Review Sheet for MATH 124 with Professor Long at UA 2
Review Sheet for MATH 124 with Professor Long at UA 2
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Date Created: 02/06/15
Wil eyPLUS iWileyPLUS Home Hel Contias Loout HughesHallett Calculus Single and Multivariable 5e MATH 124 129 223 5th ed 1 Chapter 12 Functions of Several Variables riaiei vaisiai39 I Back lm v Reading content D 121 Functions of Two Variables D 122 Graphs of Functions of Two Variables u D 124 Linear Functions D 125 Functions of Three Variables D 126 Limits and Continuity DChapter Summary D Review Exercises and Problems for Chapter Twelve DCheck Your Understanding D Projects for Chapter Twelve El p Student Solutions Manual p Graphing Calculator Manual p Focus on Theory Web Quizzes 123 Contour Diagrams The surface which represents a function of two variables often gives a good idea of the function39s general behavior ifor example whether it is increasing or decreasing as one of the variables increases However it is dif cult to read numerical values off a surface and it can be hard to see all of the function39s behavior from a surface Thus functions of two variables are often represented by contour diagrams like the weather map Contour diagrams have the additional advantage that they can be extended to functions of three variables Topographical Maps One of the most common examples of a contour diagram is a topographical map like that shown in Figure It gives the elevation in the region and is a good way of getting an overall picture of the terrain where the mountains are where the at areas are Such topographical maps are frequently colored green at the lower elevations and brown red or white at the higher elevations le ClDocuments20andnnZOSettingsmathD esktopindex uni him 1 on78262009 82252 AM Wil eyPLUS racinrh Hami 1t Figure 1233 A topographical map showing the region around South Hamilton NY The curves on a topographical map that separate lower elevations from higher elevations are called contour lines because they outline the contour or shape of the land Because every point along the same contour has the same elevation contour lines are also called level curves or level sets The more closely spaced the contours the steeper the terrain the more widely spaced the contours the atter the terrain provided of course that the elevation between contours varies by a constant amount Certain features have distinctive characteristics A mountain peak is typically surrounded by contour lines like those in Figure A pass in a range of mountains may have contours that look like Figure A long valley has parallel contour lines indicating the rising elevations on both sides of the valley see Figure 1236 D a long ridge of mountains has the same type of contour lines only the elevations decrease on both sides of the ridge Notice that the elevation numbers on the contour lines are as important as the curves themselves We usually draw contours for equally spaced values of Z le ClD0cuments20andnnZOSet ngsmathD esktopindex uni mm 2 on78262009 82252 AM Wil eyPLUS Figure 1234 Mountain peak Figure 1236 Long valley Notice that two contours corresponding to different elevations cannot cross each other as shown in Figure 1237 Ifthey did the point of intersection of the two curves would have two different elevations which is impossible assuming the terrain has no overhangs Figure 1237 Impossible contour lines le ClD0cuments20andnnZOSet ngsmathD esktopindex uni htm 3 on78262009 82252 AM Wil eyPLUS Corn Production Example 1 Solution warmer than it is now le ClD0cuments20andnnZOSet ngsmathD esktopindex uni mm 4 on7 12252009 82252 AM T lemperature in 39F quot Contour maps can display information about a function of two variables without reference to a surface Consider the effect of weather conditions on US corn production Figure w gives corn production C f R T as a function of the total rainfall R in inches and average temperature T in degrees Fahrenheit during the growing season4 At the present time R 15 inches and T 76oF Production is measured as a percentage of the present production thus the contour through R 15 T 76 has value 100 that is C f15 76 100 i H 12 15 lb 21 21 Wrainfall in inches Figure 1238 Corn production C as a function of rainfall and temperature Use Figure 1238 to estimatef18 78 andf12 76 and interpret in terms of corn production The point with Rcoordinate 18 and Tcoordinate 78 is on the contour C 100 sof18 78 100 This means that if the annual rainfall were 18 inches and the temperature were 78oF the country would produce about the same amount of corn as at present although it would be wetter and The