° F is poured into a cup that is sitting a a room that is 72° F.  
                        Five minutes after being poured the coffee measures 108° F.    What is the temperature of 
                        the coffee after 15 minutes?