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Calc Bus & Soc Sci II MATH 242
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MATH 242 LECTURE 13 l SYSTEMS OF LINEAR EQUATIONS In our studies of differential and integral calculus and in much of the mathematics you studied through highschool there was essentially only one variable which was allowed to vary freely Many problems both theoretical and applied are not welladdressed by having only one quantity which can vary freely But instead of moving directly to multivariable calculus we are going to rst cover more basic topics in algebra and geometry When you rst learned algebra the equations you focussed on most were linear equations like 31 8 which you then learned to solve them by isolating the variable We will similarly choose to look at linear algebra rst and nd that similar strategies are useful De nition 1 A linear equation in two variables I and y is one of the form az by c where a b and c are given numbers A linear equation in many variables is of the form a111 a212 39 39 39 an1n 6 De nition 2 A system of linear equations in two variables is simply a collection of such linear equations A solution to a system of linear equations is a pair of numbers which satis es all equations in the system Example 3 The collection 721 y l I y 4 is a system of linear equations in two variables The pair of numbers zy 13 is a solution as can be veri ed by substitution Solutions to a system of linear equations in two variables are places where the lines of points which satisfy those equations intersect as can be illustrated with the system in the previous example This observation leads to the rst of three methods we develop for solving these systems 11 Methods for solving linear systems There are three basic methods we discuss illustrating each one with the system of Examp e Graphical Method This method is more helpful in getting approximate solutions and in verifying that answers obtained by more precise means are reasonable We simply graph the solutions of all of the linear equations involved and estimate their point of intersection Substitution We use one equation solve for one variable as a function of the other and then substitute this function in for that variable in the other equations In a system of two variables and two equations the resulting equation will have only one variable This method is very effective in solving small systems Addition and subtraction of equations This is perhaps the most interesting method We use the fact that one can always multiply add subtract and divide but not by 0 equalities and end up with equalities So we add and subtract multiples of equations so that the resulting equations are simpler Example 4 Solve the following system of linear equations using the three methods 21y l iz2y 4 1 2 MATH 242 LECTURE 13 Use a combination of substitution and adding equations to solve 172 21yiz 712y There are many problems both toy as occur on IQ tests and realworld in which systems of linear equations naturally occur Example 5 Last year I was three times as old as my sister and in four years I will be twice her age How old are we Example 6 Almonds cost 3 per pound and Cashews 8 These two types of nuts make NuttyfunTM How many of each should should be used to produce 100 pounds at 5 per pound 100 pounds at 6 per pound 120 pounds at 4 per pound 2 OUTCOMES OF LINEAR SYSTEMS AND LARGER SYSTEMS OF EQUATIONS When we talked about linear systems in two variables it seemed like they always had a single solution This isnlt always the case Example 7 Solve the system of equations 1 2y 4 31 6y 8 What happens if in the last equation 8 is replaced by 9 These results can be explained by geometry as we illustrate with this example The set of points which satis es one linear equation in two variables is a line and an intersection point corresponds to a solution If there are no intersection points or many that corresponds to no solutions or many We can use similar techniques to understand larger systems of equations Example 8 Analyze the system of equations 12y3 21735 31y8 and the system one obtains by replacing 8 in the last equation by 9 3 THE BASICS OF MATRICES Matrices are a useful piece of notation whenever arrays of numbers occur in a problem especially linear problems De nition 9 A matrix is a collection of numbers arranged indexed in a rectangular array Matrices are usually represented within brackets like 12 To specify a number we specify its row and column For example in this matrix the number 5 is in the rst row second column or in the 12 position We call a matrix an n by m matrix if it has n rows and m columns The example above is a two by two matrix We may use matrices to represent and ef ciently solve systems of equations by putting both the coef cients of the system and the values of the equations in a matrix and mimicking our usual procedure for solving them Example 10 Translate into matrix notation and solve the system I 2y 2 z 3y 5 MATH 2427 LECTURE 10 You had a chance to ponder over the weekend Example 1 Monty Hall problem Monty Hall wants to confuse you a little bit You get to pick a door Then whichever door you pick whether there is a goat behind it or a car Monty opens a di erent door that has a goat behind it He can always do this even if you chose a goat door because there are 2 goats Now Monty gives you the opportunity to switch doors to the other unopened door Should you switch Now let s recall some basic rules governing probability and use them in some other problems Most basic rule The probability of some event occurring is the number of times that event occurs over the sample space divided by the size of the sample spacer Example The probability of rolling a four on one roll of a die is g Not rule The probability of an event not occurring is one minus its probability Example The probability of not rolling a four on one roll of a die is Or rule The probability of one of two events occurring7 when those events can never both happen7 is the sum of their probabilitiesi Example The probability of rolling a one or a four on one roll of a die is 2 And rule The probability of two independent events both occurring is the product of their proba bilitiesi Example The probability of rolling a four on one roll of a die and then not rolling a four on the next is El Example 2 Use ofa condom is 98 e ective at preventing pregnancies What is the chance ofapregnancy occurring while relying on condoms twenty times onehundred times The combination of condoms with oral birth control is 9998 e ective at preventing pregnancies What is the e ectiveness of this double barrierquot protection over twenty and onehundred times ll IN DEPTH WITH FLIPPING COINS The example of ipping coins is as simple as possible but lets us see many of the features which are prominent in probability theory more genera yi Example 3 Flip a coin four times Let X be the outcome 0 What is the sample space 0 What is the probability of each outcome 0 Let Y be the random variable giving the number of heads 0 What is the sample space 0 How many outcomes in the original sample space give each Y o What are the probabilities of each Y in this sample space i Y i 0 Summary of answers to last part above Outcomes l Probability D625 lilo Digression The number of possible ways to get Y heads for N coin ips We now digress into one of the most beautiful areas of elementary mathematics The number of ways one can get Y heads in N coin ips is called a binomial coe cient7 denoted 1 2 MATH 242 LECTURE 10 Example 4 Write down the rst few binomial coe cients in a triangle Pascal s triangle