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# Review Sheet for MATH 124 at UA 3

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Date Created: 02/06/15

Wil eyPLUS WileyPLUS Home Mel Contact us Loout HughesHallett Calculus Single and Multivariable 5e MATH 124 129 223 5th ed Chapter 12 Functions ofSeveral Variables 39I v Reading content D 121 Functions of Two Variables D 122 Graphs of Functions of Two Variables D 123 Contour Diagrams D 124 Linear Functions D125 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 III Student Solutions Manual p Graphing Calculator Manual gt Focus on Theory b Web Quizzes 125 Functions of Three Variables In applications of calculus functions of any number of variables can arise The density of matter in the universe is a function of three variables since it takes three numbers to specify a point in space Models of the US economy often use functions of ten or more variables We need to be able to apply calculus to functions of arbitrarily many variables One difficulty with functions of more than two variables is that it is hard to visualize them The graph of a function of one variable is a curve in 2space the graph of a function of two variables is a surface in 3space so the graph of a function of three variables would be a solid in 4space Since we can t easily visualize 4space we won39t use the graphs of functions of three variables On the other hand it is possible to draw contour diagrams for functions of three variables only now the contours are surfaces in 3space Representing a Function of Three Variables using a Family of Level Surfaces A function of two variables f x y can be represented by a family of level curves of the form f x y c for various values of the constant c A level surface or level set of a function of three variables f x y 2 is a surface of the form f x y 2 c where c is a constant The function f can be represented by the family of level surfaces obtained by allowing C to vary The value of the function f is constant on each level surface leC Documents VnZOEnd VuZOSettingsmathDeskt0pindexunihlm 1 0f118262009 82416 AM Wil eyPLUS Example 1 The temperature in 0C at a point x y 2 is given by T f x y 2 x2 y2 22 What do the level surfaces of the function f look like and what do they mean in terms of temperature Solution The level surface corresponding to T 100 is the set of all points where the temperature is 1000C That is where f x y 2 100 so 29quot 4 22 100 This is the equation of a sphere of radius 10 with center at the origin Similarly the level surface corresponding to T 200 is the sphere with radius The other level surfaces are concentric spheres The temperature is constant on each sphere We may view the temperature distribution as a set of nested spheres like concentric layers of an onion each one labeled with a different temperature starting from low temperatures in the middle and getting hotter as we go out from the center See Figure 12 The level surfaces become more closely spaced as we move farther from the origin because the temperature increases more rapidly the farther we get from the origin T 39 Illll l quot1 1311 E J 4 u 139 IIIH 9 Figure 1266 Level surfaces ofTfx y 2 x2 y2 22 each one having a constant temperature leCVDocuments VnZOEnd VnZOSettingsmathDeskt0pindexunihtm 2 0f118262009 82416 AM Wil eyPLUS Example 2 What do the level surfaces of f x y 2 x2 y2 and gx y 2 Z y look like Solution The level surface of f corresponding to the constant C is the surface consisting of all points satisfying the equation l y 2 C Since there is no Zcoordinate in the equation Z can take any value For 0 gt 0 this is a circular cylinder of radius vquot around the ZaXis The level surfaces are concentric cylinders on the narrow ones near the ZaXis f has small values on the wider ones f has larger values See Figure Figure 1267 Level surfaces offx y 2 x2 y2 The level surface of g corresponding to the constant c is the surface z yc This time there is no x variable so this surface is the one we get by taking each point on the straight line Z y c in the yz plane and letting x vary We get a plane which cuts the yz plane diagonally the xaXis is parallel to this plane See Figure w leClDocuments VnZOEnd VnZOSettingsmathDeskt0pindexunihtm 3 0f118262009 82416 AM Wil eyPLUS Figure 1268 Level surfaces ofgx y 2 Z y ExmnMe3 What do the level surfaces of f x y 2 x2 y2 22 look like SoM on In Section E we saw that the twovariable quadratic function gx y x2 y2 has a saddleshaped graph and three types of contours The contour equation x2 y2 0 gives a hyperbola opening rightleft when 0 gt 0 a hyperbola opening updown when 0 lt 0 and a pair of intersecting lines when C 0 Similarly the threevariable quadratic function f x y 2 x2 y2 22 has three types of level surfaces depending on the value of C in the equation x2 y2 Z2 C Suppose that c gt 0 say 0 l Rewrite the equation as x2 y2 Z2 l and think of what happens as we cut the surface perpendicular to the ZaXis by holding Z xed The result is a circle x2 y2 constant of radius at least 1 since the constant 22 1 2 l The circles get larger as Z gets larger If we take the x 0 crosssection instead we get the hyperbola y2 22 l The result is shown in Figure withabcl leClD0cuments Vn20and Vu20SettingsmathDeskt0pindexunihtm 4 0f118262009 82416 AM Wil eyPLUS Figure 1269 Elliptical paraboloid 2 Figure 1271 1 Ellipsoid L1 Figure 1272 Hyperboloid of one sheet 1 q Suppose instead 0 lt 0 say 0 1 Then the horizontal cross sections of x2 y2 Z2 l are again circles except that the radii shrink to 0 at Z i1 and between Z l and Z 1 there leClD0cuments Vn20and Vu20SettingsmathDeskt0pindexunihtm 5 0f118262009 82416 AM Wil eyPLUS are no crosssections at all The result is shown in Figure 1273 with a b c l Figure 1273 