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.
5 Key excerpts on "Growth and Decay"
eBook - ePubDifferentiating Instruction in Algebra 1
Ready-to-Use Activities for All Students (Grades 7-10)
- Kelli Jurek(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...expressions, ➤ that exponential functions have a variable as the exponent, ➤ that exponential functions have Growth and Decay factors or multipliers, ➤ that exponential growth functions approach zero as x -values decrease, and ➤ that exponential decay functions approach zero as x -values increase. As a result of this unit, students will understand that: ➤ many real-life situations are modeled in exponential functions, and ➤ for an equation in the form of y = a (b) x, a represents the starting value, the value of b reflects either exponential decay or growth, and x is the time period. As a result of this unit, students will be able to: ➤ solve exponent functions and simplify expressions by applying the exponent rules; ➤ recognize and describe relationships in which variables grow and decay exponentially; ➤ describe how the values of a and b affect the graph of an equation in the form of y = a (b) x ; ➤ recognize exponential relationships in tables, graphs, and equations; and ➤. determine the Growth and Decay rates in exponential situations. Launch Scenarios ➤ After Erin graduated from college, she bought a house for $210,000. If it is estimated that real estate is appreciating in value by 5% per year, how much will the house be worth in 10 years when she plans to sell it? (Lesson 2) ➤ When Kevin's daughter is born, he and his wife invest $10,000 in an interest-bearing account for their daughter's education. If the account is earning 4% interest compounded quarterly, and no additional money is deposited into the account, how long will it take for the account to have $15,000? (Lesson 2) ➤ Katie bought a new car for $30,000. If its value is decreasing by 8% each year, how much will the truck be worth after 3 years? (Lesson 3) ➤ It is estimated that the human body can reduce caffeine in the bloodstream at a rate of 15% per hour...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...CHAPTER 8 The Calculus of Growth and Decay Processes The aim of this chapter is to show the reader how to steal the family jewels of two functions that have important applications in all branches of biology: the natural logarithm and the exponential function. This master plan for larceny unfolds by cracking open a window and sneaking into a discussion about how to integrate reciprocal functions. Solving this mystery will unlock the door for a deeper understanding of the logarithms, unmasking their secret identity, and revealing how they can be manipulated. After ransacking the belongings of logarithms and pocketing valuable rules for performing differentiation and integration, the whole house gets turned upside-down as we move our focus to the inverse process of taking an exponential. Here, we uncover a dangerous history of predators, uncontrolled growth, and unstoppable decay. After the heist, you will be able to use your ill-gotten gains to analyze a range of important processes, taking in acid–base titrations, how to survive when lost at sea, fluorescence microscopy, working out the age of a tiger shark, and how drugs are cleared from the body. Figure 8.1 An acid–base titration using the indicator methyl orange. Image courtesy of Cudmore under Creative Commons Attribution – Share Alike 2.0 Generic. Figure 8.2 Fluorescence microscopy of bovine pulmonary artery endothelial cells, with microtubules stained green. 8.1 Integrating a reciprocal function When the rule for integrating power functions of x was introduced in Section 6.2, we noted that the rule does not apply for n = −1: i.e. ∫ x n d x = x n + 1 n + 1 + c [ for n ≠ − 1 ]. (EQ8.1) It is easy to see why this restriction is applied: setting n = −1 would cause the denominator to become zero, and dividing by zero is never a good idea; in mathematical jargon this means that the result is ‘undefined’...
eBook - ePub- Charles Krebs(Author)
- 2008(Publication Date)
- CSIRO PUBLISHING(Publisher)
...Population dynamics is a quantitative subject—it requires us to calculate the rate at which populations change. But we can use graphs to illustrate the principles of population change without delving into the underlying mathematics. Let us start our analysis with a simple model of how a population might increase over time. We will then look at patterns of population change over time in nature. Geometric Population Growth Imagine a population that has been released into a favorable environment and begins to increase in numbers. What forms will this increase take, and how can we describe it quantitatively? To begin, let us consider the simple case in which generations are separate, as in annual plants, which have a single breeding season and a lifespan of one year. What happens to this population will depend on the number of female offspring that survive and reproduce the following year (see Essay 6.1). The pattern of population change is called geometric population growth (or exponential growth) because, in the simplest case, the population grows like the geometric series 1, 2, 4, 8, 16, 32, 64, and so on. Figure 6.1 shows some examples of geometric population growth in which different numbers of female offspring are produced in each generation. As you would expect, the more female offspring that are produced, the more rapidly the population increases. This is exactly the way money grows in a bank account with a constant annual rate of interest. However, populations do not continue to grow geometrically as in Figure 6.1. If they did, the world would be stacked high with elephants and oak trees. Aristotle pointed this out 2,300 years ago, and Darwin repeated it 150 years ago. Therefore, we must modify the geometric model to take into account the fact that all populations eventually stop growing. Figure 6.1. Geometric population growth in a hypothetical species of annual plant that has a constant reproductive rate...
eBook - ePub- Kenneth W. Wachter(Author)
- 2014(Publication Date)
- Harvard University Press(Publisher)
...In the figure and the table we see how the slope R has decreased interval by interval in the wake of China’s energetic program of family limitation, including a “One Child Policy” introduced after 1978, modified in practice in recent decades. We also see how even moderate slopes continue to drive substantial multiplicative growth. On the logarithmic scale, our formula for growth at each step involves simple addition: We convert back to counts by applying the exponential function, which brings us back to multiplication: 1.4 Models and Parameters Exponential growth is a model for population size as a function of time. It is the first of many formal models studied in this book, and it provides an opportunity to introduce our terminology and approach and the ingredients that models share. Every model is a model for some quantity. We call the quantity that takes the lead in the analysis the leader or, in the case of model lifetables, the leading column. The choice of leader is the first ingredient of a model. With exponential growth, the leader is population size K (t). Other quantities can often be expressed in terms of the leader and the model can tell us about them too. Examples might be slope or acceleration in population size. But the leader is at the forefront of study. Our second ingredient is the model formula. It expresses the leader as a function of some variable or variables, often time or age. With exponential growth, we can write the formula in a version which uses the symbol C for starting population size: The third ingredient is a set of parameters. Parameters are values which fix the choice of a particular case to which the model applies from among all the cases covered by the formula. Our formula for exponential growth, as written, has two parameters, the starting size C and the growth rate R. Starting points and growth rates differ from case to case, whereas the functional form of the posited relationship between K (t) and t remains the same...
- Craig Adam(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...3 The exponential and logarithmic functions and their applications Introduction: Two special functions in forensic science The exponential and logarithmic functions play a key role in the modelling of a wide range of processes in science and many examples fall within the forensic discipline. As these two functions are in fact the inverses of each other, their meanings are closely related and manipulation of expressions involving one usually requires dealing with the other as well. However, in practice the exponential form occurs explicitly more often than does the logarithm. Most examples of the exponential function relate directly to processes that occur at rates that depend on the amount of material present at a particular time. This generates changing growth or decay rates that are governed by this function; some examples include the chemical kinetics of processes such as the ageing of inks, the decay of fluorescence from an enhanced print or the metabolism of drugs in the human body. Other applications involve the rates of heating or cooling, which are dependent on the temperature difference between the object and its surroundings. Consequently, the exponential function is used to model the post-mortem cooling of a corpse, thus enabling the estimation of the time since death. Absorption of radiation by a material, for example an analytical sample in a UV-vis spectrometer, is also described by the exponential function, where the transmitted intensity is a function of the sample thickness. In contrast, the logarithm is usually encountered when processes that occur over a large dynamic range need to be re-scaled in a more convenient form: for example, chemical pH or sound power level...
