Mathematics
Growth and Decay
Growth and decay refer to the change in quantity over time. In the context of mathematics, growth typically involves an increase in quantity, while decay involves a decrease. These concepts are often studied in relation to exponential functions, where growth is represented by positive exponential growth and decay is represented by negative exponential decay.
Written by Perlego with AI-assistance
Related key terms
1 of 5
4 Key excerpts on "Growth and Decay"
eBook - PDF- Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
CHAPTER 5 Growth and Decay: An Introduction to Exponential Functions Overview Exponential and linear functions are used to describe quantities that change over time. Expo- nential functions represent quantities that are multiplied by a constant factor during each time period. Linear functions represent quantities to which a fixed amount is added (or subtracted) during each time period. Exponential functions can model such diverse phenomena as bacteria growth, radioactive decay, compound interest rates, inflation, musical pitch, and family trees. After reading this chapter, you should be able to • recognize the properties of exponential functions and their graphs • understand the differences between exponential and linear growth • model Growth and Decay phenomena with exponential functions • represent exponential functions using percentages, factors, or rates • use semi-log plots to determine if data can be modeled by an exponential function • use base e to understand and model continuous or instantaneous compounding 255 5.1 Exponential Growth The Growth of E. coli Bacteria Measuring and predicting growth is of concern to population biologists, ecologists, demogra- phers, economists, and politicians alike. The growth of bacteria provides a simple model that scientists can generalize to describe the growth of other phenomena such as cells, countries, or capital. Bacteria are very tiny, single-celled organisms that are by far the most numerous organ- isms on Earth. One of the most frequently studied bacteria is E. coli, a rod-shaped bacterium approximately 10 −6 meter (or 1 micrometer) long that inhabits the intestinal tracts of humans and other mammals. 1 The cells of E. coli reproduce by a process called fission: The cell splits in half, forming two “daughter cells.” See course software “E2: Exponential Growth & Decay” for a dramatic visualization of Growth and Decay. Applet 1 Most types of E. coli are beneficial to humans, aiding in digestion.
- Lorenzo Sadun(Author)
- 2023(Publication Date)
- American Mathematical Society(Publisher)
6.3. Simple Models of Growth and Decay 151 Exponential Growth and Decay. The solution to (6.53) is easy. We have already seen that the derivative of () = is ( ), and that (0) = 0 = , so = 0 . In other words, the solution to (6.53) is (6.55) () = 0 . The IVP (6.53) and the solution (6.55) are incredibly common. Not only should you memorize them, but getting from one to the other should become a spinal reflex for you. If our initial condition occurs at a time 0 ≠ 0, we just adjust our formula to count the time since we started. If / = and ( 0 ) = 0 , then (6.56) () = 0 (− 0 ) . For instance, suppose that a city’s population is growing at 2.5%/year, and that its population in the year 2020 was 1,000,000 people. How many people will be in the city in the year 2030? When will the population hit 2,000,000? To figure out (2030), we just plug 0 = 1,000,000, = 0.025, and = 2030 into equation (6.56): (6.57) (2030) = 1,000,000 0.025(2030−2020) = 1,000,000 0.25 ≈ 1, 284, 000. To figure out when the population will hit 2,000,000, we solve () = 2,000,000, 1,000,000 (− 0 ) = 2,000,000, (− 0 ) = 2, ( − 0 ) = ln(2), = 0 + ln(2) ≈ 2020 + 0.693 0.025 = 2047.7. (6.58) When a quantity is growing exponentially, we call the time it takes for the quantity to double the doubling time. When a quantity is shrinking exponentially, we call the time it takes to drop to half its initial value the half-life. The solutions to = 2 and − = 1/2 are both = ln(2)/. Since ln(2) is approximately 0.70, this leads to a very simple rule of thumb. The Law of 70: The doubling time (or half-life) is approximately 70 divided by the growth (or decay) rate in percent. If a city (or an investment, or an economy) grows at 2%/year, it will double in about 35 years.
eBook - PDFIntermediate Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
The percentage change is often called a growth or decay rate. Use the following Concept Investigation to compare two different growth patterns. Exponential Growth and Decay Rates and Compounding Interest LEARNING OBJECTIVES Define a growth rate. Discover the relationship between growth rates and exponential functions. Use the formulas for compounding interest. 5.5 In this investigation, we consider two salary options for a company’s employees. Option 1: Starting salary of $50,000 with a $2000 raise each year Option 2: Starting salary of $50,000 with a 4% raise each year 1. Complete the following table of calculations. Years on the Job Option 1 Salary (dollars) Option 2 Salary (dollars) 0 50,000 50,000 1 50,000 1 2,000 5 52,000 50,000 1 50,0001 0.042 5 52,000 2 52,000 1 2,000 5 54,000 52,000 1 52,0001 0.042 5 54,080 3 4 5 6 What pattern grows faster? CONCEPT INVESTIGATION Connecting the Concepts How does our salary grow? When calculating both of these salary options, remember to use the previous year’s salary each time we do the calculation. With option 2, we take the previous year’s salary and add the 4% growth to it. We are calculating the 4% growth by multiplying the previous year’s salary by 0.04. Because the 4% is based on a new amount each year, the dollar amount the salary grows is larger each year. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C H A P T E R 5 E x p o n e n t i a l F u n c t i o n s 524 2.
No longer available |Learn more- J. N. Kapur(Author)
- 2023(Publication Date)
- Mercury Learning and Information(Publisher)
partial differential equations .Mathematical models in terms of ordinary differential equations will be studied in this and the next two chapters. Mathematical models in terms of partial differential equations will be studied in Chapter 7 .2.2 LINEAR Growth and Decay MODELS
2.2.1 Populational Growth Models
Let x (t ) be the population size at time t and let b and d be the birth and death rates, i.e., the number of individuals born or dying per individual per unit time, then in time interval (t, t + Δt ), the numbers of births and deaths would be bx Δt + 0(Δt ) and dx Δt + 0(Δt ), where 0(Δt ) is an infinitesimal which approaches zero as Δt approaches zero, so thatso that dividing by Δt and proceeding to the limit as Δt → 0, we getIntegrating Eqn. (2 ), we getso that the population grows exponentially if a > 0, decays exponentially if a < 0, and remains constant if a = 0 (Figure 2.1 ).FIGURE 2.1(i) If a > 0, the population will become double its present size at time T , whereorT is called the doubling period of the population and it may be noted that this doubling period is independent of x (0). It depends only on a and is such that the greater the value of a
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
