Review Sheet for MATH 124 at UA 2
Review Sheet for MATH 124 at UA 2
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Date Created: 02/06/15
Wil eyPLUS WileyPLUS Home Hel Contact us Loout HughesHallett Calculus Single and Multivariable 5e MATH 124 129 223 5th ed lChapter 12 Functions ofSeveral Variables 3939 v Reading content D121 Functions of Two Variables u D 123 Contour Diagrams D 124 Linear Functions D 125 Functions of Three Variables D126 Limits and Continuity DChapter Summary DReview Exercises and Problems for Chapter Twelve DCheck Your Understanding D Projects for Chapter Twelve El Student Solutions Manual gt Graphing Calculator Manual Focus on Theory Web Quizzes m 122 Graphs of Functions of Two Variables The weather map is one way of visualizing a function of two variables In this section we see how to visualize a function of two variables in another way using a surface in 3space Visualizing a Function of Two Variables using a Graph For a function of one variable y f x the graph of f is the set of all points x y in 2space such that y f x In general these points lie on a curve in the plane When a computer or calculator graphs f it approximates by plotting points in the xyplane and joining consecutive points by line segments The more points the better the approximation Now consider a function of two variables The graph of a function of two variables f is the set of all points x y 2 such that Z f x y In general the graph of a function of two variables is a surface in 3space Plotting the Graph of the Function fx y x2 y2 To sketch the graph of f we connect points as for a function of one variable We first make a table of values off such as in Table Q Table 123 Table ofValues af x W y 3 2 1 0 1 2 3 3 18 13 10 9 10 13 18 2 13 8 5 4 5 8 13 1 10 5 2 1 2 5 10 x 0 9 4 1 0 1 4 9 1 10 5 2 1 2 5 10 2 13 8 5 4 5 8 13 leClDocuments20and20SettingsmathDesktopindex uni htm 1 of168262009 82149 AM Wil eyPLUS 31813109101318 Now we plot points For example we plot 1 2 5 becausef1 2 5 and we plot 0 2 4 because f 0 2 4 Then we connect the points corresponding to the rows and columns in the table The result is called a wireframe picture of the graph Filling in between the wires gives a surface That is the way a computer drew the graphs in Figure and Q As more points are plotted we get the surface in Figure called a paraboloid Figure 1211 Wire frame picture offx y x2 y2 for 3 Sx S 3 3 Sy S 3 Figure 1212 Wire frame picture offx y x2 y2 with more points plotted 5 i l I Figure 1213 Graph offxy x2 y2 for 3 SxS3 3 SyS 3 You should check to see if the sketches make sense Notice that the graph goes through the origin since x y 2 0 0 0 satis es Z x2 y2 Observe that if x is leCiDocuments20andZOSettingsmathDesktopindex uni htm 2 of168262009 82149 AM Wil eyPLUS held xed and y is allowed to vary the graph dips down and then goes back up just like the entries in the rows of Table E Similarly if y is held xed and x is allowed to vary the graph dips down and then goes back up just like the columns of Table New Graphs from Old We can use the graph of a function to visualize the graphs of related functions Example 1 Let f x y x2 yz Describe in words the graphs of the following functions 1 gm y x2 y2 3 b hx y 5 x2 y2 c kx y x2 y 12 So I utio n We know from Figure 1213 that the graph of f is a paraboloid or bowl with its vertex at the origin From this we can work out what the graphs of g h and kwill look like a The function gx y x2 y2 3 fx y 3 so the graph ofg is the graph off but raised by 3 units See Figure 1214 0413 Figure 1214 Graph ofgx y x2 y2 3 b Since x2 y2 is the negative of x2 yz the graph of x2 y2 is an upside down paraboloid Thus the graph of hx y 5 x2 y2 5 fx y looks like an upside down paraboloid with vertex at 0 0 5 as in Figure leClDocuments20and205ettingsmathDesktopindex uni htm 3 of168262009 82149 AM Wil eyPLUS 4 111151 Figure 1215 Graph ofhx y 5 x2 y2 C The graph ofkx y x2 y 12 fx y l is a paraboloid with vertex at x 0 y 1 since that is where k x y 0 as in Figure 1216 0 Lu 39 N Figure 1216 Graph ofkxy x2 0 12 Example 2 Describe the graph of Gm V 2 112 j What symmetry does it have x Solution Since the exponential function is always positive the graph lies entirely above the xy plane From the graph of x2 y2 we see that x2 y2 is zero at the origin and gets larger as we move farther from the origin in any direction Thus eg xzf vj is l at the origin and gets smaller as we move away from the origin in any direction It can39t go below the xy plane instead it attens out getting closer and closer to the plane We say the surface is asymptotic to the xy plane See Figure 1217 leClDocuments20andZOSettingsmathDesktopindex uni hm 4 of168262009 82149 AM Wil eyPLUS 4 IIAI L H Figure 1217 GTaph OfG gx2y3j A y Now consider a point x y on the circle x2 y2 