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Exponential Growth and Decay: Continuity Law of Natural Growth and Law of Natural Decay

This chapter explores the concepts of exponential growth and decay, focusing on the continuity law of natural growth (k > 0) and the law of natural decay (k < 0). It provides solutions to initial-value problems and applies these concepts to population growth, radioactive decay, and compound interest.

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Exponential Growth and Decay: Continuity Law of Natural Growth and Law of Natural Decay

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  1. Exponential Growth and Decay CHAPTER 2 2.4 Continuity Law of Natural Growth(k>0)& (Law of natural decay (k<0)):dy/dt = ky The solution of the initial-value problem dy/dt = ky, y(0) = yois y(t) = yo ekt. Population Growth: In the context of population growth, we can write dP/dt = kP or (1/p)(dP/dt) = k, where (1/p)(dP/dt) is the growth rate divided by the population size,and it is called the relative growth rate.

  2. CHAPTER 2 Example: A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 100 cells. a) Find the relative growth rate. b) Find an expression for the number of cells after t hours. c) Find the number of cells after 10 hours. d) When will the population reach 10,000 cells? 2.4 Continuity

  3. CHAPTER 2 Example: A bacteria culture grows with constant relative growth rate. The count was 400 after 2 hours and 25,600 after 6 hours. a) What was the initial population of the culture? b) Find an expression for the population after t hours. c) In what period of time does the population double? d) When will the population reach 100,000? 2.4 Continuity

  4. Radioactive Decay: Radioactive substances decay by spontaneously emitting radiation. If m(t) is the mass remaining from an initial mass moof the substance after time t, then the relative decay rate –(1/m)(dm/dt) has been found experimentally to be constant. It follows that dm/dt = k m where k is a negative constant. The mass decays exponentially: m(t) = moek t. Physicistsexpress the rate of decay in terms of half-life, the time required for half of any given quantity to decay.

  5. CHAPTER 2 Example: Polonium-210 has a half-life of 140 days. a) If a sample has a mass of 200 mg, find a formula for the mass that remains after t days. b) Find the mass after 100 days. c) When will the mass be reduced to 10 mg? d) Sketch the graph of the mass function. 2.4 Continuity

  6. CHAPTER 2 Example: After 3 days a sample of radon-222 decayed to 58% of its original amount. a) What is the half-life of radon-222? b) How long would it take the sample to decay to 10% of its original amount? 2.4 Continuity

  7. Continuously Compound Interest: CHAPTER 2 Example: How long will it take an investment to double in value if the interest rate is 6% compounded continuously? 2.4 Continuity Example: If $500 is borrowed at 14% interest, find the amounts due at the end of 2 years if the interest is compounded: i) annually, ii) quarterly, iii) monthly, iv) daily, v) hourly, vi) continuously.

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