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.

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AP Calculus Ms. Battaglia

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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

Solving a Differential Equation

Solve the differential equation


Solving a differential equation1

Solving a Differential Equation

Solve the differential equation


Growth and decay models

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

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 the table

Complete the Table


Complete table for savings account in which interest is compounded continuously

Complete Table for Savings Account in Which Interest is Compounded Continuously


Compound interest

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


Compound interest1

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.


Classwork homework

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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