Intro to Mathematical Modeling
Intro to Mathematical Modeling MATH 1101
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This 15 page Class Notes was uploaded by Mrs. Kara Jacobs on Monday October 12, 2015. The Class Notes belongs to MATH 1101 at Georgia College & State University taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/221930/math-1101-georgia-college-state-university in Mathematics (M) at Georgia College & State University.
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Date Created: 10/12/15
Mat J quot a Math 1101 Section 12 notes Functions Given by Tables Notice in the table below that with each year there is a cor r 139 US r r 39 quot This is 39J J a functional relationship The year in this case would be considered the input and the population would be considered the out ut I Year d 1950 1960 1970 1980 1990 2000 Population in 15187 17932 20330 22654 24871 28142 Millions Nd 1 Express in functional notation the population in 1960 2 What does N1990 mean Averaging When dealing with tables one useful mathematical tool is averaging two values to estimate a value inbetween 3 Approximately what was the population in 1975 Average rates of change Another useful mathematical tool is being able to calculate an average rate of change For example if we want to calculate the population of the Us in 1972 from the information in the table above we could use an average rate of change Change in output over the interval Average rate of change over an interval units will be output units per input unit Change in input over the interval 4 a Find the average rate of change on the population of the US in the 1970 s b Use the average rate of change in part a to estimate the population in 1972 MATH 1101 Math Modeling Section 23 notes 23 Linear Equations A linear equation is the simplest type of equation Linear equations do not have exponents square roots variables in the denominator etc For example Examples of linear equations Examples of nonlinear equations y 5 y 5x y J 2x 3 9 y x2 9x6l93x 56342 y10x 5 y1e Solve the following linear equation for x 5x 9 16 2x Example 1 Sam uses long distance service from ATampT that costs 2 per month plus 7 cents per minute His brother Bob uses long distance service from MCI that costs only 3 cents per minute but 5 a month Which plan is better a Write a function S of the cost in cents of calling x minutes with Sam s plan b Write a function B of the cost in cents of calling x minutes with Bob s plan c Write both functions S and B in units of dollars 1 If I talk 0 minutes a month with Sam s plan I still pay 7 a month If I talk 0 minutes a month with Bob s plan I still pay 7 a month 6 What amount of minutes wouldl pay the same amount of money for both MATH 1101 Math Modeling Section 32 notes The formula for a linear function is y x or y x Example 1 A long distance company charges its customers 5 a month and 7 cents per minute used for a certain long distance plan a Write a linear function for the long distance plan De ne your input and output variables b What is the slope include units i 7 Interpret the slope in practical terms c Explain the initial value or yint in practical terms 1 Graph the equation for up to 300 long distance minutes used Example 2 Suppose that you bought a car in 2002 for 18000 On average the car s value decreases at a rate of 2800 per year a Write a linear equation that models this situation De ne your input and output variables b If you let the input variable be de ned as x the year what would be your yint c Graph the above equation Over what years do you think the equation is good for Examgle 3 udent39s chance nf fmlmg the cnuxse as snan tn the graph 3 Esumate andmlexprelthe slaps nfthe graph Chznw nt tannn Luamm mu ExamgleS a 1 Interpretthe slope ofthxs equation tn context b C F 160 Interpretthe slope ofthxs equation tn context Math Modeling Math 1101 Section 13 notes Function given by graphs In this sect on we will learn some tems to descibe various characteristics of graphs y A B C D E x Decreasino quot 39 and 39 l 39 39 from 2 D 39 from 3 Absolute min at 4 Absolute max at Concavitv Concave up Concave down JKU Back to the graph at top of page 5 Concave down from 6 Concave up from In ection point Point above is an point An in ection point is where a graph changes concavity or in other words changes from concave up to concave down or concave down to concave up MATH 1101 Math Modeling 33 Modeling Data with Linear Functions Test the following two data sets to see whether they are linear Section 33 notes x 1 3 5 7 x 1 3 5 7 y 11 16 23 32 l y l 11 16 21 26 Change in From 1 to 3 From 3 to 5 change in From 1 to 3 From 3 to 5 x x Change in Change in y y Linear Linear For the data above that is linear create a linear model Example 1 As a simple example consider a newspaper delivery team that makes w weekly deliveries of newspapers and devotes their Saturday mornings to selling new subscriptions Suppose that 5 new customers are added each week Observe the following table showing the number of customers over a 7 week period Weeks 0 1 2 3 4 5 6 7 Number 80 85 90 95 100 105 110 115 of Customers Calculate the differences 7 7 7 7 SecondiFirst Also observe a scatter plot of the above data How to get a scatter plot of the above data Go to STAT 7 1Edit In L1 enter the input values weeks In L2 enter the output values the number of customers Go to Y and press enter on Plotl to make it highlighted HeP Nf News39mner cusmmeus 5 Press ZOOM and number 9 or go down to ZoomStat u 1 2 3 a 5 a 7 a a m 11 12 13 a 15 6 You will see