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# Topics in Math MATH 693

WCU

GPA 3.58

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This 10 page Class Notes was uploaded by Alphonso Thompson on Thursday October 29, 2015. The Class Notes belongs to MATH 693 at Western Carolina University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/230966/math-693-western-carolina-university in Mathematics (M) at Western Carolina University.

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Date Created: 10/29/15

Objectives 1 Students will be able to create and use exponential equations to model population growth and decay 2 Students will be able to use technology to study the population of a country type of animal or a state Standards 1 National Science Education Standards Science in Personal and Social Perspectives Population Growth Populations grow or decline through the combined effects of births and deaths and through emigration and immigration Populations can increase through linear or exponential growth with effects on resource use and environmental pollution Various factors in uence birth rates and fertility rates such as average levels of af uence and education importance of children in the labor force education and employment of women infant mortality rates costs of raising children availability and reliability of birth control methods and religious beliefs and cultural norms that in uence personal decisions about family srze Populations can reach limits to growth Carrying capacity is the maximum number of individuals that can be supported in a given environment The limitation is not the availability of space but the number of people in relation to resources and the capacity of earth systems to support human beings Changes in technology can cause significant changes either positive or negative in carrying capacity 2 North Carolina Biology Standards Competency Goal 4 401 Identify the interrelationships among organisms populations communities ecosystems and biomes 405 Assess and explain human activities that in uence and modify the environment Human Population Growth 3 North Carolina Algebra 1 Standards Competency Goal 3 304 Graph and interpret in the context of the problem relations and functions on the coordinate plane Include linear equations and inequalities quadratics and exponentials 4 North Carolina Algebra 2 Standards Competency Goal 3 315 Write and graph exponential functions of the form fx a bx To Return To The Main Page Click the Back Button TEACHER NOTES Day One 1 Review the concept of exponents Review converting percents to decimals and decimals to percents 2 Using Worksheet Activity 1 review the simple interest formula and how it can be used to compute compound interest Discuss student answers to the simple interest and compound interest questions Students complete the tenyear activity in small groups As they use Monopoly dollars and real pennies to visualize the growth process they may begin to wonder what the account balance will look like with a larger initial investment Allow enough time for students to try to derive a formula for question 2 3 Use Worksheet Activity 2 to introduce the basic exponential growth formula and graph Emphasize to students that this basic formula applies to other situations that model exponential growth such as population growth Students should be able to change the sign to to derive the formula for a constant rate of depreciation Problem 4 requires students to determine whether the value in parentheses represents growth or decay For problem 5 students may need help with the final example fx 3x One explanation could be y 112x which would represent the value of 1 in x years at a growth rate of 200 Day Two 1 Using Worksheet Activity 3 students choose an automobile from the online version of Kelley Blue Book After determining the value of the vehicle in 2003 and 2002 they complete a table of depreciation and identify the exponential function involved 2 In problem 3 students access the spreadsheet link and manipulate the data to see the effect on the exponential graph and account balance Be sure students change the initial amount and the interest rate in the area above the graph only Students then create their own exponential scenario and determine the account balance in 10 years 3 Finally the students use the spreadsheet or graphing calculator to discover the initial investment required to enjoy 1000000 by age 70 Since they are required to use current interest rates they may access the moneyrates link or use local bank brochures Even if you plan to use the web site to obtain the rates students will enjoy comparing rates stated by local banks Resources Additional information on exponential functions may be found in the following textbooks 2004 Glencoe Mathematics Algebra I New York McGrawHill Chapter 10 2004 Prentice Hall Mathematics Algebra 1 NJ Pearson Education Inc Chapter 8 WORKSHEET ACTIVITY 1 Simple Interest Simple Interest is computed by using the formula I prt in which the following is true I the interest per time period p the principal or initial amount of money r the rate per time period t the time period Example 1 Suppose you place 500 in a saVings account that pays 3 simple interest annually If you make no deposits or withdrawals how much money would you have in the account at the end of 2 years 1st Year I 500031 15 Account Balance 500 15 515 2nd Year I 500031 15 Account Balance 515 15 530 How much will be in the account at the end of 3 years 545 What type of growth does the account balance exemplify Why Explain your answers in complete sentences This is linear growth because each year accumulates the same amount of interest Compound Interest Simple interest