point with Rcoordinate 12 and T coordinate 76 is about halfway between the C 80 and the C 90 contours sof12 76 2 85 This means that if the rainfall fell to 12 inches and the temperature stayed at 760 then com production would drop to about 85 of what it is now Wil eyPLUS Example 2 Use Figure 1238 to describe in words the crosssections with T andR constant through the point representing present conditions Give a common sense explanation of your answer Solution To see what happens to corn production if the temperature stays xed at 76oF but the rainfall changes look along the horizontal line T 76 Starting from the present and moving left along the line T 76 the values on the contours decrease In other words if there is a drought corn production decreases Conversely as rainfall increases that is as we move from the present to the right along the line T 76 corn production increases reaching a maximum of more than 110 when R 21 and then decreases too much rainfall oods the fields If instead rainfall remains at the present value and temperature increases we move up the vertical line R 15 Under these circumstances corn production decreases a 20 increase causes a 10 drop in production This makes sense since hotter temperatures lead to greater evaporation and hence drier conditions even with rainfall constant at 15 inches Similarly a decrease in temperature leads to a very slight increase in production reaching a maximum of around 102 when T 74 followed by a decrease the corn won39t grow if it is too cold Contour Diagrams and Graphs Contour diagrams and graphs are two different ways of representing a function of two variables How do we go from one to the other In the case of the topographical map the contour diagram was created by joining all the points at the same height on the surface and dropping the curve into the xyplane How do we go the other way Suppose we wanted to plot the surface representing the corn production function C f R T given by the contour diagram in Figure 1238 Along each contour the function has a constant value if we take each contour and lift it above the plane to a height equal to this value we get the surface in Figure 1239 le ClD0cuments20andnnZOSet ngsmathD esktopindex uni mm 5 on78262009 82252 AM Wil eyPLUS ll39H J contour raised lI39JU units 1 incontour raised J 1 I unnls Figure 1239 Getting the graph of the com yield function from the contour diagram Notice that the raised contours are the curves we get by slicing the surface horizontally In general we have the following result Contour lines or level curves are obtained from a surface by slicing it with horizontal planes Finding Contours Algebraically Algebraic equations for the contours of a function f are easy to nd if we have a formula for f x y Suppose the surface has equation f y A contour is obtained by slicing the surface with a horizontal plane with equation Z 0 Thus the equation for the contour at height 0 is given by le ClDocuments20andnnZOSettingsmathD esktopindex uni mm 5 on78262009 82252 AM Wil eyPLUS Example 3 Find equations for the contours of f x y x2 y2 and draw a contour diagram for f Relate the contour diagram to the graph off Solution The contour at height 0 is given by f 3quot 91 3 This is a contour only for c 2 0 For 0 gt 0 it is a circle of radius v For 0 0 it is a single point the origin Thus the contours at an elevation of c l 2 3 4 are all circles centered at the origin of radius 1 VIE Va 2 The contour diagram is shown in Figure The bowlshaped graph of f is shown in Figure H Notice that the graph of f gets steeper as we move further away from the origin This is re ected in the fact that the contours become more closely packed as we move further from the origin for example the contours for c 6 and c 8 are closer together than the contours for c 2 and c 4 Figure 1240 Contour diagram for f x y x2 y2 even values of 0 only le ClD0cuments20andnnZOSet ngsmathD esktopindex uni htm 7 on78262009 82252 AM Wil eyPLUS Figure 1241 The graph offx y x2 y2 Example 4 fl and relate it to the graph off Draw a contour diagram for Jquot Ex y 4x2 Solution The contour at level 0 is given by f A y 5le C For 0 gt 0 this is a circle just as in the previous example but here the radius is 0 instead of if For 0 0 it is the origin Thus if the level 0 increases by l the radius of the contour increases by 1 This means the contours are equally spaced concentric circles see Figure which do not become more closely packed further from the origin Thus the graph of f has the same constant slope as we move away from the origin see Figure w making it a cone rather than a bowl le ClD0cuments20andnnZOSet ngsmathD