What patterns do you notice 0 What do the rows add up to o What do you notice reading them right to left versus left to right 0 How can you nd the values in a row from the previous row 0 What are other immediate patterns Odds and evens Now for the fun part plot the values of the rows as a histogram where the blocks each have area 2 so that the total area is one What do we see The Bell Curve is approximating the rows of Pascal7s triangle and thus modeling coin ips1 Or conversely the numbers in Pascal7s triangle are telling us about the Bell Curve normal distribution 12 Back to coin ips Example 5 Let Y be the number of heads after 10 successive coin flips The probabilities for Y are Yo1224557891o Probability 001 010 044 117 205 245 205 117 044 010 001 What is the probability of between 5 and 7 heads Example 6 All 80 of us flip coins at once Let Y be the total number of heads 0 We probably won t get exactly 40 heads The probability of that is 089 0 But P35 S Y S 45 1781 and o P30 S Y S 50 1982 Note that as the number of trials gets very large the probability of any one outcome gets very small It starts being less relevant to ask for example what is the probability that you would get exactly 31 heads out of 80 The answer is just too small to matter What works better is asking the chances that the number of heads will be in some range This change of emphasis from individual measurement to ranges is important when we look at continuous variables 21 CONTINUOUS VS DISCRETE RANDOM VARIABLES o If a random variable X takes on a nite number of possible values it is called a discete random variable All of our examples so far have been discrete o If a random variable X takes on a range of possible values it is called a continuous random variab e o The sample space is still the collection of possible values for X o It no longer makes sense to ask what PX a is but rather what is Pa S X S b Example 7 Let X be a number randomly chosen between 0 and 1 Note that it is hard to randomly choose numbers in this fashion 0 What is PX 13 o If we assume all choices between 0 and l are equally likely what is PX 2 15 o What is PX S 13 orX 216 The probabilities end up corresponding to ratios of lengths within these state spaces Probabilities can also correspond to ratios of areas Example 8 If a dartboard is 8 inches in radius and the center circle is inch in radius what is the probability that a dart thrown at random will hit the center square What is the probability that a dart thrown at random will hit in the 20point wedge MATH 242 LECTURE 12 l REVENUE STREAM EXAMPLES The simplest kind of revenue stream calculations are those for which the payments are constant Example 1 Find the present value of 1000 per month paid over ten years assuming a prevailing in ation rate of 4 Last time we also looked at revenue streams involving linear increases or decreases of payments We provide details this time Example 2 If you invest Rt 2000 400t dollars per year in a retirement account earning nine percent per year how much will you have after forty years How much of that is principal and how much is interest Answer The future value is fem 2000400te0390940quotdt Using the fact that fen and f tequot 7 87 we can evaluate this integral or we can just use a numerical integrator The value is 40 40 3 6 200067009t 3 6 764092 7009t E I E 39 7009 7009 7 70092 0 The answer is 2371230 which in current dollars assuming a 4 rate of in ation is 478743 enough to live on for a good number of years The amount of principal deposited is fem 2000 400tdt 400 000 which leaves roughly two million of interest the lions share Note that even the rst 2000 deposited becomes 73200 It is somewhat surprising that revenue streams involving exponential functions can be easier to deal with than linear functions such as 2000 400t Example 3 How much would you save in total you saved Rt 2000e0391 dollars per in the previous problem Once you have the basic formulae down setting up the problems is the most difficult part Example 4 Assume you make 40 err year in your new jobcareer with a typical annual raise of 5 Suppose you choose to set aside 10 of your earnings in a retirement account which earns 5 interest Model these assumptions by continuous streams and calculate what you would save over 10 years as well as its present value What percentage of your earnings would you need to set aside to save up 200K in ten years MATH 242 LECTURE 5 o 1 A little pmctice with ersoolres Example 1 Suyyose we nal yyooe Manges m oomuzy ducnlmcw39wmz 2127mm o v1 ma39 o scandam39 oeyyomoo 08 What some do you 1222ch 72 m we coy 25 o moms Wm ymmm me you in y you some o 60 ANALYleG TWO VARIABLES scanlemon CORRELATIONS AND causes The lnvallol assumpllon lhal oollelallon lmplles oallse ls plobabh among lhe lwo ol lhlee mosl sellous and common ellols ol human leaso eSlephen Jay Gould The Mlsmeasule ol Man So lal we ve been anamlng a olala lol a slngle vanable We now anahze lwo vanables whele lhlngs ale halolel lo lnlelplel As belole we will develop some ol lhe mole baslo lools anol lnlolmal language nlsl lhloughl b ol examples A w l allow us lo be mole pleolse ln oul language Example 2 Consida39r 112 gzyaccmcy m 417 eounmes M mm wanes omega 5 ma39 75 PM women n youes bgmam 53 and ma39 women The zye ayaccmwy oy mm 32 y by country ln lhal oounlly One way lo leplesenl lhls ls by a soawemlot a ploy on lwo axes as abme In lhls plol eaoh polm leplesenls a oounlly The uooolollnale ol lhal polnl leplesenu lhe lemale llle expeolanoy ln lhe oounlly The yooolollnale leplesenls lhe male llle expeolanoy ln lhe oounlly al mm Nolloe lhal lhese vallable seem lelaleol They lall mole ol less along a llne Does lhls mean a longllle xpeolanoy lol men ol Vice velsa7 ol ls ll lusl a comcldence7 y some lhllol lhlng7 Plobabh lhe lasl These vallables We lelaleol anol lhey ale solongh eomloceo we can mole ol less pleollol one llom lhe olhel But nellhel one ls eouseo by lhe olhel 1 1 Scattelplofs De nmon s A seoomplocys o yyoymu my39resmmmm o lwo yummomye youoozes ossoeloceo co 022 5mm 5 c o mmyyouus 2 MATH 242 LECTURE 5 In our previous example the individuals are the 40 countries The two quantitative variables are female life expectancy and male life expectancy Suppose we have a set of data with 6 variables country name life expectancy number of people per television number of people per doctor female life expectancy male life expectancy Which variables could be plotted in a scatterplot Example 4 Consider a scatterplot with people per television on the r azris and people per doctor on the y azris These two variables are also related but much less neatly There are examples North Korea at 90 370 with few TVs and lots of doctors There are others Thailand at 11 4883 with many TVs and very few doctors More noticeable is just that the wealthy countries are clustered in the lower left After some analysis we find that although these two variable are related it seems unlikely that either is a partial cause for the other In this case both are at least partially related to the wealth of the country This is called a lurking variable because it doesn t appear directly in our scatterplot but is responsible for a lot of the variation 12 Explanatory and Response Variables Often when comparing two variables we suspect that the variation in one variable is partly caused by variation in the other We call the variable causing the change the explanatory variable and the variable being changed the response variable Example 5 We compare the of economically disadvanted children in elementary schools in Kalamazoo Michigan with the of students at those