Ch 5 Integration Techniques Detailed Notes

Ln y kt c integrate ln y kt c e e exponentiate ln y

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ln y kt C Integrate. ln y kt C e e Exponentiate. ln y kt C e e e Separate right side C kt y e e Exponents on left and right must equal. kt y Ae A becomes the amount at time zero. 0 kt y y e Another way to write equation above. The solution to the differential equation is kt y Ce . 0 k represents decay, while 0 k represents growth. Example The rate of bacterial growth is proportional to the number of colonies. If 10 colonies are planted, and 15 minutes later the number of colonies is 15, how many colonies will there be one full hour after the original 10 were planted? Notice that it never said the colony was doubling, tripling etc. ln1.5 60 15 15 15 15 10 10 1.5 50.625 ln 1.5 15 51 ln1.5 0.027 15 kt kt k k e y Ce y Ce e y e e y k y colonies k
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Ch 5 AP Integration Techniques LP 46 Radioactivity If 0 y is the number of radioactive nuclei present at time zero, the number still present at any later time t will be 0 kt y y e where k > 0. Half-life The half-life of a radioactive substance with rate constant k (k>0) is ln 2 half life k Example A radioactive element has a half life of 35 days. How long will it take for a 2 kg sample to decay to 50g? ln 2 35 k   ln2 35 50 2000 t e ln 2 35 k ln2 35 0.025 t e ln 2 ln 0.025 35 35ln 0.025 186.3 ln 2 t t days Newton’s Law of Cooling (See Notes Next Two Pages) A cup of coffee is 100 F in a room at 75 F . Ten minutes later the coffee is 90 F . What will the temperature be in 15 more minutes. 0 0 10 0.1ln0.6 25 10 10 90 75 100 75 100 75 75 15 25 81.97 0.6 ln 0.6 10 0.1ln 0.6 kt kt S s s S k k o k T T T T e T T T e T e T e e T F e k k    
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Ch 5 AP Integration Techniques LP 47
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Ch 5 AP Integration Techniques LP 48
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Ch 5 AP Integration Techniques LP 49 Assignment : Page 338 Blue Book 11, 13, 17, 19
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Ch 5 AP Integration Techniques LP 50 Answers for Page 338 Blue Book
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