Review Sheet for MATH 124 with Professor Long at UA
Review Sheet for MATH 124 with Professor Long at UA
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Date Created: 02/06/15
WileyPLUS MATH 124 129 223 5th ed iWileyPLUS Home Hall Contact us Loout HughesHallett Calculus Single and Multivariable 5e Chapter 12 Functions ofSeveral Variables stanmwew39 l l l ll 3939 v Reading content D 121 Functions of Two Variables D 122 Graphs of Functions of Two Variables D 123 Contour Diagrams n D 125 Functions of Three Variables D126 Limits and Continuity DChapter Summary Review Exercises and Problems for Chapter Twelve DCheck Your Understanding D Projects for Chapter Twelve III gt Student Solutions Manual Graphing Calculator Manual Focus on Theory gt Web Quizzes 124 Linear Functions What is a Linear Function of Two Variables Linear functions played a central role in onevariable calculus because many onevariable functions have graphs that look like a line when we zoom in In twovariable calculus a linear function is one whose graph is a plane In Chapter we see that many twovariable functions have graphs which look like planes when we zoom in What Makes a Plane Flat What makes the graph of the function Z f x y a plane Linear functions of one variable have straight line graphs because they have constant slope On a plane the situation is a bit more complicated If we walk around on a tilted plane the slope is not always the same it depends on the direction in which we walk However at every point on the plane the slope is the same as long as we choose the same direction If we walk parallel to the xaXis we always find ourselves walking up or down with the same slope the same is true if we walk parallel to the yaXis In other words the slope ratios AZAx with y fixed and AZAy with x xed are each constant Example 1 A plane cuts the ZaXis at Z 5 has slope 2 in the x direction and slope l in the y direction What is the equation of the plane Solution Finding the equation of the plane means constructing a formula for the Zcoordinate of the point on the plane directly above the point x y in the xyplane To get to that point start from the point above the origin where Z 5 Then walk x units in the x direction Since the slope in the x direction is 2 the height increases by 2x Then walk y units in the y direction since the slope in the y direction is l the height decreases by y units leMCIDocuments VnZOand VnZOSettingsmath Desktopindex uni htm 1 of128262009 82336 AM WileyPLUS Since the height has changed by 2x y units the Zcoordinate is 5 2x y Thus the equation for the plane is z 5 it 21 y For any linear function if we know its value at a point x0 yo its slope in the x direction and its slope in the y direction then we can write the equation of the function This is just like the equation of a line in the onevariable case except that there are two slopes instead of one If a plane has slope m in the x direction slope n in the y direction and passes through the point x0 yo 20 then its equation is z 24 l mt ix In Ir m1quot DJ This plane is the graph of the linear function 3 y 23 I mi m 32 iv vl If we write 0 20 mxo nyo then we can write f x y in the equivalent form f r rlr FR it y Just as in 2space a line is determined by two points so in 3space a plane is determined by three points provided they do not lie on a line Example 2 Find the equation of the plane passing through the points 1 0 1 1 1 3 and 3 0 1 Solution The first two points have the same xcoordinate so we use them to nd the slope of the plane in the ydirection As the y coordinate changes from 0 to l the Zcoordinate changes from 1 to 3 so the slope in the ydirection is n AZAy 3 l l 0 2 The first and third points have the same y coordinate so we use them to find the slope in the xdirection it is m AZAx l l3 l 1 Because the plane passes through 1 0 1 its equation is z ix l 2L U or z 2 x You should check that this equation is also satisfied by the points 1 l 3 and 3 0 l Example 2 was made easier by the fact that two of the points had the same x coordinate and two had the same ycoordinate An alternative method which leCl Documents20and20Settingsmath Desktopindex uni mm 2 of128262009 82336 AM WileyPLUS works for any three points is to substitute the x y and Zvalues of each of the three points into the equation Z c mx ny The resulting three equations in c m n are then solved simultaneously Linear Functions from a Numerical Point of View To avoid ying planes with empty seats airlines sell some tickets at full price and some at a discount Table w shows an airline39s revenue in dollars from tickets sold on a particular route as a function of the number of full price tickets sold f and the number of discount tickets sold 1 Table 1210 Revenuefmm Ticket Sales Dollars Fullprice tickets f 100 200 300 400 200 39 63 87 111 700 600 500 400 400 55 79 103 127 Distance 500 400 300 200 tickets 600 71 95 119 143 d 300 200 100 000 800 87 111 134 158 100 000 900 800 1000 102 126 150 174 900 800 700 600 In every column the revenue jumps by 15 800 for each extra 200 discount tickets Thus each column is a linear function of the number of discount tickets sold In addition every column has the same slope 15800200 79 dollarsticket This is the price of