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by: Evert Christiansen


Evert Christiansen
Texas A&M
GPA 3.92


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Class Notes
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This 15 page Class Notes was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Class Notes belongs to MATH 141 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/226046/math-141-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15
Math 141 Business Math I FALL 2006 Section 520 Blocker 158 Tuesday and Thursday 355510 Week 11 8183 part 2 1 Review meanaverage New median and mode Example 11 Professor Xavier s calculus class had 10 students who had AP calculus and 10 students who had never seen any calculus before The AP students had a mean of 85 and the others had a mean of 70 What is the class mean One year later Professor Xavier s calculus class had 10 students who had AP calculus and 20 students who had never seen any calculus before Each group s scores increased by 5 points Now the AP students had a mean grade of 90 and the others had a mean of 75 What is the new class mean To nd the median of a list of numbers arrange them in increasing order and pick the middle number If there is no middle number that is when the list has an even number of items take the average of the two middle numbers On the calculator it7s LIST MATH 4 Median To nd the mode of a list of numbers pick the number or numbers that occur the most often When all numbers occur an equal number of times we say that there is no mode Example 12 What are the mean median and mode for the following list 36363 9 What are the mean median and mode for the following list 361463 9 What are the mean median and mode for the following list 36141007 9 2 Expected Value Variance Standard Deviation Example 21 When trying to print the following histogram the computer didn t include the last column What must it be What is the expected value of this distribution What type of random variable is X Sometimes we need a way of determining how far from the expected value our data is We can do this with the variance and standard deviation The variance of X is found in the following way Let X be a random variable that takes on values x1 x2 x3 with associated probabilities p17p27p3 pm If the expected value is EX u then the variance is VarX p1x1 7 u2 p2x2 7 u2 7 u2 The standard deviation is written a and is de ned as the square root of the variance 0 VarX On calculator you can nd these by putting the X7s into L1 and the probabilities into L2 use STAT7 EDIT for this Then do STAT7 CALC7 l Var Stats L17 L2 This will give you a list of information a is the mean we is the standard deviation Med is the median Example 22 What are the variance and standard deviation for the above example A spiffy way to relate this to nding probabilities is via Chebychevls Inequality Let X be a random variable with expected value M and standard deviation 0 Then the probability that a randomly chosen outcome of the experiment lies between M 7 ha and M ho is at least 1 7 MW We can write this 1 PM7ho X Mko217 Example 23 The distribution of heights of American women aged 18 to 24 has a mean of 655 inches and standard deviation of 25 inches How large must the probability that a randomly chosen woman is between 605 and 705 inches be If we measured the heights X with in nite precision what kind of random variable would X be Note It turns out that heights are distributed in a very nice wayi they follow a normal distribution777 that will allow us to make a better approxirnation later We7ll learn more about these in a couple of weeks Example 24 Raisiny Goodness Cereal is calculating the amount of raisins in their cereal They found that the amount of raisins per bop has a mean weight of 5 oz and a standard deviation of 04 oz For what values of W can the manufacturer of the machinery claim that the amount of raisins has a weight between 5W and 5 W oz with probability at least 9375 A fair game is a game where the expected value of the net winnings is zero Example 25 From an old eparn A game of chance is played where the player can draw and keep two bills from a bag containing ten 100 bills and two 2000 bills How much should be charged to play this game for it to be a fair game Round your answer to the nearest penny Example 26 Suppose we have the following probability distribution z PX a 0 1 2 5 4 1 6 05 8 25 Is 02408 a uniform sample space What is PX gt 3 What isP2 Xlt6 Find M and o Example 27 quotFour in ve voters who approve of Bush s performance were supporting Re publicans in House races while an equal number of those who disapprove of his performance were supporting Democrats But those who disapprove of the way the president has handled his job outnumber those who approve by about three to two 7 Washington Post on line version November 8 2006 Assume for simplicity that voters are supporting either Democrats or Republicans What is the probability that a voter approves of Bush s performance What is the probability that a voter supported a Republican in House races Suppose a voter supports a Democrat in the house race What is the probability that he or she approved of Bush s