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AP Calculus Ms. Battaglia

6-2 Differential Equations: Growth and Decay (Day 2) Objective: Use separation of variables to solve a simple differential equation; use exponential functions to model growth and decay. AP Calculus Ms. Battaglia. Solving a Differential Equation. Solve the differential equation.

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AP Calculus Ms. Battaglia

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  1. 6-2 Differential Equations: Growth and Decay (Day 2)Objective: Use separation of variables to solve a simple differential equation; use exponential functions to model growth and decay. AP Calculus Ms. Battaglia

  2. Solving a Differential Equation Solve the differential equation

  3. Solving a Differential Equation Solve the differential equation

  4. Growth and Decay Models In many applications, the rate of change of a variable y is proportional to the value of y. If y is a function of time t, the proportion can be written as follows. is Rate of change of y proportional to y. If y is a differentiable function of t such that y > 0 and y’ = ky for some constant k, then y = Cekt. C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0.

  5. Using an Exponential Growth Model The rate of change of y is proportional to y. When x=0, y=6, and when x=4, y=15. What is the value of y when x=8?

  6. Complete the Table

  7. Complete Table for Savings Account in Which Interest is Compounded Continuously

  8. Compound Interest Find the principal P that must be invested at rate r, compounded monthly, so that $1,000,000 will be available for retirement in t years. r = 7.5% and t = 20

  9. Compound Interest Find the time necessary for $1000 to double if it is invested at a rate of 7% compounded (a) annually (b) monthly (c) daily and (d) continuously.

  10. Classwork/Homework • AB: Read 6.2 Page 420 #1-12, 21, 23, 25-28 • BC: Read 6.2 Page 420 #7-14, 21, 25-28, 33, 34, 57, 58, 73, 75-78

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