a discount ticket Similarly each row is a linear function and all the rows have the same slope 239 which is the price in dollars of a fullfare ticket Thus R is a linear function of f and 1 given by R quot I 7 91 We have the following general result A linear function can be recognized from its table by the following features 0 Each row and each column is linear 0 All the rows have the same slope All the columns have the same slope although the slope of the rows and the slope of the columns are generally different leCVDecuments VnZOand VnZOSettingsmath Desktopindex uni mm 3 of128262009 82336 AM WileyPLUS Example 3 The table contains values of a linear function Fill in the blank and give a formula for the function x 15 20 y 2 05 15 3 05 7 Solution In the rst column the function decreases by 1 from 05 to 05 as x goes from 2 to 3 Since the function is linear it must decrease by the same amount in the second column So the missing entry must be 15 1 05 The slope ofthe function in the xdirection is 1 The slope in the ydirection is 2 since in each row the function increases by 1 when y increases by 05 From the table we get f 2 15 05 Therefore the formula is flimy 05 x 2 21 15 15 I 2y What Does the Contour Diagram of a Linear Function Look Like The formula for the airline revenue function in Table 1210 is R 239f 79d where f is the number of fullfares and d is the number of discount fares sold Notice that the contours of this function in Figure are parallel straight lines What is the practical significance of the slope of these contour lines Consider the contour R 100000 that means we are looking at combinations of ticket sales that yield 100000 in revenue Ifwe move down and to the right on the contour the f coordinate increases and the d coordinate decreases so we sell more fullfares and fewer discount fares This is because to receive a fixed revenue of 100000 we must sell more fullfares ifwe sell fewer discount fares The exact tradeoff depends on the slope of the contour the diagram shows that each contour has a slope of about 3 This means that for a fixed revenue we must sell three discount fares to replace one fullfare This can also be seen by comparing prices Each full fare brings in 239 to earn the same amount in discount fares we need to sell 23979 2 303 2 3 fares Since the price ratio is independent of how many of each type of fare we sell this slope remains constant over the whole contour map thus the contours are all parallel straight lines leCiDocuments VnZOand VnZOSettingsmath Desktopindex uni htm 4 of128262009 82336 AM WileyPLUS ml IHII Bill iHIl il lll Fullquot Figure 1262 Revenue as a function of full and discount fares R 239f 79d Notice also that the contours are evenly spaced Thus no matter which contour we are on a fixed increase in one of the variables causes the same increase in the value of the function In terms of revenue no matter how many fares we have sold an extra fare whether full or discount brings the same revenue as before Example 4 Find the equation of the linear function whose contour diagram is in Figure 1263 W 5 la 7quot f quot w IL Figure 1263 Contour map of linear function f x y Solution Suppose we start at the origin on the Z 0 contour Moving 2 units in the y direction takes us to the Z 6 contour so the slope in the y direction is AZAy 62 3 Similarly a move of leCVDocuments VnZOand VnZOSettingsmath Desktopindex uni mm 5 of128262009 82336 AM WileyPLUS 2 in the xdirection from the origin takes us to the Z 2 contour so the slope in the x direction is AZAx 22 1 Since f0 0 0 we havefx y x 3y Exercises and Problems for Section 124 Exercises Problems 1 and 2 each contain a partial table of values for a linear function Fill in the blanks 1 xy 00 10 00 10 20 30 50 2 xy 10 00 10 20 40 30 30 50 Which of the tables of values in Exercises 3 A Q and g could represent linear functions 3 y 0 1 2 0 0 1 4 x 1 1 0 1 2 4 1 0 4 y 0 1 2 0101316 x16912 leCi Documents20and20Settingsmath Desktopindex uni mm 5 of128262009 82336 AM WileyPLUS 5 y 012 0 0 510 x12 712 2 4 914 6 y 01 2 0 5 7 9 x16 912 2 71115 Which of the contour diagrams in Exercises 1 and could represent linear functions 7 1r l 8 I1 9 Find the equation of the linear function Z c mx my whose graph contains the points 0 0 0 0 2 l and 3 0 4 10 Find the linear function whose graph is the plane through the points 4 0 0 0 3 0 and 0 0 2 11 Find an equation for the plane containing the line in the xyplane where y l and the line in the xZplane where Z 2 leCl Documents20and20Settingsmath Desktopindex uni mm 7 of128262009 82336 AM WileyPLUS 12 Find the equation of the linear function Z c mx my whose graph intersects the xZplane in the line Z 3x 4 and intersects the yZplane in the linezy4 13 Suppose that Z is a linear function of x and y with slope 2 in the x direction and slope 3 in the y direction a A change of 05 in x and 02 in y produces what change in 2 b Ifz 2 when x 5 andy 7 what is the value ofz whenx 49 and y 72 14 a Find a formula for the linear function whose graph is a plane passing through point 4 3 2 with slope 5 in the xdirection and slope 3 in the ydirection b Sketch the contour diagram for this function P rob lems 15 A store sells CDs at one price and DVDs at another price