schools passing the MEAP Michigan Educational Assesment Program So our individuals are 14 elementary schools in Kalamazoo Our variables are percent of economically disadvantaged students and percent passing the MEAP What is the relationship between these variables Is there an explanatory variable and a response variable 13 Postive association and negative association De nition 6 Two variables are positively associated if larger values in one variable tend to accompany larger values in the other MATH 242 LECTURE 5 3 Two variables are negatively associated smaller values in one variable tend to accompany larger values in the other Example 7 Let s examine our three previous examples to see there are any associations and so whether they seem weak or strong 14 Measuring strength of association By looking at a scatterplot we get a subjective impression of strength of association One objective measure is given by the correlation Suppose we have a data set with n individuals and two variables I and y The correlation measures how close the linear relationship is between I and y De nition 8 The correlation is usually written r and the formula for it is 1 n 12quot 91quot r 7 7H MT gt In this formula T is the mean of the Ii and sac is the standard deviation of the Ii Similarly for the y s Lets not worry about how to compute this value and just look at our three examples 0 We get r 979 in our example comparing menls and women s life expectancy in 40 countries That tells us that the points on the scatterplot are close to a line sloping up 0 We get r 620 in our example comparing people per TV77 with people per doctorl77 This tells us that the points on the scatterplot are roughly approximated by some line sloping up 0 We get r 7 735 in our example comparing percent economically disadvantaged with percent passing MEAP77 This tells us the points on the scatterplot are roughly approximated by a line sloping down How do these values relate to what we said about positive and negative association above Here are some general facts to remember about correlationl o 71 S r S 1 0 When r l the points on the scatterplot lie on a line sloping up which means that the variables are positively correlated When r is near one but below the points are close to being on a line sloping up 0 When r 71 the points on the scatterplot lie on a line sloping down which means that the variables are negatively correlated If r close to 71 the points are close to being on a line sloping own o If r is close to 0 there is a very weak relationship 0 To measure r both variables must be quantitative o r tells you nothing about cause and effect just about whether the two variables are related 0 r is not resilientl It can be strongly effected by outliersl MATH 242 LECTURE 17 1 SOLVING SYSTEMS OF EQUATIONS USING INVERSES OF MATRICES Matrices are collections of numbers which behave in some ways just like numbers themselves We can add subtract and multiply them and there is a zero matrix The similarities only go so far though only square matrices can both be added and multiplied the multiplication is complicated and it depends on the order in which the matrices appear Matrices arise in a very wide range of applications from computer graphics to econometrics to population dynamics to quantum physics As a pleasant digression let s see how they are used in a simpli ed example of computer graphics Example 1 Digressive example You can use a matrix to specify the vertices of a line drawingquot For example the matrix 3 represents a letter C Using matrix multiplication can shrink rotate 3 1 1 1 1 4 4 etc such a shape For example we can multiply by Bl What happens to this line drawing when the 098 702 matrix multiplication is performed Consider also repeated multiplication by 0 2 0 98 Our next step will be to learn how to use matrices to solve linear systems by dividing 7 matrices Remember that for numbers to solve the equation 3x 2 8 we rst need to subtract to get 3x 6 and then we divide both sides by 3 the key stepl to get x 2 11 Identity matrices Division by x is just multiplication by i so rst we must understand what 1 1 1 1s for matr1ces Your rst guess m1ght be the matrix 1 1 But how would we know this 1s r1ght The key fact about 1 is that 1 y y for any y We can check whether satis es this key fact E So it doesn7tl Fortunately there is a matrix which satis es this key property De nition 2 The n by n identity matrix is the square matrix with 1 s for the 11 22 nn entries 1 0 0 andO s everywhere else For example the 3 by 3 identity matrix is 0 1 0 Identity matrices are often 0 0 1 denoted by the letter I which then does not specify the size of the matrix Theorem 3 For any identity matrix I and square matrix M of the same size M MI M Example 4 We verify this theorem for the 2 X 2 identity matrix We have already seen the identity matrix when we rst looked at solving systems of equations using matrix notation In fact linear equations expressed in terms of the identity matrix couldn7t be any easier y 5 1 Example 5 Translate the matrix equation 2 MATH 242 LECTURE 17 12 Inverses of matrices De nition 6 The inverse of a square matrix M is one which we call M 1 as opposed to where M M 1 I M 1 a b 7d 7 Theorem 7 The inverse of the 2 X 2 matrix a l is ladch 50 c d 7 adibc adibc Note that the number ad 7 bc which appears everywhere in this formula has its own special name it is the determinant of the matrix If it is zero then an inverse does not exist Example 8 Verify this theorem for the matrices and There are formulae for inverting larger matrices but they are complicatedl Fortunately we do not have to invert matrices by hand Many graphing calculators including the TlSS have a function to invert a matrix 13 Using inverses to solve linear systems Let M be a square matrix X be a column vector of variables and C be a column vector of constants of compatible sizes To solve the equation MX C we rst nd M 1 if it exists and then multiplying both sides by M 1 we get M lMX M IC or IX M IC or X 10 So the steps to solve MX C are 1 Find M 1 2 Compute the product M IC 3 For thoroughness check that your answer works Example 9 Use matrix algebra to solve the system of equations 5x 7y 3 2x By 71 One great advantage to matrix methods is in solving related systems Example 10 Solve the matrix equation 3 g I 5 What we replace 5 by 1 9 By 71 9 5 5 y 6 6 2 7r With these techniques and calculators in hand we can move on to swiftly solving problems with more variables Example 11 The average yield on Alzonds is 5 on Blzonds is 7 and on C bonds is 10 Because of a hedging scheme you must invest twice as much money in A bonds as C bonds Find the amounts to invest for the following desired outcomes 0 25K invested with an annual return of 18K 0 30K invested with an annual return of 22K 0 40K invested with an annual return of 29 K MATH 242 LECTURE 6 Today we go back and more carefully develop the de nite integral as a total or aggregate amount We pay particular attention to how one would program a computer or calculator to approximate it We do so even though it may be calculated in some cases by the Fundamental Theorem because 0 The Fundamental Theorem cannot always be applied Some functions such as e 2 do not have antiderivatives which are easy to describe 0 To better understand why the Fundamental Theorem is true it is important to understand what the de nite integral is on its own separate from antiderivatives 0 In most realworld applications de nite integrals are computed using sums with the help of computers 1 RIEMANN SUMS If we are given an amount by which something changes we may use Riemann sums