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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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