Hyperboloid of two sheets Z1 39I v y When 0 0 we get the equation x2 y2 22 Again the horizontal crosssections are circles this time with the radius shrinking down to exactly 0 when Z 0 The resulting surface shown in Figure with a b c l is the cone 2 nix Jr 33 studied in Section 123 together with the lower 1 Fi ure 1274 J 3 2 9 Cone 7 2 kg 2 a e39 If leClD0cuments Vn20and Vu20SettingsmathDesktopindexunihtm 6 0f118262009 82416 AM Wil eyPLUS Figure 1275 Plane axbyczd Figure 1276 Cylindrical surface x2 y2 a2 Figure 1277 Parabolic cylindery m2 A Catalog of Surfaces For later reference here is a small catalog of the surfaces we have encountered These are viewed as equations in three variables x y and 2 How Surfaces Can Represent Functions of Two Variables and Functions of Three Variables You may have noticed that we have used surfaces to represent functions in two different ways First we used a single surface to represent a two variable function f x y Second we used a family of level surfaces to represent a threevariable function gx y 2 These level surfaces have equation gx y 2 c What is the relation between these two uses of surfaces For example consider the function r w 392 f m1 y 2 y I 3 De ne leClD0cuments Vn20and Vu20SettingsmathDesktopindexunihim 7 0f118262009 82416 AM Wil eyPLUS 363212 2r2 I132 i 3 z The points on the graph offsatisfy Z x2 y2 3 so they also satisfy x2 y2 3 Z 0 Thus the graph offis the same as the level surface gXJ39 zquot If r 3 z D In general we have the following result A single surface that is the graph of a twovariable function f x y can be thought of as one member of the family of level surfaces representing the threevariable function 5 51 3919 y z w inf 2 The graph off is the level surface g 0 Conversely a single level surface gx y 2 0 can be regarded as the graph of a function f x y if it is possible to solve for Z Sometimes the level surface is pieced together from the graphs of two or more twovariable functions For example if gx y 2 x2 y2 22 then one member of the family of level surfaces is the sphere XE 47312714 1 This equation defines Z implicitly as a function of x and y Solving it gives two functions quotI if EV 1 x yz and Z vl1rquot yz The graph of the first function is the top half of the sphere and the graph of the second function is the bottom half Exercises and Problems for Section 125 Exercises 1 Match the following functions with the level surfaces in Figure 1278 3 fx y Z y2 22 b hx y 2 x2 22 Figure 1278 leCiDocuments VnZOEnd VnZOSettingsmathDeskt0pindexunihim 8 0f118262009 82416 AM WileyPLUS 2 Match the functions with the level surfaces in Figure 1279 a fxy ZxZy2Z2 b gx y 2 x2 22 n Figure 1279 3 Write the level surface x 2y 32 5 as the graph of a function f x y 4 Find a formula for a function f x y 2 whose level surface f 4 is a sphere of radius 2 centered at the origin 5 Write the level surface X2 y pg 1 as the graph of a function f x y 6 Find a formula for a function f x y 2 whose level surfaces are spheres centered at the point a b c 7 Which of the graphs in the catalog of surfaces is the graph of a function of x and y Use the catalog to identify the surfaces in Exercises 2 10 and ll 8 x2y2Z0 9 x2y222l 10 xyl 11 x2y2422l In Exercises Q Q E and Q decide if the given level surface can be expressed as the graph of a function f x y 12 Zx23y20 l3 2x3y52100 14x2y222l0 15 22c24r3y2 leCVDOcuments Vn20and Vu20SettingsmathDeskt0pindexunihtm 9 0f118262009 82416 AM Wil eyPLUS Problems 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 In Exercises m u and Q represent the surface whose equation is given as the graph of a twovariable function f x y and as the level surface of a threevariable function gx y 2 c There are many possible answers The plane 4x y 22 6 The top half of the sphere x2 y2 z2 10 0 The bottom half of the ellipsoid x2 y2 222 1 Find a function f x y 2 whose level surface f l is the graph of the function gx y x 2y Find two functions f x y and gx y so that the graphs of both together form the ellipsoid x2 y24 229 1 Find a formula for a function gx y 2 whose level surfaces are planes parallel to the plane Z 2x 3y 5 The surface S is the graph of f u 1 T3 v2 a Explain why S is the upper hemisphere of radius 1 with equator in the xyplane centered at the origin b Find a level surface gx y 2 0 representing S The surface S is the graph of Iquot 2 11 fl V3 a Explain why S is the upper half of a circular cylinder of radius 1 centered along the xaxis b Find a level surface gx y 2 0 representing S A cone C with height 1 and radius 1 has its base in the xZplane and its vertex on the positive yaxis Find a function gx y 2 such that C is part of the level surface gx y 2 0 Hint The graph of f x y is a cone which opens up and has vertex at the origin Describe in words the level surfacefx y 2 x24 Z2 1 Describe in words the level surface gx y 2 x2 y24 Z2 l Hint Look at crosssections with constant x y and Z values Describe in words the level surfaces of the function gx y z x y Z Describe in words the level surfaces offx y 2 sin x y 2 Describe the surface x2 y2 2 sin Z2 In general if f 2 2 0 for all Z describe the surface x2 y2 fz2 What do the level surfaces of f x y 2 x2 y2 22 look like Hint Use crosssections with y constant instead of crosssections with Z constant 7 3 391 Describe in words the level surfaces of oh v 739 Vd q zu Sketch and label level surfaces of hx y 2 82 39y for h l e 82 leClDocuments VnZOEnd VnZOSettingsmathDeskt0pindexunihtm 10 0f118262009 82416 AM Wil eyPLUS 33 Sketch and label level surfaces offx y 2 4 x2 y2 Z2 forf 0 l 2 34 Sketch and label level surfaces ofgx y 2 l x2 y2 forg 0 l 2 leCVDOcuments Vn20and Vn20SettingsmathDesktopindexunihtm 11 0f118262009 82416 AM

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