to approximate the total quantity over that period of time We will look again at the formalism around these sums after looking at a couple of examples Example 1 Suppose a ball thrown up in the air has velocity vt 732t 100 feet per second What is its velocity at 3 31 and 32 seconds Using its velocity at 31 seconds estimate how far it travels between 31 and 32 seconds Using approximations at tenths hundredths and thousandths of a second estimate how far it travels between three and four seconds Example 2 Suppose the marginal cost for producing CD s at afactory is cents for the xth CD Write down sums with four ten and twenty terms which approximates how much costs to make 900 CD s In these examples we see how closely related antiderivates are to these Riemann sums De nition 3 If f is a function over some interval ab then the Riemann sum of f with n terms is de ned as RSnflz Arl ro fI1 flt1nl7 where Ax and xi a iAx If f represents the change in some quantity the Riemann sum approximates the total amount of that quantity Example 4 Identify the parameters f a b n Ax and x3 in each of the previous examples Example 5 Compute the Riemann sum for the function x2 7 5 over the interval from 0 to 4 with n 8 Riemann sums are easy to do with the help of a computer Example 6 Use Excel to compute the Riemann sum with everything as in the previous example but with n 20 2 AREAS AND RIEMANN SUMS We have seen that Riemann sums arise natural when one is trying to compute a total displacement from velocity data total cost from marginal cost data etc They rst arose arose historically in computing areas under curves a rst step to computing areas of arbitrary shapes 1 2 MATH 242 LECTURE 6 Example 7 Use rectangles to approximate the area of the trapezoid which is under the graph of y 00 7 32x above the xaxis and between the lines x and x 4 Compare this question with the rst example above Use geometry to get an exact answer As we see in this example7 the areas of individual rectangles used to approximate area are exactly the terms Which occur in a Riemann sumi This relationship is so natural7 we won t muddy the waters by trying to formalize it Well just give one more example Example 8 Write down and evaluate a Riemann sum with twenty terms which approximates the area under the curve y 122 between x 1 and x 3 MATH 242 LECTURE 11 l VALUES OF REVENUE STREAMS Often in business and economics money is paid back in installments as with student loans mortgages and retirement accounts Standard application of the Fundamental Theorem allows us to extract a total amount paid given the rate Theorem 1 If money is paid at a rate of Rt dollars per unit of time the total amount paid between time a and b is f Rtdt Example 2 Stupid example What does this theorem say when Rt is constant say 100 per month between the second and sixth months of some year But as any pensioner or lottery winner will tell you getting a constant amount of money means that your purchasing power will go down over time How can we determine the current or future value of installment payments Example 3 100 deposited per month in an account earning 5 interest has a present value of 100 00e 03905 100e 039051 b dollars What if instead of deposited as 100 per month 1200 was deposited as a continuous income stream evenly over the year Then over a short fraction of the year At th of a year we would deposit 1200At dollars lts present value would be 1200Ate 0395 where t is a time at which this money is deposited The sum of all of these pieces would e 1200Ate 0395 0 1200Ate 0395 1 We have gone through the steps of setting up a Riemann sum leading to an integral by passing to a limit and using the de nite integral we see that the present value will be 0 1200e0395 Note how much this income stream is worth making payments more frequently can make a big difference Theorem 4 The present value of a revenue stream Rt between times a and b is fRteT quotdt The future value of a revenue stream Rt between times a and b is f Rterbquotdt Letls justify this again by analyzing the present and future value of the money deposited over a short period of time Then we re ready for an example Note that the term e or erb which occurs can be pulled out77 of the integral Example 5 What is the present value of a million dollars paid constantly over twenty years assuming a prevailing inflation rate of four percent per year But what happens if the revenue function isnlt constant but is even a linear function We run into integrals like that of ten These are done in section 71 but we will just use a general formula 1 2 MATH 242 LECTURE 11 Theorem 6 tner dt 1 7 1 finer 7 2tnilert 7 01 3 tn72ert7 7 7 7 nn71n721 n niTe 0 We are mainly concerned in the cases of n 0 or 17 in Which case we can verify this by taking the derivative We can use this for more complicated revenue stream problemsl Example 7 If you invest Rt 2000 400t dollars per year in a retirement account earning nine percent per year how much will you have after forty years How much of that is principal and how much is interest The other class of revenue streams for Which exact answers are Within are reach are those described by exponential functions Example 8 The revenues of Startupcom is modeled by 10e1392 7 20 millions of dollars Adjusting for inflation of 4 estimate the present value of its revenues over its rst ve years MATH 242 LECTURE 3 ll TECHNIQUES OF INTEGRATION So far our method for nding antiderivatives has been to guess check by taking the derivative of our guess and modify until the answer is correct It would be better to have a more systematic approach Unfortunately while differentiation requires only a few rules to take the derivatives of most familiar functions antidifferentiation is a much more complicated game involving many more rules so many that at MIT they hold an Integration Bee Moreover there are many functions such as e752 which have no simple antiderivative The rst way to be systematic about integration is to nd analogues of the rst rules we used for differentiation For example if Fz is an antiderivative for so that Fz and similarly Gz 91 then Fz So that Fz Cz is an antiderivative for This is the rst of the rules listed belowl All of these rules may be checked by differentiating II First integration rules 0 1 y1d1 fdeI f91d1 o For any constant number 5 f cfzdz cf o fzndzn n1Cifn7 7ll Looking at the rst two rules do you think there will be an easy rule involving f gzdz Example 1 Evaluate the following inde nite integrals o f213 7 51 3dz o f esxdz o f dz 12 Integration by substitution While we might not have thought about it the chain rule was by far the rule we used most frequently in nding derivatives In the world of integration the cousin of the chain rule is the technique of u substitution for integration We start with a couple examples which we attempt before formalizing the process Example 2 Evaluate 21 7 31 20dx Example 3 Evaluate 12 7 l5101dz and fz12 7 l5dz We make this systematic through steps similar to the steps we took when applying the chain rule To evaluate ffzdz 0 First we identify a function traditionally called hence the name u substitution which we use as a building block to make the integrand In particular we want to have some appearing naturally as part of the integrand 0 Then we take the derivative of uz which is part of the chain rule 0 We substitute the variable u in for this is easy and then here7s the rub try to match what s remaining with du which is Zigdz Sometimes here some constants need to be moved in front of the integral sign 1 MATH 242 LECTURE 3 o If the substitution rneshes7 we have