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picture1_Calculus Pdf 170135 | Ap Calc Bc Chap 7 Sect 4 Intro


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File: Calculus Pdf 170135 | Ap Calc Bc Chap 7 Sect 4 Intro
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 ...

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              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?
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...Advanced placement calculus differential equations and mathematical modeling chapter section exponential growth decay essential question explain how the laws of change are used in applications compound interest radioactivity netwon s law cooling resistance proportional to velocity student objectives students will be able solve problems involving a variety vocabulary compounded continuously continuous rate half life newton radioactive constant key ideas formulas formula y t cekt where c initial amount time k is nt r n ending number times collected year years ekt temperature at an given stabilizing period being measured kt v e m stopping distance lims any mass moving object resisting force sample questions certain substance hours what percent if left after days can placed into account that earn quarterly for or would which most money much did this coffee measures f poured cup sitting room five minutes...

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