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Chap3_Sec8 - DIFFERENTIATION RULES 3.8 Exponential Growth...

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3.8 Exponential Growth and Decay In this section, we will: Use differentiation to solve real-life problems involving exponentially growing quantities. DIFFERENTIATION RULES
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Use the fact that the world population was 2,560 million in 1950 and 3,040 million in 1960 to model the population in the second half of the 20 th century. (Assume the growth rate is proportional to the population size.) square4 What is the relative growth rate? square4 Use the model to estimate the population in 1993 and to predict the population in 2020. POPULATION GROWTH Example 1
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We measure the time t in years and let t = 0 in 1950. We measure the population P ( t ) in millions of people. square4 Then, P (0) = 2560 and P (10) = 3040 POPULATION GROWTH Example 1
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Since we are assuming dP / dt = kP , Theorem 2 gives: 10 ( ) (0) 2560 (10) 2560 3040 1 3040 ln 0.017185 10 2560 kt kt k P t P e e P e k = = = = = POPULATION GROWTH Example 1
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The relative growth rate is about 1.7% per year and the model is: square4 We estimate that the world population in 1993 was: square4 The model predicts that the population in 2020 will be: 0.017185 ( ) 2560 t P t e = 0.017185(43) (43) 2560 5360 million P e = 0.017185(70) (70) 2560 8524 million P e = POPULATION GROWTH Example 1
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The graph shows that the model is fairly accurate to the end of the 20th century. square4 The dots represent the actual population. POPULATION GROWTH Example 1
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So, the estimate for 1993 is quite reliable. However, the prediction for 2020 is riskier. POPULATION GROWTH Example 1
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Radioactive substances decay by spontaneously emitting radiation. square4 If m ( t ) is the mass remaining from an initial mass m 0 of a substance after time t , then the relative decay rate has been found experimentally to be constant. square4 Since dm / dt is negative, the relative decay rate is positive. 1 dm m dt - RADIOACTIVE DECAY
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It follows that where k is a negative constant. square4 In other words, radioactive substances decay at a rate proportional to the remaining mass.
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