successfully translated the integral f fgudui o If we can integrate fgudu Cu C then we can substitute for u to nd the original integrali The only way to learn this technique is through plenty of practice Its a bit of an art Example 4 Evaluate the following integrals o 14 21 71 5213 1dz MATH 242 LECTURE 15 l LINEAR PROGRAMMING IN APPLIED PROBLEMS Linear programming and our steps for nding maxima and minima are applicable in a number of kinds of applied optimization problems As we practice these problems we will see that setting them up accurately can be more dif cult than solving the mathematics problem which arises which for bounded regions is pretty straightforward given the steps we outlined last lecture Example 1 Farmer Lynn raises chickens and goats She wants to raise no more than 16 animals including no more than 10 chickens She spends 5 to raise a chicken and 15 to raise a goat She has 180 available to spend Each chicken generates 6 in pro t and each goat 20 How many of each animal should she raise in order to maximize pro ts Example 2 You want to invest up to 10000 in the stock market Dorf shares sell for 50 each yield a dividend of 5 and have a risk index of 20 each Cubik shares sell for 25 each yield a dividend of 8 and have a risk index of 30 each How many shares of each stock should you buy if you want your dividends to be at least 500 and you want to minimize your total risk index 2 UNBOUNDED FEASIBILITY REGIONS As stated when we gave the steps for solving linear programming problems if the feasibility region is unbounded we need to check the values of the objective function at boundary points which go to in nity More formally we 39 quot H 39 t the feasibility region by bounded feasibility regions which grow to ll the original as is best seen in a picture Example 3 Find the maxima and minima if they exist of the function Fxy 2x 4y over the zy22 constraint region y 2 0 Repeat the problem for the function Fxy 4x 4y 7x 2y S 4 In the second half of this example we see that an optimum value can occur along all of the boundary points in a boundary line when the boundary line is parallel to the lines of constant value for the objective function Example 4 Suppose that in a production facility the proportion of two components is never allowed to be more than 2 to 1 so they are always ordered together in less than 2 to 1 proportions to one another Suppose the rst component costs 20 each and the second 30 each and orders must be at least 500 What is the minimum number of components which can be purchased 3 THE BASICS OF MATRICES We have seen that solving systems of linear equations is essential even in de ning feasibility regions for multivariable problems Before moving on to more general multivariable problems we will discuss the most ef cient tool for dealing with linear systems in general namely matrices De nition 5 A matrix is a collection of numbers arranged indexed in a rectangular array 1 2 MATH 242 LECTURE 15 l 5 Matr1ces are usually represented w1th1n brackets like 2 1 To speclfy a number we speclfy 1ts row 7 i and column For example in this matrix the number 5 is in the rst row second column or in the 12 position We call a matrix an n by m matrix if it has n rows and m columns The example above is a two by two matrixl We may use matrices to represent and ef ciently solve systems of equations by putting both the coef cients of the system and the values of the equations in a matrix and mimicking our usual procedure for solving theml Example 6 Translate into matrix notation and solve the system I 2y 2 z 3y 5 Matrix notation has uses well beyond solving systemsl Sill Arithmetic of matrices Before doing involved applications of matrices we develop ways to ma nipulate them and combine them algebraicallyl Matrices add by adding their entries 1 2 4 72 7 5 0 3 4 3 75 7 6 fl lmportant note you can only add matrices of the same size Matrices also subtract in the way you would expect 123427177304 4567375 1105 Example 7 Example 8 Addition and subtraction of matrices obey the same rules as for numbers The zero matrix has zero for all of its entriesl Matrices do not change when the zero matrix is add to or subtracted from theml We will see next time that the most natural way to multiply matrices is not what you would expect 4 VECTORS Matrices with only one column or row are called column or row vectorsl Vectors naturally sit on a line plane threespace etcl Example 9 and 3 4 5 are vectors which we can represent by points in the plane or in space Because they are special cases of matrices they can be added and subtractedl Vectors also have a fundamental way to multiply each other 121 b2 De nition 10 The dot product of a row vector a1 a2 an and a column vector is the sum bn a1111 a2112 39 39 39anbn Note that the dot product of two vectors is not another vector but a number The dot product arises in many contexts especially geometryl MATH 242 LECTURE 15 3 Theorem 11 The dot product ofv and w is zero if and only if the line between 0 and v is perpendicular to the line between 0 and w where O is the origin the point 07 o o o 70 Example 12 Verify this theorem when 1 1 2 and w o MATH 242 LECTURE 16 ll THE BASICS OF MATRICES We have seen that solving systems of linear equations is essential even in de ning feasibility regions for multivariable problems Before moving on to more general multivariable problems we will discuss the most ef cient tool for dealing with linear systems in general namely matricesi De nition 1 A matrix is a collection of numbers arranged indexed in a rectangular array Matrices are usually represented within brackets like 712 To specify a number we specify its row and column For example in this matrix the number 5 is in the rst row second column or in the 12 position We call a matrix an n by m matrix if it has n rows and m columns The example above is a two by two matrixi We may use matrices to represent and ef ciently solve systems of equations by putting both the coef cients of the system and the values of the equations in a matrix and mimicking our usual procedure for solving themi Example 2 Translate into matrix notation and solve the system I 2y 2 z 3y 5 Matrix notation has uses well beyond solving systems 2 ARITHMETIC OF MATRICES Before doing involved applications of matrices we develop ways to manipulate them and combine them algebraicallyi Matrices add by adding their entries 1 2 4 72 7 5 0 3 4 3 75 6 71 Important note you can only add matrices of the same size Matrices also subtract in the way you would expect 1 2 3 4 2 71 7 73 0 4 l4 5 6l l3 75 gl li 10 5 Addition and subtraction of matrices obey the same rules as for numbers The zero matrix has zero for all of its entriesi Matrices do not change when the zero matrix is add to or subtracted from themi We will see that the most natural way to multiply matrices is not what you would expect 1 Example 3 Example 4 2 MATH 242 LECTURE 16 3 VECTORS Matrices with only one column or row are called column or row vectors Vectors naturally sit on a line plane threespace etc 1 Example 5 2 and 3 4 5 are vectors which we can represent by points in the plane or in space Because they are special cases of matrices they can be added and subtracted Vectors also have a fundamental way to multiply each other 111 b De nltlon 6 The dot product of a row vector a1 a2 an and a column vector I 39239 is the sum bn a1171 a2172 39 39 39anbn Note that the dot product of two vectors is not another vector but a number The dot product arises in many contexts especially geometry Theorem 7 The dot product of v and w is zero if and only if the line between 0 and v is