esktopindex uni mm 8 on78262009 82252 AM Wil eyPLUS Flgure1243 The graph off 3 lxf a39 4 y le ClD0cuments20andnnZOSet ngsmathD esktopindex uni mm 9 on78262009 82252 AM In both of the previous examples the level curves are concentric circles because the surfaces have circular symmetry Any function of two variables which depends only on the quantity x2 y2 has such symmetry for example GEx g Lx y or Hut y smwxg Wil eyPLUS Example 5 Draw a contour diagram for f x y 2x 3y 1 Solution The contour at level 0 has equation 2x 3y l c Rewriting this as y 23x c l3 we see that the contours are parallel lines with slope 23 The yintercept for the contour at level 0 is c l3 each time 0 increases by 3 the yintercept moves up by l The contour diagram is shown in Figure 1244 Figure 1244 A contour diagram forfx y 2x 3y 1 Contour Diagrams and Tables Sometimes we can get an idea of what the contour diagram of a function looks like from its table leClDocuments20andnnZOSettjngsmathDesktopindex uni htm 10 on78262009 82252 AM Wil eyPLUS Example 6 Relate the values of f x y x2 y2 in Table 124 to its contour diagram in Figure 1245 Table 124 Table of Values off x y x2 y2 3 0 5 8 9 8 5 0 2 5 0 3 4 3 0 5 1 8 3 0 1 0 3 8 y 0 9 4 1 0 1 4 9 1 8 3 0 1 0 3 8 2 5 0 3 4 3 0 5 3 0 5 8 9 8 5 0 3 2 1 0 1 2 3 x 39 3 1 Figure 1245 Contour map offx y x2 y2 Solution leClDocuments20andnnZOSettjngsmathDesktopindex uni mm 11 on78262009 82252 AM Wil eyPLUS One striking feature of the values in Table Q is the zeros along the diagonals This occurs because x2 y2 0 along the lines y x andy x So the Z 0 contour consists of these two lines In the triangular region of the table that lies to the right of both diagonals the entries are positive To the left of both diagonals the entries are also positive Thus in the contour diagram the positive contours lie in the triangular regions to the right and left of the lines y x and y x Further the table shows that the numbers on the left are the same as the numbers on the right thus each contour has two pieces one on the left and one on the right See Figure As we move away from the origin along the xaxis we cross contours corresponding to successively larger values On the saddleshaped graph of f x y x2 y2 shown in Figure H this corresponds to climbing out of the saddle along one of the ridges Similarly the negative contours occur in pairs in the top and bottom triangular regions the values get more and more negative as we go out along the yaxis This corresponds to descending from the saddle along the valleys that are submerged below the xy plane in Figure Notice that we could also get the contour diagram by graphing the family of hyperbolas x2 y2 0 i2 i4 Figure 1246 Graph offx y x2 y2 showing plane Z 0 Using Contour Diagrams The CobbDouglas Production Func on Suppose you decide to expand your small printing business How should you expand Should you start a night shift and hire more workers Should you buy more expensive but faster computers which will enable the current staff to keep up with the work Or should you do some combination of the two Obviously the way such a decision is made in practice involves many other considerationsgsuch as whether you could get a suitably trained night shift or whether there are any faster computers available Nevertheless you leClDocuments20andnnZOSettjngsmathDesktopindex uni him 12 on78262009 82252 AM Wil eyPLUS might model the quantity P of work produced by your business as a function of two variables your total number N of workers and the total value V of your equipment How would you expect such a production function to behave In general having more equipment and more workers enables you to produce more However increasing equipment without increasing the number of workers will increase production a bit but not beyond a point If equipment is already lying idle having more of it won39t help Similarly increasing the number of workers without increasing equipment will increase production but not past the point where the equipment is fully utilized as any new workers would have no equipment available to them Example 7 Explain why the contour diagram in Figure 1247 does not model the behavior expected of the production function whereas the contour diagram in Figure 1248 does leClDocuments20andnnZOSettjngsmathDesktopindex uni htm 13 on78262009 82252 AM Wil eyPLUS Figure 1248 Correct contours for printing production Solution Look at Figure 1247 Fixing Vand letting N increase corresponds to moving to the right on the contour diagram As you do so you cross contours with larger and largerP values meaning that production increases inde nitely On the other