...Population Growth – Exponential and Logistic Models vs. Complex Reality I. Exponential Population Growth 1a. Suppose a single bacterium is placed in a flask that contains lots of food for bacteria. In this flask, each bacterium grows and divides in two every 30 minutes. Therefore, the number of bacteria in the population doubles every 30 minutes. How many bacteria do you think there willbe by 5 hours after the single bacterium is placed in the flask (just guessing)? ______ 1b. Complete the table to calculate how many bacteria there will be at each time. 1 bacterium at the beginning = 0 minutes bacteria by 30 minutes bacteria by 1 hour bacteria by 1 hour and 30 minutes bacteria by 2 hours bacteria by 2 hours and 30 minutes bacteria by 3 hours bacteria by 3 hours and 30 minutes bacteria by 4 hours bacteria by 4 hours and 30 minutes bacteria by 5 hours 2.Plot the number of bacteria at each time; connect the points to show the population growth. Number of Bacteria Time (hours) 3a. How long would it take for the population of bacteria to increase from 1 bacterium to 500 bacteria? 3b. How long would it take for the population to increase from 500 bacteria to 1000 bacteria? Notice that, when a population doubles in each time interval, the number of bacteria in the population increases faster and faster as the population gets larger. This kind of population growth is called exponential population growth. 4.For these bacteria, population...
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...MA131 0 : Module 2 Exponential a nd Logarithmic Functions Exercise 2 .2 Solving Exponential and Logarithmic Equations 1 Answer the following questions to complete this exercise: 1. Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents: 6 x = 216 2. Solve the following exponential equation: e x = 22.8 Express the solution in terms of natural logarithms. Then, use a calculator to obtain a decimal approximation for the solution. 3. Solve the following logarithmic equation: log 7 x = 2 Reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. 4. Solve the following logarithmic equation: log ( x + 16) = log x + log 16 Reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. 5. The population of the world has grown rapidly during the past century. As a result, heavy demands have been made on the world's resources. Exponential functions and equations are often used to model this rapid growth, and logarithms are used to model slower growth. The formula 0.0547 16.6 t Ae models the population of a US state, A , in millions, t years after 2000. a. What was the population in 2000? b. When will the population of the state reach 23.3 million? 6. The goal of our financial security depends on understanding how money in savings accounts grows in remarkable...
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...can be expressed in terms of addition, subtraction, multiplication, division, and the taking of roots of variables and constants are called algebraic functions. In exponential & logarithmic functions we introduce and investigate the properties of exponential functions and Logarithmic functions. These functions are not algebraic; they belong to the class of transcendental functions. Exponential and logarithmic functions are used to model a variety of realworld phenomena: growth of populations of people, animals, and bacteria; radioactive decay; epidemics; absorption of light as it passes through air, water, or glass; magnitudes of sounds and earthquakes. We consider applications in these areas plus many more in the sections very important. As a part of our BBA course, we are required to submit a term paper for every subject each semester. As our Advance Business Mathematics faculty Associate Professor Lt. Col. Md. Showkat Ali has asked us to submit a term paper on a topic upon our will. So, we have decided to choose “Exponential & Logarithmic Functions”. to graph exponential functions to evaluate functions with base e to learn the use of compound interest formulas to learn the changing from logarithmic to exponential form to learn the changing from exponential to logarithmic form to learn the evaluation of logarithms to learn the use of basic logarithmic properties to learn the use of graph logarithmic functions to find the...
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...Population Growth and Ecological Footprints The population size (N) of a species at any point in time (t) is determined by its size at (t-1), its per capita birth rate (b), its per capita death rate (d), and immigration and emigration. Each of these values is, in turn, affected by a huge suite of biotic and abiotic conditions. Human populations are governed by these same variables. In this laboratory, you will use models of population growth to understand how population growth is affected by changes in b and d, and why the actual values of b and d are important. You will also explore some dimensions of human impacts on the environment. A model is a mathematical description of how one thinks a system works. For a population growing at a constant rate, where b and d never change, an appropriate model for population growth is the exponential growth model: dN/dt = rN where r = b – d, assuming that immigration and emigration are negligible. This is a density-independent model since neither b nor d change with increasing population size. Conversely, the logistic model is a density-dependent model, where b and d change with changes in the population size: dN/dt = rN [(K-N)/K] where K = carrying capacity of the environment. In this lab you will use a simple Excel spreadsheet model to explore how population size (N) changes as b and d change under the two different growth models, exponential and logistic. The spreadsheets have areas that you...