perpendicular to the line between 0 and w where O is the origin the point 0 0 g 4 MATRIX MULTIPLI CATION Example 8 Verify this theorem when v 1 2 and w The multiplication of matrices often throws people for a loop since it seems unnatural It builds on the dot product of vectors which is a helpful starting point De nition 9 The product M N of two matrices is the matrix whose entry in the ith row and jth column is the dot product of the ith row ofM and the jth column of N when this is de ned Example 10 Try to perform the following matrix multiplications 1 2 1 0 12 1 01 12 3 5 2 3 4 72 4 5 6 1 4 5 6 1 3 So the matrices M and N can only be multiplied if the number of columns of M equals the number of rows of N So in the second example while the two matrices given can be multiplied they could not be if their order is reversed Reversing order of matrices is a problem even when the multiplication is still 0 1 2 de ned Compare our rst example with the product 71 2 3 4 i 7 Matrix multiplication is not commutative In general M N is not equal to N M even when both are de ned You should be asking at this point why should 1 multiply matrices like this Why can7t 1 say X 57 251 7 At least that would be commutative The answer is that you could but you would not nd it as useful as the matrix multiplication de ned using the dot product Experience with many kinds of problems ranging from geometry and computer graphics to iterative processes and political science has shown that the matrix multiplication just de ned is a remarkably expedient MATH 242 LECTURE 16 Example 11 The system of equations 2x 3y 5 x 7 2y 3 can be written in matrixvector notation as 2 3 x 7 5 1 72 y 7 3 Because matrix 39 quot 39 is 39 quot 7 some quot quot can be developed cleanlyl Example 12 Suppose that the number of vampires and slayers in Sunnydale changes from one year to o the next according to Vnw 1 2 726 V0 anal la H i Interpret this equation Then after ten year you would expect 12 726 1 2 726 1 2 726 V0 Tao lilm ll39la lilsol vampires and slayers to be on the prowl Because matrix multiplication is associative this can be written 12 726 1 W10 1 SD This example is a taste of population modeling using matrices For a more complete story7 take Math and computed more succinctly as 341 MATH 2427 LECTURE 23 l REGRESSION ANALYSIS De nition 1 The vertical deviation of a function from some collection of data points is the sum 91 f112v2 f122quot39yi f1i2quot39 Example 2 Find the line whose vertical deviation from the points 11 2 3 and 34 is minimal Some important features of this example 0 Though it looks like I and y should be our variables7 the slope m and yintercept b of the line are the real variables Much as the coefficients of a quadratic polynomial are variables when we t a parabola to data 0 Ultimately7 the critical point is found as a solution of a system of linear equations De nition 3 The linear regression line for a collection of data is the linear function whose vertical deviation from that collection is minima The ability to nd such lines is programmed into statistical and data analysis software7 as well as your calculators The book gives explicit formulae which you may use for the homework7 but for the exam it will be more important that you understand the way in which minimization techniques are employed This is another case where we are learning exactly what our calculators are doing behind the scenes Theorem 4 The linear regression line for the collection of data rhyl zgy2zkyk is the function mm I such that the sum 11mb7y12zkmbiyk2 is minimized Taking the partial deiivatives and setting them to zero leads to a system of two linear equations in the variables m an One of the main applications of linear regression is to ll in predict extrapolate values for a function from known values Example 5 Because of a computer error some of the sales gures for a real estate company were lost The sales gures measured in millions of dollars which are available for Gary Gladhand are 1998 1999 200 2005 09 15 19 2 sold in 2000 and what he will sell in 2004 Graph these and then use the regression line to estimatepredict what he 2 CONSTRAINED OPTIMIZATION AND LAGRANGE MULTIPLIERS 21 Motivation the need for additional tools in constrained optimization In multivariable optimization7 it is often the case that there is some equation which imposes relations among the variables under consideration Such constraint equations arise naturally in at least two distinct ways 0 The equation represents a relation intrinsic to the problem7 as for example when the variables represent money spent and there is one xed limited source for the funds We saw such problems at the end of last term 2 MATH 242 LECTURE 23 0 When optimizing a multivariable function over a region one must check not only relative maxima and minima but values on the boundary of the region as we did in linear programming The boundary of a region is a curve de ned by some equation and we must focus our attention on that curve Before continuing general discussion we clarify what we mean by constraint equations by looking at examples which are manageable with techniques developed last termi Example 6 Minimize the function 12 y2 my subject to the constraint y 73 Minimize it subject to the constraint 1 y 5 What we see in these examples is that in these cases we can use a constraint equation to solve for one variable in terms of the other substitute that expression into our function and thus obtain a onevariable function to optimize We were able to do such problems last term because we ultimately had a onevariable function to optimize But what ifl wanted to minimize the fz y from the example subject to the constraint z5y773zy2 2 I could not just solve for one variable in terms of the other so a new method is needed 22 Tangencies of level curves and the Lagrange multiplier equations In order to understand the fundamental idea behind the Lagrange equations we investigate simple examples paying close attention to the level curves at and near the optimum pointi Example 7 Find the minimum and graph the level curves and the constraint curve near that minimum for the function 12 y2 constrained by z y 2 and y 12 7 2 What we see is that at the optimum point the level curve for the function and the constraint curve are tangentl This makes sense geometrically as we can see with graphical illustrationsi This observation leads to a way to nd optimum points because of the following theorem which we will not be able to justify Theorem 8 The slope of the level curve fzy c for the function f at any point zy is given by m 75 Therefore the level curve of fz y is tangent to the constraint curve gz y c when 7 7 or y y f4 73 If we call the number that they are both equal to A pronounced lambduh then we have a y that f7 Aggc and fy Agyi These are known as the rst two Lagrange equationsi Theorem 9 The maximum and minimum values for the function fz y subject to the constraint gz y h occur at points zy for which the following three equations hold 1 hwy Ag ryy 2 May Ayy yy 3 9Iyy k We rst apply this theorem to see that it gives the same results we found in our previous simple examp esi MATH 2427 LECTURE 19 01 Level curves We saw when rst looking at multivariable functions that it is helpful to look at the sets of points where fz7 y has a xed value like 07 I7 357 Such collections are depicted in topographical maps as contour lines7 and on weather maps as isotherms lines of constant temperature De nition 1 A level curve for afunction is the collection of allpoints Ly such that c for a given c Example 2 What are some level curves of the