AP Calculus Ms. Battaglia. The strategy is to rewrite the equation so that each variable occurs on only one side of the equation. This strategy is called.

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Presentation transcript:

AP Calculus Ms. Battaglia

The strategy is to rewrite the equation so that each variable occurs on only one side of the equation. This strategy is called separation of variables. Ex: y ’ = 2x/y

a. b. c.

A constant rate of growth applied to a continuously growing base over a period of time

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 t=0, y=2, and when t=2, y=4. What is the value of y when t=3?

The rate of change of y is proportional to y. When t=0, y=6, and when t=4, y=15. What is the value of y when t=8?

Suppose that 10 grams of the plutonium isotope 239 Pu was released in the Chernobyl nuclear accident. How long will it take for the 10 grams to decay to 1 gram? Radioactive decay is measured in half-life- the number of years required for half of the atoms in a sample of radioactive material to decay. The rate of decay is proportional to the amount present. Uranium ( 238 U)4,470,000,000 years Plutonium ( 238 U)24,100 years Carbon ( 14 C)5715 years Radium ( 226 Ra)1599 years Einsteinium ( 254 Es) 276 days Nobelium ( 257 No)25 seconds

Suppose an experimental population of fruit flies increases according to the law of exponential growth. There were 100 flies after the 2 nd day of the experiment and 300 flies after the 4 th day. Approximately how may flies were in the original population?

Four months after it stops advertising, a manufacturing company notices that its sales have dropped from 100,000 units/month to 80,000 units per month. If the sales follow an exponential pattern of decline, what will they be after another 2 months?

 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