performance Example 28 Find a set of4 numbers whose mean is 3 median is 2 and mode is Z Math 141 Business Math I FALL 2006 Section 520 Blocker 158 Tuesday and Thursday 355510 Week 5 3133 51 notes In this chapter were going to discuss o How to graph inequalities that involve lines o How to graph a region of points that satisfy multiple inequalities o How to set up a collection of inequalities from a word problem and o How to use the graph of those inequalities to help us solve the word problem 1 Graphing linear inequalities 2 variables When you were younger you learned how to graph things like z 2 3 on a number line First you would nd the point z 3 and put a dot there Then you7d color in the part of the line to the right of 3 If instead you wanted to graph z gt 3 you7d draw a little circle at 3 Then you7d color in the part of the line to the right of 3 Graphing linear inequalities is similar to this Suppose you wanted to graph z y 2 3 First you graph the line z y 3 Then you pick a point above the line Does that point satisfy z y 2 3 If yes color the region above the line If not color the region below the line If instead you wanted to graph z y gt 3 you draw a dotted line instead of a solid one Then proceed as before We can do this for more than one inequality at once We graph each inequality on the same graph7 and then take the intersection of the spaces Example 11 Sketch the solution set for the system z 2 71 2x73y S 2 Example 12 Sketch the solution set for the system l V l H 2x73y xy Vl ow Example 13 Sketch the solution set for the system x S 71 y S 0 zy gt 0 De nition 14 A solution set for a system of linear inequalities is bounded if you can enclose it by a circle Otherwise it is unbounded 2 Setting upsolving linear inequality word problems De nition 21 A linear programming problem consists of a linear objective function to be maccimized or minimized subject to certain constraints in the form of linear equations or inequalities Often the objective function will be cost or pro t The constraints are the inequalities that we graph Let7s set one up Example 22 It takes Mike 10 minutes to do an oil change and 20 minutes to do a tire change He s working for 60 minutes today He only has enough tires in stock to change them on two cars If he can earn 10 per oil change and 40 per tire change how many of each should he do to maccimize pro ts Step 1 Make a table of information Step 2 Determine the equation to maximize or minimize Step 3 Set up the system of linear inequalities How can we solve a problem like this well need to nd the set of possible solutions the feasible solutions and pick the best one the optimal solution Then we use the method of corners Step 4 Graph the inequalities This is our feasible set We can look at an isopro t line line where all points on it yield the same pro t If we look at all of these7 we7ll nd that our maximum in the feasible set occurs at a corner point This motivates the next step In general7 we will only need to look at the corners Step 5 Find the corner points vertices Step 6 Calculate the objective function at these points Step 7 Pick the maximum or minimum This will always work when our set is bounded Note If the set is unbounded there might not be a solution Note If you cant create a feasible set7 then the problem is infeasible also called incon sistent For example7 if I want to spend at least 8 hours sleeping tomorrow and at least 20 hours working tomorrow we have Example 23 A bakery makes bread arid cookies It takes them 12 miriutes to bake a batch of cookies arid 24 miriutes to bake a batch of bread It takes them 20 miriutes to decorate arid package a batch of cookies but orily 5 miriutes to decorate arid package a batch of bread They cart sperid at most 120 miriutes usirig the overt each morning They cart sperid at most 60 miriutes decoratirig arid packagirig They earri 20 per batch of cookies arid 30 per batch of bread How much or each should they bake iri order to maccimize the pro t Step 1 Make a table of information Step 2 Determine the equation to maximize or minimize Step 3 Set up the system of linear inequalities Step 4 Graph the inequalities Step 5 Find the corner points vertices Step 6 Calculate the objective function at these points Step 7 Pick the maximum or minimum Example 24 I m tryirig to learn Polish usirig a workbook arid some cassette tapes Each workbook page takes me 12 art hour to do arid each cassette takes 1 hour to listeri to I have 50 workbook pages arid 4 cassette tapes I cart iricrease my uocabulary by 5 words for each workbook page I do arid by 12 words for each cassette I listeri to If I cart sperid 10 hours tryirig to learn Polish this week what should I do to maccimize my vocabulary We have leftover resources when there are items that we do not use in the optimal solution What are the leftover resources here Mazcza Dzosc Apul 22 2005 l Streaming Video As An Instructional Tool Week in Reviews Week in Reviews were previously offered one night each week for two hours With the week in reviews now available online for MATH 141 150 151 152 166 171 172 and 365 the students are able to review highlights of important topics 24hrday and 7daysweek They can spend