Chap3_Sec8

# Chap3_Sec8 - DIFFERENTIATION RULES 3.8 Exponential Growth and Decay In this section we will Use differentiation to solve real-life problems

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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.) s What is the relative growth rate? s Use the model to estimate the population in 1993 and to predict the population in 2020. POPULATION GROWTH Example 1
We measure the time t in years and let t = 0 in 1950. We measure the population P ( t ) in millions of people. s 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
The relative growth rate is about 1.7% per year and the model is: s We estimate that the world population in 1993 was: s 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. s The dots represent the actual population. POPULATION GROWTH Example 1
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. s 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. s Since dm / dt is negative, the relative decay rate is positive. 1 dm m dt - RADIOACTIVE DECAY
where k is a negative constant. s In other words, radioactive substances decay at a rate proportional to the remaining mass. s

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## This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.

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Chap3_Sec8 - DIFFERENTIATION RULES 3.8 Exponential Growth and Decay In this section we will Use differentiation to solve real-life problems

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