r2 Since GM 5 siftquot 9T the value of the function G is the same at all points on this circle Thus we say the graph of G has circular symmetry CrossSections and the Graph of a Function We have seen that a good way to analyze a function of two variables is to let one variable vary at a time while the other is kept xed For a function f x y the function we get by holding x xed and letting y vary is called a cross section of f with x xed The graph of the cross section of f x y with x c is the curve or crosssection we get by intersecting the graph of f with the plane x c We de ne a crosssection of f with y fixed similarly For example the crosssection offx y x2 y2 with x 2 isf2 y 4 y2 The graph of this crosssection is the curve we get by intersecting the graph off with the plane perpendicular to the xaxis at x 2 See Figure 1218 Surlace s graph 01 flji39 u Iquot 1 5quot Curve 5 gragh of MOSSSECll n filly 14 u Figure 1218 Crosssection of the surface 2 x b the lane 3 y P x 2 leClDocuments20andZOSettingsmathDesktopindex uni hm 5 of168262009 82149 AM Wil eyPLUS Figure 1219 shows graphs of other crosssections of f with x xed Figure 1220 shows graphs of crosssections with y xed Surface Curve 1 r yll I fr 1 yl Figure 1219 The curves Z fa y with a constant crosssections with x xed Surface i39ICr39 y I Figure 1220 The curves Z fx b with b constant crosssections with y xed Example 3 Describe the crosssections of the function gx y x2 y2 with y fixed and then with x xed Use these crosssections to describe the shape of the graph of g Solution The crosssections with y xed at y b are given by 3015 X3 532 Thus each crosssection with y xed gives a parabola opening upward with minimum Z b2 The crosssections with x xed are of the form 391 z z y v which are parabolas opening downward with a maximum of Z 12 See Figures 1221 and 1222 The graph ofg is shown in Figure 1223 Notice the upward opening parabolas in the xdirection and the downward opening parabolas in the ydirection We say that the surface is saddleshaped leCDocuments20and20ettingsmathDesktopindex uni htm 5 of168262009 82149 AM Wil eyPLUS Figure 1221 Crosssections ofgx y x2 y2 with y xed Figure 1222 Crosssections ofgx y x2 y2 with x xed Figure 1223 Graph ofgx y x2 y2 showing cross sections Linear Functions Linear functions are central to single variable calculus they are equally important in multivariable calculus You may be able to guess the shape of the graph of a linear function of two variables It39s a plane Let39s look at an example leCl39Documents20and20SettingsmathDesktopindex uni hm 7 of168262009 82149 AM Wil eyPLUS Example 4 Describe the graph offx y l x y Solution The plane x a is vertical and parallel to the yZplane Thus the crosssection with x a is the line Z l a y which slopes downward in the ydirection Similarly the plane y b is parallel to the xZplane Thus the crosssection with y b is the line Z l x b which slopes upward in the x direction Since all the crosssections are lines you might expect the graph to be a at plane sloping down in the ydirection and up in the xdirection This is indeed the case See Figure Q P anem r Figure 1224 Graph ofthe plane Z 1 x y showing crosssection with x a When One Variable is Missing Cylinders Suppose we graph an equation like Z x2 which has one variable missing What does the surface look like Since y is missing from the equation the crosssections with y xed are all the same parabola Z x2 Letting y vary up and down the y axis this parabola sweeps out the troughshaped surface shown in Figure Q The crosssections with x xed are horizontal lines obtained by cutting the surface by a plane perpendicular to the xaxis This surface is called a parabolic cylinder because it is formed from a parabola in the same way that an ordinary cylinder is formed from a circle it has a parabolic crosssection instead of a circular one leClDocuments20and20SettingsmathDesktopindex uni htm s of168262009 82149 AM Wil eyPLUS Figure 1225 A parabolic cylinder Z x2 Example 5 Graph the equation x2 y2 l in 3space Solution Although the equation x2 y2 1 does not represent a function the surface representing it can be graphed by the method used for Z x2 The graph ofx2 y l in the xyplane is a circle Since Z does not appear in the equation the intersection of the surface with any horizontal plane will be the same circle x2 y2 1 Thus the surface is the cylinder shown in Figure M Figure 1226 Circular cylinder x2 y2 l Exercises and Problems for Section 122 leClDocuments20andZOSettingsmathDesktopindex uni hm 9 of168262009 82149 AM Wil eyPLUS Exercises 1 Without a calculator or computer match the functions with their graphs in Figure 1227 a z2x2y2 b 22xZy2 0 Z 2062 32 d Z22xy e 22 till i M l Figure 1227 2 Without a calculator or computer match the functions with their graphs in Figure 