6 2 Differential Equations Growth and Decay Day

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. Rate of change of y is 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 the Table Isotope 226 Ra 14 C Half-Life (in years) Initial Quantity 1599 20 g 1599 Amount After 1, 000 Years Amount After 10, 000 Years 1. 5 g 1599 0. 1 g 5715 3 g

Complete Table for Savings Account in Which Interest is Compounded Continuously Initial Investment Annual Rate $4000 6% $18, 000 5. 5% $750 $500 Time to Double Amount After 10 Years 7¾ yr $1292. 85

Compound Interest Find the principal P that must be invested at rate r, compounded monthly, so that $1, 000 will be available for retirement in t years. r = 7. 5% and t = 20

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 � AB: � BC: Read 6. 2 Page 420 #1 -12, 21, 23, 25 -28 Read 6. 2 Page 420 #7 -14, 21, 25 -28, 33, 34, 57, 58, 73, 75 -78