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Glacier National Park, Montana Photo by Vickie Kelly, 2004 Exponential Growth and Decay Greg Kelly, Hanford High School, Richland, Washington Ex. Find the equation of the curve in the xy-plane that passes through the given point and whose tangent at a given point (x, y) has the given slope y2 slope = 3 x Point: (1, 1) The number of bighorn sheep in a population increases at a rate that is proportional to the number of rabbits present (at least for awhile.) So does any population of living creatures. Other things that increase or decrease at a rate proportional to the amount present include radioactive material and money in an interest-bearing account. If the rate of change is proportional to the amount present, the change can be modeled by: dy ky dt dy ky dt 1 dy k dt y 1 y dy k dt Rate of change is proportional to the amount present. Divide both sides by y. Integrate both sides. ln y kt C 1 y dy k dt Integrate both sides. ln y kt C ln y e kt C e y e e C kt Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication. ln y e kt C e y e e C kt Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication. y e e C kt y Ae kt Since eC is a constant, let e C A. y e e C kt y Ae kt y0 Ae Since eC is a constant, let e C A. 1 k 0 At t 0 , y y0 . y0 A y y0 e kt This is the solution to our original initial value problem. Exponential Change: y y0 e kt If the constant k is positive then the equation represents growth. If k is negative then the equation represents decay. Population Growth In the spring, a bee population will grow according to an exponential model. Suppose that the rate of growth of the population is 30% per month. a) Write a differential equation to model the population growth of the bees. b) If the population starts in January with 20,000 bees, use your model to predict the population on June 1st.