1228 21 b 2 x2y2 c Zx2y3 d Zy2 6 Zx3 siny leClDocuments20andZOSettingsmathDesktopindex uni hm 10 of168262009 82149 AM Wil eyPLUS Iilll 3 W Figure 1228 In Exercises 3 Q L 2 and amp sketch a graph of the surface and brie y describe it in words Z 3 x2 y2 22 9 Z x2 y2 4 Z 5 x2 y2 Z y2 226 4y 32 12 x2 y2 4 10 x2 22 4 FOWHQU39FP Problems Problems g Q and E concern the concentration C in mg per liter of a drug in the blood as a function of x the amount in mg of the drug given and t the time in hours since the injection For 0 S x S 4 and t 2 0 we have C f x t te t 5 quot 11 Find f 3 2 Give units and interpret in terms of drug concentration 12 Graph the following two single variable functions and explain their significance in terms of drug concentration a f 4 t h f x 1 leClDocuments20and205ettingsmathDesktopindex uni him 11 of168262009 82149 AM Wil eyPLUS 13 Graphfa t for a l 2 3 4 on the same axes Describe how the graph changes as 1 increases and explain what this means in terms of drug concentration 14 Consider the function f given by f x y y3 xy Draw graphs of cross sections with a x xed atxlx0 andx l b y xedatyly0andy l 15 Without a computer or calculator match the equations aii with the graphs IHIX x g Z COS 90 b 11 Z MM 0 w 1 737 z 23quot 7 Jal quot 139 I vl39 u H l V i H 139 III leClDocuments20andZOSettingsmathDesktopindex uni htm 12 of168262009 82149 AM Wil eyPLUS IV VI VII VIII IX leC ocuments20and VnZOSet ngsmath sktopindex uni htrn 13 of168262009 82149 AM Wil eyPLUS 16 Figure 1229 contains graphs of the parabolas Z fx b for b 2 1 0 l 2 Which of the graphs of Z f x y in Figure 1230 best ts this information Figure 1229 Ill 1 1 nv Figure 1230 17 For each of the following functions decide whether its graph could be a bowl a plate or neither Consider a plate to be any fairly at surface and a bowl to be anything that could hold water assuming the positive ZaXis is up a Zxzy2 b 21xZy2 c xyzl d z Hyu5X2y3 e 23 18 For each function in Problem sketch crosssections For Problems g amp A and 2 give a formula for a function whose graph is described Sketch it using a computer or calculator 19 A bowl which opens upward and has its vertex at 5 on the ZaXis 20 A plane which has its x y and Z intercepts all positive 21 A parabolic cylinder opening upward from along the line y x in the xy plane 22 A cone of circular crosssection opening downward and with its vertex at the origin leClDocuments20and205ettingsmath sktopindex uni htm 14 of168262009 82149 AM Wil eyPLUS 23 By setting one variable constant nd a plane that intersects the graph of 24x2y2lina a Parabola opening upward b Parabola opening downward c Pair of intersecting straight lines 24 By setting one variable constant nd a plane that intersects the graph of Zx2 l sinyxy2ina a Parabola b Straight line c Sine curve 25 You like pizza and you like cola Which of the graphs in Figure represents your happiness as a function of how many pizzas and how much cola you have if a There is no such thing as too many pizzas and too much cola b There is such a thing as too many pizzas or too much cola c There is such a thing as too much cola but no such thing as too many pizzas ll Ilappne s H nami ess I AA sale I cola pizza pa 39I 39r39s l 1 mm Wk haspmess pizza pizza Figure 1231 26 For each of the graphs IIV in Problem 2 draw a Two crosssections with pizza xed b Two crosssections with cola xed leClDocuments20andZOSettingsmath sktopindex uni hm 15 of168262009 82149 AM Wil eyPLUS 27 A wave travels along a canal Let x be the distance along the canal t be the time and Z be the height of the water above the equilibrium level The graph ofz as a function ofx and t is in Figure 1232 a Draw the pro le of the wave for t l 0 l 2 Put the xaxis to the right and the Zaxis vertical b Is the wave traveling in the direction of increasing or decreasing x c Sketch a surface representing a wave traveling in the opposite direction Figure 1232 28 At time t the displacement of a point on a vibrating guitar string stretched between x 0 and x TB is given by the function I205 sinx U 2 lt1 U Ii a Sketch the crosssections of this function with I xed at t 0 TE4 and the crosssections with x xed at x TE4 TE2 b What is the value offifx 0 or x 7 Explain why this is to be expected c Explain the relation of the crosssections to the surface representing f Represent the surfaces in Problems 2 Q Q and Q as graphs of functions f x y and as level surfaces of the form gx y 2 c There are many possible answers 29 Paraboloid obtained by shifting Z x2 y2 vertically 5 units 30 Plane with intercepts x 2 y 3 Z 4 31 Upper half of unit sphere centered at the origin 32 Lower half of sphere of radius 2 centered at 3 0 0 leClDocuments20and205ettingsmathDesktopindex uni htm 15 of168262009 82149 AM
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