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...Human Population Growth SCI 230 August 16, 2013 Tammy Botten The term population growth refers to how the numbers of individuals in a population increases or decreases, with time. This growth is controlled by the rate of which new individuals are added to the population. Between the years 1800 and 1930 human population has doubled from one billion to two billion and then doubled again by the year 1975. This substantial growth in population has led to a decrease in population and biodiversity in many other species. The two types of population growth that can both be seen in human population growth are exponential growth and logistic growth. In population showing exponential growth the individual are not limited by food or disease. However in most real population both food and disease becomes important as condition becomes crowded. Logistic growth model is a more accurate representation of population growth because it takes into account that there are outside factors that may affect the growth. Logistic growth this is frequently used to model biological growth pattern where there’s an initial exponential growth period followed by leveling off as more as the population is infected so as the food supply or some other factors that limits further growths. Exponential growth means that some quality grows a fixed percentage rate form one year to the next. If the population has a constant birth rate through...
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...pursuing Bachelor of Business Mathematics with Honors in UUM. This program can also be seen as a method to integrate the theoretical knowledge with the hands-on training. In this way, students would have an opportunity to apply their understanding of theories into the real working environment within industries. Practicum program provides opportunities for students to improve two vital skills which are soft kills and hard skills. It aims to help student to apply these two skills in interpersonal relationship with employees from various ages and skill levels. At the end of this program, students are expected to have a strong individual personality with a variety of technical and soft skills and able to present themselves as a dynamic role model in family, community and country. 1.1.1 PRACTICUM OBJECTIVES After practical training, students will be able to:...
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...Question: How will you use the power of exponential technologies to bring about exponential impact? Answer: Embrace the idea of a Future that looks very different from Today. Before talking bigly about how the power of exponential technologies can bring about exponential impact, let’s take one step backward and see what the word “exponential” means to us first. The term “exponential” first popped up at 1965, it can be dated back to when Gordon Moore introduced the Moore’s law, which predicts the number of components per integrated circuit would double every year. With the establishment of Moore’s law, it left several insights to the world which are worth to mention: 1. It is predictable that the performance of technologies will double in very rapid way 2. The price of technologies will also decrease by half in very rapid pace. In terms of these prevailing conditions, we can see the exponential technology self-expands in its two traits to the world: Breadth and Depth. So how could these two expansions traditionally make exponential impacts in century? Take camera as an example, it has been evolved from the world’s first digital camera – Kodak Digital Camera at 1975 which cannot take video-taped to the nowadays digital single-lens reflex camera at 2016 which can offer superb image quality and video recording at 4K resolution. In terms of performance, instead of using camera to capture best moments every day, people now use camera: To show the rest of the “fan community”...
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...actually a symbiotic relationship between an algae and a fungus. And they can't live by themselves. You can see the green lichen, but you can also see the orange lichen growing. They are both exploiting a different niche or a different job. Did you know that one of the first scientists to come up with a good definition for a niche was named George Hutchinson. He talked about putting on one side some biotic factor or abiotic factor. Coming to the end of the video basically what it talked about was how an organism makes a living and they play within the ecosystem. So they are constantly in competition with each other. In the second video he talked about ecological succession. That means succession is simply a series of events by which life comes back. So he had a picture of a forest, but this was a year after there was a forest fire. And so you can see a lot of the charred burnt trunks. So basically a photographer came and took pictures of over years of the forest going back to what it looked like before the fire. What you see is life coming back and changes taking place. So that is called succession. And it tends to occur at a predictable rate. Also using a number of predictable species. So basically succession we could break down into two different parts. The first one is called primary succession. Primary succession occurs when you have bare rock. In other words no soil left. On the video there was a picture of lava flow, basically the lava flowed down. And you could see they kind...