hand in Figure 1248 as you move in the same direction you move nearly parallel to the contours crossing them less and less frequently Therefore production increases more and more slowly as N increases with Vfixed Similarly if you x N and let Vincrease the contour diagram in Figure 1247 shows production increasing at a steady rate whereas Figure 1248 shows production increasing but at a decreasing rate Thus Figure 1248 ts the expected behavior of the production function best Formula for a Production Function Production functions are often approximated by formulas of the form P f N V1 CWT quot where P is the quantity produced and 0 0L and B are positive constants 0 lt 0L lt 1 and 0 lt B lt 1 Example 8 Show that the contours of the function P CWVB have approximately the shape of the contours in Figure 1248 Solution The contours are the curves where P is equal to a constant value say P0 that is where NW3 Pg Solving for Vwe get i 1 quotJ F iii i DUI 0f a II 5 Al Thus Vis a power function of N with a negative exponent so its graph has the shape shown in leClDocuments20andnnZOSettingsmathDesktopindex uni him 14 on78262009 82252 AM Wil eyPLUS Figure 1248 The CobbDouglas Production Model In 1928 Cobb and Douglas used a similar function to model the production of the entire US economy in the first quarter of this century Using government estimates of P the total yearly production between 1899 and 1922 of K the total capital investment over the same period and of L the total labor force they found that P was well approximated by the Cobb Douglas production function P 1 JiL 75KE3935 This function turned out to model the US economy surprisingly well both for the period on which it was based and for some time afterward Exercises and Problems for Section 123 Exercises For each of the surfaces in Exercises 1 g g and A sketch a possible contour diagram marked with reasonable Z Values Note There are many possible answers 1 leCiD0cuments20andnnZOSettingsmathDesktopindex uni him 15 on78262009 82253 AM Wil eyPLUS In Exercises i g 1 2 Q Q Q and Q sketch a contour diagram for the function with at least four labeled contours Describe in words the contours and how they are spaced 5 fxyxy 6 fxy3x3y 7 fxyxzy2 8fxyx2y21 9fxyxy 10 fxyyx2 11fxyxz2y2 leClD0cuments20andnnZOSettingsmathDesktopindex uni mm 15 on78262009 82253 AM Wil eyPLUS 12 V I39A q j 21quot VIIf2 2y 13 J my cos its 39 14 Find an equation for the contour of f x y 3ny 7x 20 that goes through the point 5 10 15 a For Z fx y xy sketch and label the level curves 2 i1 Z i2 b Sketch and label crosssections offwith x i1 x i2 c The surface Z xy is cut by a vertical plane containing the line y x Sketch the crosssection 16 Match the surfaces ae in Figure 1249 with the contour diagrams 1V in Figure 1250 a39l 10 Idl Figure 1249 leClD0cuments20andnnZOSettjngsmathDesktopindex uni mm 17 on78262009 82253 AM Wil eyPLUS 111 y I 2 um 11 WI 11 Figure 1250 17 Match Tables 125 126 127 and 128 with the contour diagrams 17IV in Figure 1251 Table125 y 1 0 1 w 2 1 2 1 0 1 0 1 1 2 1 2 leCD0cuments20andnnZOSettingsmathDesktopindex uni mm 18 on78262009 82253 AM Wil eyPLUS Table 126 y 1 0 x 0 1 1 0 1 2 1 0 1 Table 127 y 1 0 x 2 0 1 0 2 0 1 2 0 Table 128 y 1 0 x 2 2 1 0 0 0 1 2 2 leCD0cuments20andnnZOSettingsmathDesktopindex uni mm 19 on78262009 82253 AM Wil eyPLUS u I i J Ij V f U Figure 1251 18 Total sales Q of a product is a function of its price and the amount spent on advertising Figure 1252 shows a contour diagram for total sales Which aXis corresponds to the price of the product and which to the amount spent on advertising Explain 3 w 4000 L fHHJIJ 3 DI39JUU g 20110 Figure 1252 leCD0cuments20andnnZOSettingsmathDesktopindex uni mm 20 on78262009 82253 AM Wil eyPLUS Problems 19 Figure 1253 shows contours of f x y lOOex 50y2 Find the values of f on the contours They are equally spaced multiples of 10 4I lll39i mg I Figure 1253 20 Figure 1254 shows the level curves of the temperature H in a room near a recently opened window Label V the three level curves with reasonable values of H if the house is in the following locations a Minnesota in winter where winters are harsh b San Francisco in winter where winters are mild c Houston in summer where summers are hot d Oregon in summer where summers are mild Wit luaur y Figure 1254 leClDocuments20andnnZOSettjngsmathDesktopindex uni mm 21 on78262009 82253 AM Wil eyPLUS 21 Figure 1255 shows a contour map ofa hill with two pathsA and B a On which path A or B will you have to climb more steeply b On which path A or B will you probably have a better View of the surrounding countryside Assuming trees do not block your View