is computed on the original principal only Compound interest however pays interest on the principal plus any preViously earned interest Example 2 Use the same situation as Example 1 but this time the interest is compounded annually How much money is in the account at the end of 2 years 1st Year I 500031 15 Account Balance 500 1500 515 2nd Year I 515031 1545 Account Balance 515 1545 53045 How much will be in the account at the end of 3 years 54636 rounded to the nearest cent What type of growth does the account balance exemplify Why Explain your answers in complete sentences Students should recognize that this is NOT linear growth they may assume it is quadratic growth Lead them to consider a new type of growth Group Activity You deposit 1 in an account that pays a whopping 15 interest compounded annually You make no deposits or withdrawals Complete the following table to determine the account balance at the end of 10 years Round to the nearest cent Use Monopoly money and pennies to visually represent the account balance at the end of each year 1 Look at the 10 stacks of money Does this account show linear growth quadratic growth Explain your answers in complete sentences Again students should recognize it is NOT linear growth and may not be quadratic 2 Derive a formula for the above situation that would eliminate the need to first compute the interest and then add it back in each time In other words nd a formula in which you change the value for time to get the account balance Think about how you compute year 2 based upon year 1 Show your work below Let At the account balance at the end of year t Then AO principal 100 AO 1 Al A01 15 1115 A2 A11 15 1115115 11152 A3 A21 15 11152115 11153 A4 A31 15 11153115 11154 AlO A911511159115111510 Thus At A01 1 rf where A0 the initial amount or principal WORKSHEET ACTIVITY 2 The mnnetary gmth mndels are pannfa e1ass nrfunehnnsknnwn as expmrmu39al functinmx Thebasx Expnnenhal grmmh fm mulaxs gvm by A0 Anl r Atxsthe t be subjecttn the rate m neehnal rnrrn r m 10 years by ea1eu1anng the fullnwmg A1011 15 11 15 4 05 rnunded tn the nearest eent 1 NW graph y 11 15Xnn ynur graphrng ea1eu1atnr Use Znnm Standard and then znnrn ut nne tarne sketeh the graph belnw and desmbe rts appearanee m enmplete sentenees 2 Graphy 1000 0annynurgaphngea1eu1atnr Ynumayneentn adjustme WAnan a Inenhfyeaeh nthe fnllnwmg xmualmvestment 100 rate tarne x b Use the traee feature nfynur calcmatnrtn estarnate tn the nearest dullarthe aeenunt ha1anee meears apprnx 015070160 e ancalculatethebalancem 2 years rnunnentn thenearest eent 159 32 3 antnmnhue that enst 20000 when new mm a t t 22 b an graph the fmmula nn a graphrng ea1eu1atnr and sketeh rts graph belnw Cnrnp are rts appearaneetn the graphrn 1 abnve use enmplete sentenees 4a enponentra1 growth or exponenaa1 deeay depreeratron Jusufy your answer wrth a sketeh y1ooo 5y fx200104X y150075x fx3x decal rowth decal rowth Graphs depend on the wmdow used Look at the quantrty rnsrde the parentheses quantities greater than 1 growth and quantrtres less than 1 deeay 5 Desenbe a srtuauon that eould be modeled by eaeh funetron m nurnber4 Answers wru vary WORKSHEET ACTIVITY 3 Use the following web site to access the online version of Kelley Blue Book httpwwwkbbcom Find the category quotUsed Car Valuesquot and click underneath on quotRetail and Private Party Valuesquot 1 Find the 2003 and 2002 suggested retail value on a car of your choice Estimate the rate of depreciation to the nearest whole percent by using the following formula for percent of change Show your work below percent decrease 2002 value 2003 value 2002 value Remind students of the general formula to find percent of change percent increasedecrease amount of changeinitial amount Automobile Make Automobile Model Depreciation 2 Complete the table below by using an exponential formula for depreciation Round your answers to the nearest dollar VALUE IN Write the specific exponential function you used to complete the table 3 Use the spreadsheet provided to manipulate the data and answer the following questions a To the nearest dollar what is the account balance at the end of 20 years if you change the rate to 10 673 Describe the change in the graph when you changed the rate The graph rises more steeply b Change the initial amount to 500 and leave the rate at 10 To the nearest dollar what is the account balance at the end of 20 years 3364 Describe the change in the graph when you changed the initial amount The graph begins rising sooner c Is it better to triple the initial investment or triple the interest rate Explain It is better to triple the interest rate which is affected by the exponent d How can you show exponential decay depreciation of 15 on the spreadsheet Use a negative sign in front of the rate e Create a scenario that you think models exponential growth or decay other than the ones we have already used in this lesson Describe the scenario giving initial amount and rate Use the spreadsheet to determine the amount in 10 years Answers will vary 4 Now that you know how money grows with compound interest it is time for you to put your knowledge to good use You would like to retire at age 70 with 1000000 in your savings account You have to abide by current interest rates and assume they will remain relatively stable until you retire Use bank brochures or access httpwwwmonevmt htm to find a current compound interest rate The rate is If you start today the number of years your money will grow until your 70th birthday is years Estimate the initial amount you need to place in the bank today to become a millionaire by age 70 Show your work below Answers will vary Comments

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