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...Ecology The study of how organisms interact with their environments. Ecosystem All the organisms that live within a given area and all the abiotic features of their environment. Exponential growth A model of population growth in which a population grows at a rate proportional to its size. Life history strategy The position a population of organisms occupies on the continuum between producing a large number of “inexpensive” offspring and a small number of “expensive” offspring. Logistic growth A model of population growth in which growth slows as the population approaches the carrying capacity of its habitat. Niche The total set of biotic and abiotic resources a species uses within a community. Population A group of individuals of a single species that occupies a given area. Producer An organism that makes organic molecules from inorganic materials and energy. Symbiosis A situation in which individuals of two species live in close association with one another. Trophic level One of the feeding levels in a food chain, including producers, primary consumers, secondary consumers, tertiary consumers, and so forth. Review Questions 21.1 Organisms and Their Environment 1. What is ecology? 2. Does an organism’s environment include only nonliving components? yes 3. What is the difference between a community and an ecosystem? The difference between a community and an ecosystem is one (community) is an area where organisms live and the other...
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...Populations, demography, population models, competition, predation Dr. Dorn Introduction to Populations: (chapter 8) A. What is a population? What is a species? Stickleback (fish) in lakes: Primary population parameters – B I D E B. Methods of estimating populations 1. Census Method 2. Sampling Methods Census of sample areas N = C/ N = population estimate C = survey count = proportion of area 3. Relative Abundances – CPUE (catch per unit effort) 4. Mark-Recapture Estimates of population size Lincoln-Peterson (closed populations) Assumptions 5. Removal Design (closed population) Examining the number of animal “removed” and cumulatively removed during successive catches Gear set 1 2 3 4 Catch 76 54 35 24 Cumulative 0 76 130 165 Demographic Techniques (chapter 8) A. What is Demography? B. Life Table Analysis of populations Cohort: Parts of a life table analysis: 1. x and nx 2. Survivorship schedule: lx a. Per capita mortality rate: qx b. Survivorship patterns: Type I Type II Type III 3. Getting the data: cohort life tables static life tables 3. Fertility schedule bx Life history strategies: - Semelparous - Iteroparous Costs: 5. The Intrinsic rate of increase (r) a. Net Reproductive Rate (Ro): b. Generation Time (G): c. r ~ ln...
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...Outline Global Company Profile: Tupperware What is Forecasting? Forecasting Time Horizons The Influence of Product Life Cycle Types of Forecasts The Strategic Importance of Forecasting Human Resources Capacity Supply-Chain Management Seven Steps in the Forecasting System 4-2 Outline - Continued Forecasting Approaches Overview of Qualitative Methods Overview of Quantitative Methods Time-Series Forecasting Decomposition of Time Series Naïve Approach Moving Averages Exponential Smoothing Exponential Smoothing with Trend Adjustment Trend Projections Seasonal Variations in Data Cyclic Variations in Data 4-3 Outline - Continued Associative Forecasting Methods: Regression and Correlation Analysis Using Regression Analysis to Forecast Standard Error of the Estimate Correlation Coefficients for Regression Lines Multiple-Regression Analysis Monitoring and Controlling Forecasts Adaptive Smoothing Focus Forecasting Forecasting in the Service Sector 4-4 Learning Objectives When you complete this chapter, you should be able to : Identify or Define: Forecasting Types of forecasts Time horizons Approaches to forecasts 4-5 Learning Objectives - continued When you complete this chapter, you should be able to : Describe or Explain: Moving averages Exponential smoothing Trend projections Regression and correlation analysis ...