just a few minutes to see problems 2 10 and 20 for example or review the entire set of problems posted In addition to weekly reviews online reviews for major exams gives the student an excellent source of problems to review as they prepare for the exam The Night Before Drills are still provided where students can show up and watch numerous problems being explained but students are very appreciative that reviews are also available online httpwwwmathtamueduteachinghelpsessionstreaminghtml httpwwwmathtamueduJaniceEpstein141reviewhtml httpwwwmathtamuedukahlig166WlRpagehtml httpwwwmathtamueduNdmanuelmathl71wirindexhtm httpwwwmathtamueduNdmanuelmathl72wirindexhtm httpwwwmathtamueduNaustinwirmath151html httpwwwmathtamueduNaustinwirmath152html httpwwwmathtamuedumarciadrost365wirspr05html httpwwwmathtamuedumarciadrost150wirnew02html The web addresses listed above give you a quick link to view some of the reviews offered to the students here at Texas AampM University If you are interested in producing a Video for your students I have included a list of steps below to help you get started Note These steps are the directions to follow when producing Streaming Video on the Fujitsu laptop Type the list of problems you wish to review and expand the type 18 pt is suggested Add extra space we suggest 2 or 3 problems per page On LaTex distill the le and move it to publichtml and chmod to change permissions Then on the laptop wireless connection required turn on computer and login Open Mozilla open lepdf and save Close Mozilla Mazcia Dzosc Apzi 22 2005 2 Open Windows Journal and click on le import My documents Marcia insert your name here lepdf click on The le will open in Windows Journal SAVE This will automatically save the le as a jnt le Open Camtasia Studio Click on little green ball Click on Camtasia Recorder small Blue Box open on the right Underneath the red ball is a microphone symbol Make sure the microphone button is depressed It should have a box frame around it Slide the blue box to the far right part of your screen with the 0 Start pause I Stop and Delete in the Viewing window On the background Camtasia Studio click on B to shrink but keep open Plug in the microphone and the order is critical On the bottom left is the symbol for opening the keyboard Click on the keyboard symbol and in the upper right hand corner is V 77speech tools Click on speech tools Click on microphone adjustment Read and follow the directions record 77This papaya tastes perfect next Close keyboard Rotate the lid of the laptop and lay at Button 3 lower right hand side of the laptop will rotate the laptop 1800 Slide Camtasia recorder rectangle to the far right Note When you click once on the red ball green corners appear showing the portion of the screen about to be Visable to the students With the stylus move corners to the correct position over the Windows Journal le When ready to record click on the red ball The green corners will ash when recording Mazcia Dzosc Apzi 22 2005 To pause click on This is useful when changing the pen color7 erasing7 or scrolling a long way to the next problem Click on the red ball to start taping again7 after each pause Suggestion Tape only about 5 problems at a time7 or 10 15 minutes then stop When you click on stop I type in a le name reopen the keyboard by clicking on the icon on the lower left Close the keyboard Save the le Suggestions I chose 150wir8ptA which is 10 15 min long 150wir8ptB also 10 15 min long 150wir8ptC 15 min long Produce Video later when asked for a le name I will use 150wir8A which will include all parts that can t in under 55min Reopen Camtasia Studio that we earlier minimized Each le segment appears in the Clip Bin Drag and drop each segment in order to the time line at the bottom of the screen There is a small open space on the right of the time line for you to drop each segment after the rst segment Fifty ve minutes appears to be the upper limit Note After dropping the rst clip7 the time line will look like this l X with a small space on the right for the next clip7 AND the total time will show If you closed Camtasia Studio earlier follow these steps instead Open Camtasia Studio lmport import Video 150wir8ptA import Video 150wir8ptB import Video 150wir8ptC Click on Produce Click on produce Video as These will automatically appear 0 Macromedia Flash SWF movie le 16 Bit High Color 5 Frame Rate Mazcia Dzosc Apzi 22 2005 4 Encode Audio MP3 11025kHZ7 Mono7 16k Bitssec Advanced Details High Color 16 Bit 5 800x600 size MP3 0 largest Video size 800x600 next Destination automatic Documents and Settings MyDocumentsMarcz a File Name swf change the pre x before swf to new name7 ie 150wir8A in this case After taping all parts of Video7 and producing all segments7 close Camtasia Open Documents click on Camtasia Studio Open My Computer semer1V deoRAID Click on the le name of the course you are putting the streaming Video in Go back to Camtasia Studio7 click7 and highlight the FOUR les produced under each le name Drag and drop them into VideoRAlD Go to your web page and update the links to allow access You7ve now produced your rst Video Congratulations


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