a scatter plot of the data 22 m slope Interpret Initial value Interpret Let w the number of weeks and Cw the number of newspaper customers after w weeks Write a linear formula for the data above Objective To see that we can model the above data in the table as a linear function Why can we model the above data with a linear function Is the above data nearly linear or perfectly linear GRAPH FUNCTION ON CALCULATOR Mat J quot a Math 1101 Section 11 notes A Function Of One Variable Example 1 Suppose Joe gets paid 8 an hour Create a formula for this situation giving the amount A Joe gets paid as a function of the number of hours x that he works a week Input and Output variables The is x To defme we write a brief description like x is the number of hours Joe works in a particular week The is A or Ax To defme we write a brief description like A or Ax is the amount Joe gets paid in a particular week A Model then would include the following 3 items 1 2 3 Therefore a complete model for the above context would be A Function Of One Variable Example 2 Joe really wants to know how much money he will take home a week after taxes Assume that taxes taken out add up to 25 of Joe s pay Modify the above equation to approximate Joe s take home pay T as a function of the number of hours x that he works a week Write out a COMPLETE MODEL which means define your input and output variables Functional Notation Fx is functional notation It represents the output value F if the input is x 1 Calculate and interpret A30 2 Calculate and interpret T30 3 Express in functional notation Joe s take home pay if he works for 10 hours one week A Function of Several Variables Example 1 Express the Total Revenue from a play as a function of the number of tickets sold if a child costs 5 Adult costs 10 and senior citizens cost 8 First de ne all variables What are the variables So we need to de ne all four of these as variables Let c a S Then Tc a s l Interpret T209844 2 Calculate T209844 3 Express in functional notation the revenue from a play where 52 children 70 adults and zero senior citizens attend MATH 1101 Math Modeling Section 35 notes Systems of Equations 3x 2 y 14 5x 2 y 18 In a system of equations we treat the equations as a group The solution to a system of linear equations is a point x y that satis es equations in the system The following is a system of equations Verify that the point 4 l solves the above system Solve each equation for y and graph the equations so that you see the intersection Find the intersection and verify that it is 41 Example 1 You re mixing blue paint with yellow paint to get a total of 10 gallons of the mixture You want to use 3 times as much yellow paint as blue paint to get a certain light green color How many gallons of each should you use a Create 2 equations from the context above de ning your input and output variables b Solve the system of equations Explain what each value means in practical terms Example 2 A stack contains 400 worth of paper money consisting of twentydollar bills and ftydollar bills There are 11 bills altogether How many twentydollar bills and how many ftydollar bills are in the stack a Create 2 equations from the context above de ning your input and output variables b Solve the system of equations Explain what each value means in practical terms MATH 1101 Math Modeling Section 25 notes 25 Optimization Example 1 Imagine a cannon elevated at an angle of 45 degrees Mathematically we can model the ight of the cannonball with the following formula if we ignore wind resistance 2 x yxg m where x is the horizontal distance in feet the cannonball has traveled y is the height in feet of the cannonball g is the acceleration due to gravity in feet per second per second and m is the muzzle velocity in feet per second The acceleration due to gravity near the surface of the Earth is about 32 feet per second per second Let s assume the cannonball is red with a muzzle velocity of 250 feet per second Therefore we have a When the cannonball is 0 feet downrange horizontal distance the height of the cannon ball is7 This just means we graphically start the cannonball at the origin b Graph the entire ight of the cannonball Xmin 7 Xmax777 Ymin77 Ymax777 c Find EXACTLY the max height of the cannonball to 3 decimal places by doing the following To nd a maximum or minimum point on a graph 1 Press 2nd Trace for Calc 2 Choose 3 or 4 for either minimum or maximum 3 Left bound cursor must be anywhere to the left of the max or min then press enter 4 Right bound cursor must be anywhere to the right of the max or min then press enter 5 Guess just hit enter 6 The max or min point will be displayed 1 How far downrange to 3 decimal places was the cannonball at its max height 77 77 e What was the total distance the cannonball traveled correct to 3 decimal places 7 7 7 f When was the cannonball exactly 300ft off the ground round to 3 decimal places 77 7 7 MATH 1101 Math Modeling Section 21 notes 21 Getting Tables from Formulas The goal in this section is to learn to use the table feature on the calculator This will assist us in understanding various information about a function Example 1 As a simple example we will example the following model N x 100 4x x the number of minutes since class started Nx the number of MampM39s left x minutes into class It is easy to see that there was MampM39s at the beginning of class Then each minute there was MampM s less than the previous minute Therefore ll in the following table without the calculator x 0 5 I10 I15 IMO I We can have the calculator give