functions fz7 y 12 y2 12 7 y2 and exy One of the most prominent uses of level curves is in economics The utility function U of two or more variables measures satisfaction also called utility a consumer derives from having different amounts of goods Level curves in this setting are called indi erence curves7 because consumers are supposed to be indifferent to choices which lead to the same level of satisfaction 1 PARTIAL DERIVATIVES De nition 3 If is a function of two va7iables its partial derivative with respect to z denoted either or f7 Ly is the function obtained by treating y as a constant and di erentiating with respect to 1 Similarly the partial de7ivative with respect to y denoted gig or fy is obtained by treating I as a constant and di erentiating with respect to y So7 the game is to treat one variable as a constant while differentiating with respect to the other variable Example 4 Find the partial derivatives of the following functions 0 fz7 y ISyQ 7 7zy3 V312 5 o my 0 f y 111962 f 11 Geometric and practical interpretations Recall that one of the rst and most important inter pretations of the derivative was that it was the slope of the tangent line to a curve There is a similar rst interpretation of the partial derivative As we illustrate on the overhead7 the partial derivative with respect to z is the slope7 within the plane where y is xed at some c and z and 2 are allowed to vary7 of the tangent line to the graph of the function Thus7 practically speaking7 if you were walking along some surface which you could then think of as the graph of a twovariable function7 then the partial derivatives tell you about how steep your climb or descent will be if you walk parallel to the z or y axes due northsouth or eastwest You might wonder how you could nd out about the steepness of the climb or descent if you travel northeast that7s the topic of the gradient of a function 12 Higherorder partial derivatives As in the case of a single variable7 we are free to take a derivative of a derivative The notation works as follows 2 De nition 5 The partial derivative with respect to z of the partial with respect to z is f or 27 2 The partial de7ivative with respect to y of the partial with respect to z is f7y or 32 The partial de7ivative with respect to z of the partial with respect to y is fygc or 517 The partial de7ivative with respect to y of the partial with respect to y is fyy or 2 MATH 242 LECTURE 19 Example 6 Find all four secondorder partial derivatives of z2y3e1y and lnzy What do you notice Theorem 7 The partial derivatives fgcy and fygc are equal for a large class of functions including composites and products of polynomial exponential and logarithmic functions MATH 242 LECTURE 25 l APPLICATIONS OF LAGRANGE MULTIPLIERS For use throughout this lecture we should record the Lagrange equations To optimize the function fzy subject to the constraint gzy h it suffices to solve the following system of three equations which we put in the most succinct form in the variables I y and fx gx fy gy 919 Lagrange multipliers allow us to solve much more general optimization problems than we were able to do with singlevariable techniques Example 1 Find the optimal dimensions for a sh tank it is to hold fty thousand cubic centimeters of water is supposed to have a total surface area of ve thousand square centimeters and costs one dollar per square centimeter for the base and cents per square centimeter for the sides In economics Lagrange multipliers are often used to maximize utility functions In the idealized world of theoretical economics a utility function gives a numerical measure of satisfaction one experiences when one has given amounts of various goods lntuitively there is some utility function for beer and nachos whose maximum involves a combination of both if you have nachos and no beer you feel kind of thirsty beer and no nachos and you7re inebriated and hungry Example 2 Suppose beer costs 4 for a pint and nachos 6 per platter Suppose a group of customers has 120 to spend and they derive utility from 1 pints and y platters according to a CobbDouglas utility function 1010395y0394 How much beer and nachos should this group of customers order to maximize their utility 11 Application to geometry If we remember the interpretation of the optimum point as the place where the level curves of f and g are tangent we can adapt this technique for certain geometry problems 2 2 Example 3 Optimize the function fz y z y subject to be constrained on the ellipse 7 y l in order to nd a tangent line to this ellipse of the form I y c 12 Optimization over entire regions As a bonus topic we end our lectures by placing all of the topics over the past three weeks plus topics from linear programming into one important context namely that of optimizing a nonlinear function over some region in the plane Instead of stating a theorem we state a method which will work for reasonable functions To nd the maximum and minimum of a function fz y go through the following steps 0 Find all critical points inside the region and evaluate the function at those critical points 0 Optimize the function constrained to each of the boundary curves of the region and evaluate the function at those optima 0 Evaluate the function at the corner points of the region 0 Collect all values from these steps The greatest is the maximum over the region and the smallest is the minimum An example utilizing this technique is a tting culmination of our efforts in learning new techniques in calculus in Math 241 and 242 Example 4 Find the maximum and minimum of the function fz y 1312 5y2 il6zy7101 6y2 over the triangle whose vertices are 00 40 and 03 1 MATH 242 LECTURE 5 Today we practice using the Fundamental Theorem of Calculus FTC We start by rounding out the development of the FTC o The FTC replaces a complicated limit which computes area by a simple evaluation of the anti derivative an amazing savings of effort when it applies For many applied problems needing only approximate answers it can be easier for a computer to use rectangular approximation with a large number of rectangles Moreover some functions cannot be antidifferentiated so the rectangular method is all that7s available 0 The area between the graph of a function and the zaxis which sits over the curve when the function is negative counts as negative area77 when computing through the fundamental theorem This isn7t as strange as it sounds in examples we will see when we compute with revenue streams negative marginal revenues do decrease total revenue To count all area between the graph and the zaxis as positive you need to use absolute value andor break up the function as we will see in examples below o It is often helpful to sketch the region whose area is being computed by an integral In some cases one can get an idea of what the answer should be so that there s some chance you can catch an error in computation 0 Notation it is customary to denote the area of the region between the graph of and the z axis between I a and z b counted as negative area when is negative area as f This is also called the de nite integral from a to b of with a and b known as the limits of integration not to be confused with taking some limit In this notation the FTC is stated as follows Theorem 1 IfFz is an antiderivative for then f Fb 7 Fa Example 2 0 Find the total area counting all area as positive between the graph of 12 71 and the zaxis for r values between I 71 and z 2 0 Find the total area between the graph of e 7 2 and the zaxis from I 0 and z 2 0 Find the area of a triangle with side lengths 3 and 5 by evaluating 05 dz 