AP Calculus Ms. Battaglia. Solve the differential equation.

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Presentation transcript:

AP Calculus Ms. Battaglia

Solve the differential equation

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 = Ce kt. 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.

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?

IsotopeHalf-Life (in years) Initial Quantity Amount After 1,000 Years Amount After 10,000 Years 226 Ra g 226 Ra g 226 Ra g 14 C g

Initial Investment Annual RateTime to Double Amount After 10 Years $40006% $18,0005.5% $7507¾ yr $500$

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

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.

 AB: Read 6.2 Page 420 #1-12, 21, 23,  BC: Read 6.2 Page 420 #7-14, 21, 25-28, 33, 34, 57, 58, 73, 75-78