Figure 1264 shows the revenue in dollars of the music store as a function of the number 0 of CDs and the number 1 of DVDs that it sells What is the price of a CD What is the price of a DVD r 3L Im If EHH Figure 1264 16 A college admissions of ce uses the following linear equation to predict the grade point average of an incoming student 2 mi 1 Ely 4 where Z is the predicted college GPA on a scale of 0 to 43 and x is the sum of the student39s SAT Math and SAT Verbal on a scale of 400 to 1600 andy is the student s high school GPA on a scale of0 to 43 The college admits students whose predicted GPA is at least 23 a Will a student with SATs of 1050 and high school GPA of30 be admitted b Will every student with SATs of 1600 be admitted c Will every student with a high school GPA of 43 be admitted d Draw a contour diagram for the predicted GPA Z with 400 S x S 1600 and 0 Sy S 43 Shade the points corresponding to students who will be admitted e Which is more important an extra 100 points on the SAT or an extra 05 of high school GPA leCVDocuments VnZOand VnZOSettingsmath Desktopindex uni mm 8 of128262009 82336 AM WileyPLUS 17 A manufacturer makes two products out of two raw materials Let q1 q2 be the quantities sold of the two products p1 p2 their prices and m1 m2 the quantities purchased of the two raw materials Which of the following functions do you expect to be linear and why In each case assume that all variables except the ones mentioned are held fixed a Expenditure on raw materials as a function of MI and m2 b Revenue as a function of ql and q2 c Revenue as a function of p1 and q1 Problems E Q and concern Table 1211 which gives the number of calories burned per minute for someone rollerblading as a function of the person39s weight and speed 18 19 20 Table 1211 Calories burned per minute Weight 8 9 10 11 mph mph mph mph 120 lbs 42 58 74 89 140 lbs 51 67 83 99 160 lbs 61 77 92 108 180 lbs 70 86 102 117 200 lbs 79 95 111 126 Does the data in Table look approximately linear Give a formula for B the number of calories burned per minute in terms of the weight w and the speed 3 Does the formula make sense for all weights or speeds Who burns more total calories to go 10 miles A 120 lb person going 10 mph or a 180 lb person going 8 mph Which of these two people burns more calories per pound for the 10mile trip Use Problem E to give a formula for P the number of calories burned per pound in terms of w and s for a person weighing w lbs rollerblading 10 miles at 3 mph For Problems A and 2 nd possible equations for linear functions with the given contour diagrams leCVDocuments VnZOand VnZOSettingsmath Desktopindex uni mm 9 of128262009 82336 AM WileyPLUS For Problems 2 and nd equations for linear functions with the given values 23 x 1 0 1 2 y 0 15 1 05 0 1 35 3 2 5 2 2 55 5 4 5 4 3 75 7 65 6 24 xy 10 20 30 40 100 3 6 9 12 200 2 5 8 11 300 1 4 7 10 400 0 3 6 9 leCl Documents20and20Settingsmath Desktopindex uni mm 10 of128262009 82336 AM WileyPLUS It is difficult to graph a linear function by hand One method that works if the x y and Zintercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure 1265 this shows the part of the plane in the octant where x 2 0 y 2 0 Z 2 0 Ifthe intercepts are not all positive the same method works if the x y and Zaxes are drawn from a different perspective Use this method to graph the linear functions in Problems 26 and E Figure 1265 25 222xy 26 Z2x2y 27 Z4x2y 28 262x3y 29 Let fbe the linear function f x y c mx ny where c m n are constants and n 72 0 a Show that all the contours of f are lines of slope mn b For all x and y showfx n y m fx y c Explain the relation between parts a and b Problems and 3l refer to the linear function Z f x y whose values are in Table 1212 Table 1212 y 4 6 8 10 12 5 3 6 9 12 15 10 7 10 13 16 19 x 15 11 14 17 20 23 20 15 18 21 24 27 25 19 22 25 28 31 leCVDocuments VnZOand VnZOSettingsmath Desktopindex uni mm 11 of128262009 82336 AM WileyPLUS 30 Each column of Table w is linear with the same slope m AZ Ax 45 Each row is linear with the same slope n AZAy 32 We now investigate the slope obtained by moving through the table along lines that are neither rows nor columns a Move down the diagonal of the table from the upper left corner 2 3 to the lower right comer Z 31 What do you notice about the changes in 2 Now move diagonally from Z 6 to Z 27 What do you notice about the changes in Z now b Move in the table along a line right one step up two steps from Z 19 to Z 9 Then move in the same direction from Z 22 to Z 12 What do you notice about the changes in Z c Show that AZ mAx nAy Use this to explain what you observed in parts a and b 31 Ifwe holdy xed that is we keep Ay 0 and step in the positive x direction we get the xslope m If instead we keep Ax 0 and step in the positive ydirection we get the yslope 11 Fix a step in which neither Ax 0 nor Ay 0 The slope in the Ax Ay direction is Rise 2 Run Length of Slope a Compute the slopes for the linear function in Table w in the direction of Ax 5 Ay 2 b Compute the slopes for the linear function in Table w in the direction ofo lO Ay 2 leCVDocuments VnZOand VnZOSettingsmath Desktopindex uni mm 12 of128262009 82336 AM
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