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...TOPIC 1. FUNDAMENTALS OF ECONOMIC FORECASTING TOPIC I TOPIC I. FUNDAMENTALS OF ECONOMIC FORECASTING Contents 1. Meaning of forecasting 2. Features, importance and limitations of forecasting 3. Forecast types 1. Meaning of forecasting Forecast is a likely, scientifically well-grounded opinion about the possible state of the events, objects or processes in the future. Forecasting is a process of making statements about events whose actual outcomes (typically) have not yet been observed. Forecasting is a process of predicting or estimating the future based on past and present data. Economic Forecasting is a process of making forecasts based on analysis of past trends and regularities of the economic processes. Economic forecasts can be carried out at a high level of aggregation – for example for GDP, inflation, unemployment or the fiscal deficit – or at a more disaggregated level, for specific sectors of the economy or even specific companies. Economic forecasting provides information about the potential future events and their consequences for the organization. It may not reduce the complications and uncertainty of the future. However, it increases the confidence of the management to make important decisions. Economic forecasting includes the following steps: 1. Identifying items to be forecast. The items of socio-economic forecasting are the economic processes (for example, inflation, demand, supply), any indicator describing the company activity (for example...
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...Question 1. What is the effect of changing lambda (λ) in the discrete model, and r in the continuous model? What is the effect of changing N0 in both models? (2 marks) When in discrete mode, a change in lambda (λ) will change the growth rate of the species. If λ increases, so does the growth rate and if λ decreases, as will the growth rate. However, the growth rate only increases if the number is above 1. 1 itself indicates no growth and below one is negative growth. The same principle applies to r. However, as apposed to 1, 0 is no growth while below 0 and above 0 is negative and positive growth, respectfully. Figure 1 and 2 show a stable, increasing and decreasing population for both continuous and discrete growth. Question 2. What are the differences between the discrete and continuous models of population growth? (2 marks) Discrete growth shows a type of growth in populations that have seasonal breeding seasons (Alstad, 2001). This means the percentage increase can be added after the season to the group. Continuous growth means that the growth of the population is continuously growing without seasons and has overlapping generations. An example of discrete growth is figure 1 and continuous growth in figure 2. Figure 1 Note that the dots are the changes in population after the previous, this graph could also show horizontal lines from one dot to the point in t matches with the above dot. The vertical line from the horizontal dot and dot further along t show the change...
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...Cheat Sheet Index laws (1)Exponents: An exponent is a number that shows how many time to use a particular number for a multiplication. For example 8^of what equals 2002. Know and apply index laws to numerical expressions What are the Index Laws The index laws are a heap of laws that are used when using calculations of indices and exponents, this helps with reducing the calculating process of potential equations that will be raised when dealing with large numbers in mathematics. Rule Example Rule Example Adding Indices when your multiply When two numbers that are the same and are been multiplied together are raised to the power a simple principle can be used to shorten the time used when achieving the problem. X^a x X^b=X^(a+b) Example:...
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...manager for the company, Alex Benson, is uncertain about how many bottles to order from the company’s German glass supplier. His decision is complicated by the possibility of a new bottle design being introduced that would compromise his existing inventory of bottles. Additionally, he is faced with storage limitations and erratic sales, all of which are impacting his decision. He is also concerned about over ordering to avoid issues from an off year, impact from government tax, tourism, and growth of exports. Assignment Question How many bottles should Alex Benson, purchasing manager of Greaves Brewery, order to support his company’s sales for the upcoming year 2004? Case Analysis Greaves Brewery was founded in 1924 by John Greaves in the Caribbean island of Trinidad. In addition to being a favorite among the local population, the beer was also popular with tourists which contributed to the export of the beer some years later. The brewery experienced positive growth between 1999 and 2003, which was attributed to peak sales periods during Carnival, Easter, Christmas and Independence. Throughout the years, the government of Trinidad introduced a number of excise taxes on beer. This negatively impacted sales at Greaves Brewery. Along with normal business challenges and other factors such as number of tourists and U.S. sales, determining the full impact of the taxes on the market made it more challenging for the company to forecast sales. Greaves...
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