c Alongside which path is there more likely to be a stream Figure 1255 22 Figure 1256 is a contour diagram of the monthly payment on a 5year car loan as a function of the interest rate and the amount you borrow The interest rate is 13 and you borrow 6000 a What is your monthly payment b If interest rates drop to 11 how much more can you borrow without increasing your monthly payment c Make a table of how much you can borrow without increasing your monthly payment as a function of the interest rate Inan ammm ml LL I IVI IJ I J 311nm x lilin m lK I 23131 1mm R l 3 leClDocuments20andnnZOSettjngsmathDesktopindex uni mm 22 on78262009 82253 AM Wil eyPLUS Figure 1256 23 Describe in words the level surfaces of the function gx y 2 cosx y z 24 Match the functions ad with the shapes of their level curves IIV Sketch each contour diagram 3 f x y x2 b fx y x2 2y2 0 fxyyx2 d fxy x2 y2 1 Lines 11 Parabolas III Hyperbolas IV Ellipses 25 Figure 1257 shows the density of the fox population P in foxes per square kilometer for southern England Draw two different crosssections along a northsouth line and two different crosssections along an east west line of the population density P kilometers north 150 IOU North p 0 kllometers east Figure 1257 leClD0cuments20andnnZOSettingsmathDesktopindex uni mm 23 on78262009 82253 AM Wil eyPLUS 26 A manufacturer sells two goods one at a price of 3000 a unit and the other at a price of 12000 a unit A quantity ql of the rst good and q2 of the second good are sold at a total cost of 4000 to the manufacturer a Express the manufacturer39s pro t TE as a function of ql and q2 b Sketch curves of constant pro t in the qquplane for TE 10000 TE 20000 and TE 30000 and the breakeven curve TE 0 27 Match each CobbDouglas production function ac with a graph in Figure 1258 and a statement DG a FL K L 025K 025 b FL K L 05K 05 c FL K L 075K 075 D Tripling each input triples output E Quadrupling each input doubles output G Doubling each input almost triples output in H H 1 x If 39 39I J l fliil H Figure 1258 leClDocuments20andnnZOSettjngsmathDesktopindex uni mm 24 on78262009 82253 AM Wil eyPLUS 28 A general CobbDouglas production function has the form n I r1 v3 1quot Aquot What happens to production if labor and capital are both scaled up For example does production double if both labor and capital are doubled Economists talk about 0 increasing returns to scale if doubling L and K more than doubles P 0 constant returns to scale if doubling L and K exactly doubles P 0 decreasing returns to scale if doublingL and K less than doubles P What conditions on 0L and B lead to increasing constant or decreasing returns to scale 29 Figure 1259 is the contour diagram of f x y Sketch the contour diagram of each of the following functions a 3f x y b f x y 10 0 fx2y2 d f x y Figure 1259 leClD0cuments20andnnZOSettjngsmathDesktopindex uni mm 25 on78262009 82253 AM Wil eyPLUS 30 Figure 1260 shows part of the contour diagram of f x y Complete the diagram for x lt 0 if a f x y fx y b f x y fx y Figure 1260 j39p 4 y 2 3 x 4 V 1 3 4 J are in Table 129 L 31 Values of f y a Find a pattern in the table Make a conjecture and use it to complete Table 129 without computation Check by using the formula for f b Using the formula check that the pattern holds for all x 2 l and y 2 l Table129 y l 2 3 4 5 6 l l 3 6 10 15 21 2 2 5 9 14 20 3 4 8 l3 19 x 4 7 12 18 5 ll 17 6 l6 leClDocuments20andnnZOSettjngsmathDesktopindex uni mm 25 on78262009 82253 AM Wil eyPLUS 32 The temperature T in 0C at any point in the region 10 S x S 10 10 S y S 10 is given by the function l a 39 TL y 1m x Viv a Sketch isothermal curves curves of constant temperature forT 1000C T 750C T 500C T 250 C and T 00C b A heatseeking bug is put down at a point on the xyplane In which direction should it move to increase its temperature fastest How is that direction related to the level curve through that point 33 Use the factored form of f x y x2 y2 x yx y to sketch the contour f x y 0 and to find the regions in the xyplane where f x y gt 0 and the regions where f x y lt 0 Explain how this sketch shows that the graph of f x y is saddleshaped at the origin 34 Use Problem g to find a formula for a monkeysaddle39 surface Z gx y which has three regions with gx y gt 0 and three with gx y lt 0 35 Use the contour diagram for f x t cos t sin x in Figure 1261 to describe in words the crosssections off with I xed and the crosssections of f with x xed Explain what you see in terms of the behavior of the string Figure 1261 leClDocuments20andnnZOSettjngsmathDesktopindex uni htm 27 on78262009 82253 AM
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