0 Set up and evaluate an integral which computes the area of a trapezoid of height 2 and bases 3 and Finally we give the following important WARNING If you use a u substitution to nd an antiderivative to evaluate a de nite integral t en you must either make sure to substitute back in for u in order to use the limits of integration for z or you have to substitute in for the limits of integration For example to evaluate e2x2dz we want to substitute u 21 We can either change the inde nite integral to feudu which has e as an antiderivative and then plug 21 in for u to get an antiderivative of e21 for our original function which we evaluate to get e2392 7 e2391 e4 7 e Or when we plug in u 21 then we also observe that when I l u 2 and when I 2 u 4 So we change the de nite integral to f2 eudu which gives the same answer Which approach you choose is a matter of taste What is INCORRECT is to change the integral to 12 eudu Example 3 0 Find the area under the graph of z2x13 8 between I 1 and z 2 0 Evaluate f dz 2 2 MATH 242 LECTURE 5 4 0 Evaluate 2 dz Finally note that the inde nite integral is useful in the process of nding the total Change of a quantity whose marginal values that is derivative is given Looking at the FTC in a different light we see that for any function f whose derivative exists and any two numbers a and b then fa 7 ff1M1 so we can recover the difference in value of f between a and b if we are given the derivative of Example 4 0 Find the total distance travelled by a car whose velocity is given by 55 3t miles per hour over three hours Check the answer against how far it would travel it travelled only at 55 or only at 64 mph 0 Find the cost of raising production from 4 to 9 units measured in thousands when the marginal cost is 5 7 q 3q2 dollars per unit MATH 2427 LECTURE 7 We continue our study of Riemann sums7 being a little more formal De nition 1 If f is afunction over some interval a7b then the left Riemann sum of f with n terms is de ned as R57le Arl ro f11 flt1nl7 b a where Ax f and xi a iAx While this might look complicated7 it is simple to understand if you take your time in a concrete example Example 2 Compute the Riemann sum for the function x2 7 5 over the interval from 0 to 4 with n 8 The point of the level of generality in de ning the Riemann sum as evidenced by the number of variables in its de nition is that it can be applied in a number of different settings Example 3 Identify all the terms f a b n Ax xi and in the Riemann sum we used last lecture to estimate the total displacement of a ball thrown in the air with a velocity of vt 732t 100 between 3 and 4 seconds Example 4 Suppose the marginal cost for producing CD s at afactory is cents for the xth CD Write down Riemann sums with four ten and twenty terms which approximate how much costs to make the second batch of 100 CD s 01 Areas and Riemann sums We have seen that Riemann sums arise natural When one is trying to compute a total displacement from velocity data7 total cost from marginal cost data7 etc They rst arose arose historically in computing areas under curves7 the rst step to computing areas of arbitrary shapes Example 5 Write down a Riemann sum with twelve terms which approximates the area under the curve y 122 between x 1 an x 4 identifying all terms a b Use Excel to evaluate this sum We can formalize What welve already done intuitively as follows Theorem 6 The left approximation using n rectangles to the area under the graph of y is equal to the Riemann sum R n P 1 THE FUNDAMENTAL THEOREM We return to the the Fundamental Theorem of Calculus b The way mathematicians usually organize these ideas is by de ning fa to be the limit as N A 00 of RSNlZ x Theorem 7 Fundamental Theorem 12 fltzgtdz M 7 Fa where F is any antiderivative for 2 MATH 242 LECTURE 7 This theorem now makes some intuitive sense7 in the case where is measuring the marginal Change in some quantity Fz think of our marginal cost examplei Adding up all of those Changes yields the total Change F02 7 Fai lndeed7 we can almost ignoring the limits see a mathematically complete proof by writing in the de nition of the derivative of everywhere appears in the Riemann sums de ning f Example 8 Use the Fundamental Theorem to nd exact answers for each of the questions we have considered so far in this lecture MATH 242 LECTURE 24 l TANGENCIES OF LEVEL CURVES AND THE LAGRANGE MULTIPLIER EQUATIONS In order to understand the fundamental idea behind the Lagrange equations we investigate simple examples paying close attention to the level curves at and near the optimum point Example 1 Find the minimum and graph the level curves and the constraint curve near that minimum for the function 12 y2 constrained by z y 2 and y 12 7 2 What we see is that at the optimum point the level curve for the function and the constraint curve are tangentl This makes sense geometrically as we can see with graphical illustrations This observation leads to a way to nd optimum points because of the following theorem which we will not be able to justify Theorem 2 The slope of the level curve fzy c for the function f at any point zy is given by 72175 Therefore the level curve of fz y is tangent to the constraint curve gz y c when 7 7 or y y f4 If we call the number that they are both equal to A pronounced lambduh then we have a that f7 Aggc and fy Agy These are known as the rst two Lagrange equations Theorem 3 The maximum and minimum values for the function fz y subject to the constraint gz y h occur at points zy for which the following three equations hold 1 fxwy Agx yy 2 May M74179 3 9I7y k We rst apply this theorem to see that it gives the same results we found in our previous simple examples plus the following Example 4 Find the minimum value of the function 12 zy y2 subject to the constraint xz2 y 5 2 USING LAGRANGE MULTIPLIERS TO SOLVE CONSTRAINED OPTIMIZATION PROBLEMS The method of Lagrange multipliers is one of the most commonly used optimization techniques For example most equilibria in basic graduatelevel economics are found using this method For applied problems it is important to identify the objective function and the constraint functions Example 5 A production team has been budgeted 60 million for the development and promotion of a new product line Market experience predicts that of 1 million dollars is spent on development and y million on promotion then 40132y units will be sold in the rst year How much money should be alloted to development and how much to promotion In economics Lagrange multipliers are often used to maximize utility functions In the idealized world of theoretical economics a utility function gives a numerical measure of satisfaction one experiences when one has given amounts of various goods Intuitively there is some utility function for beer and nachos whose maximum involves a combination of both if you have nachos and no beer you feel kind of thirsty beer and no nachos and you7re inebriated and hungry 1 2 MATH 242 LECTURE 24 Example 6 Suppose beer costs 4 for a pint and nachos 6 per platter Suppose a group of customers has 120 to spend and they derive utility from 1 pints and y platters according to a CobbDouglas utility function 1010395y039 How much beer and nachos should this group of customers order to maximize their utility Example 7 Find the optimal dimensions for a sh tank it is to hold fty thousand cubic centimeters of water is supposed to have a total surface area of ve thousand square centimeters and costs one dollar per square centimeter for the base and cents per square centimeter for the sides
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