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AP Calculus Ms. BattagliaPowerPoint Presentation

AP Calculus Ms. Battaglia

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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

Solving a Differential Equation

Solve the differential equation

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.

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?

Complete Table for Savings Account in Which Interest is Compounded Continuously

Compound Interest Compounded Continuously

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

Compound Interest Compounded Continuously

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.

Classwork/Homework Compounded Continuously

- 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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