us tables such as the one above automatically To do this follow the following directions Put this function into your y list 2 Then go to 2 window for TBLSET table setup and set up your table as follows TABLE SETUP TblStart Value you want the table to start at 0 in this case ATbl 1510 increment ofvalues in the table 5 in this case lndpnt Auto Ask Auto Ask 3 Go to 2nd graph and view the table Verify the values above One other way that we will use the table is by doing the following Go to 2 Window for tblsit 39 TABLE SETUP TblStart does not matter ATbl does not matter lndpnt Auto Ask Auto Ask Now go to 2nd graph If your table is NOT empty use the delete key to delete any values in it In this mode we can enter any value into the table and therefore evaluate the function N x 100 4x at any value of x Put the following values in for x 3 125 I25 I 302 x N00 Example 2 Renting Canoes A small business has 20 canoes that it rents for oat trips down the Chattahoochee river The pricing structure offers a discount for group rentals One canoe rents for 35 two rent for 34 each three rent for 33 each and in general the group rate per canoe is found by taking 1 off the base of 35 for each extra canoe rented a How much money is taken in if 4 canoes are rented to a group b Write a formula that gives the price charged for each canoe if n canoes are rented to a group c Find a formula Rn that shows how much money is taken in from renting n canoes to a group d Objective We want to make a table of values from the formula above in order to be able to discuss how the Revenue R changes as the amount of canoes rented to a group changes Set up the table on the calculator to see the revenue starting from 0 canoes rented and increasing in increments of 5 canoes e Using the table how large a rental to a single group will bring the most income hint use the table f Does this function have a limiting value MATH 1101 Math Modeling Chapter P notes u F quot roundimr minus sign 2 and 71 e and TE are two special mathematical numbers that we will use occasionally Calculate the following Round ONLY your nal answer to 2 decimal places 1 You can nd 6 by pushing 2quotd and then the LN button Do this and put 601 and write down what 6 is to 2 decimal places 2 You can nd TE by pushing 2quotd and then the A button Do this and write down what TC is to 2 decimal places 1675e2 5 37 3J05 4 23000 X 005 X 100560 7239 1 60 60 Simple percentage increase or decrease 1 A computer that sells for 850 is being discounted by 12 What is the discounted price 2 A shirt has a regular price of 50 and is discounted by 20 The discounted price is then written on the tag It doesn t sell so the store puts it on a rack that states 40 off lowest marked price What is the price now and how much was the discount off the regular price Compound Interest The formula F P1 r can be used to nd the future value F if the present value P is invested for time t at an interest rate r as a decimal If time is in months the interest rate has to be a monthly interest rate If time is in years then the interest rate has to be an annual rate 1 If a savings account has 10000 in it how much would you earn after 12 months if the interest rate was 3APR how about 7APR 2 Sam has 7000 on a credit card with a 192APR very high If no payments were made for a year how much would Sam owe 3 How much would it take the above 10000 to double at 3 or 7 APR We could use the formula above to solve this problem but there is a simple rule called The Rule of 72 that gives a rough estimate as to how long an investment will take to double at a certain APR The Rule of 72 72r where r is the APR as an integer not a decimal Use this rule to roughly estimate how long it would take 10000 to double at 3 and 7 APR MATH 1101 Math Modeling Section 22 notes Example 1 Last class we observed the following context N x 100 4x x the number of minutes since class started Nx the number of MampM s left x minutes into class Our goal in this section is to graph functions practically Practically speaking what values of x would make sense in the context above From if to Graph the function below for x values in the interval above Before graphing ll in the following table to assist you in graphing the function accurately X 0 5 10 15 20 25 Nx N x ymax ymin xmin xmax Now let s do the same thing we did above the table and graph on the calculator First use the table feature on the calculator to get a table automatically that looks like the one above Next press the Window button and set the window The table helps us set our xmax xmin ymax and ymin intelligently Xmin if Xmaxiii Ymin if Ymax 7 Find exact output values on a graph Find the exact value of N4 on the graph by pressing trace then 4 and enter 7 if Example 2 Recall the 2 d example in the 21 notes A small business has 20 canoes that it rents for oat trips down the Chattahoochee river The pricing structure offers a discount for group rentals One canoe rents for 35 two rent for 34 each three rent for 33 each and in general the group rate per canoe is found by taking 1 off the base of 35 for each extra canoe rented We found a formula Rn that shows how much money is taken in from renting n canoes to a group Rn 36nn dollars made from renting n canoes From the context above it is clear that Xmin and Xmax Setup a table for the above x values to determine the Ymin and Ymax Set the window and graph the equation By tracing what is the largest Rn value you can get from this pricing sturcture Interpret this v alue
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