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1 You may know these...

2 Chapter 1: Multivariables Functions 1.1 Functions of Two Variables Function representations D Coordinate System Graph of two variable functions Sketching of the function (3-D *Level Curves Domain and Range 1. Functions of Three Variables 1..1 Domain and Range 1.. Level Surfaces 3

3 1.1 Functions of Two Variables z f ( x, y) Means that z is a function of x and in the sense that a unique value of the dependent variable z is determined by specifying values for the independent variables x and y. and x (, xy) Domain z Range and y : the two different independent variables z : the dependent variable Domain (D) : the set of all possible inputs (, x y) of the function f ( xy, ) that is y 4

4 Range (R) : the set of output result when xy the domain D For example, 1. f ( x, y) x y f (1,1) 1 1 (, ) z that varies over Function of two variables Substitute 1 for x and 1 for y. xy z f( x, y) 64x e f (1, 0) f (, 3) 64 4 e 60 e 6 6 5

5 1.1.1 Function Representation of z f ( x, y) 3-D coordinate system Coordinate Planes f ( x, y) is a rule that assigns a unique real number to each point (x,y) in same set D in the xy-plane 6

6 D Coordinate system 3D coordinate system has 3 main planes:- xy plane or z 0 ( xy,,0) xz plane or y 0 ( x,0, z) yz plane or x 0 (0, y, z) 10

7 The orientation of xyz-axis 11

8 1.1.3 Graph of a Function of Two Variables The graph of the function f of two variables is the set of all points ( x, y, z) in three-dimensional space, where the values of (x, y) lie in the domain of f and z f ( x, y). 1

9 The graphs of z f ( x, y) is called a surface 3 in 3D system or three-space ( ). It looks like a blanket! Four types of surface in space: Planes Example 1 z 0, y 0, x0 x3, y 1, z 5 Given as a constant equation with onevariable. Example y x6, y 4z5, zx 4 Given as a linear equation with twovariable. Example 3 Tetrahedron 13

10 yx y 1 z 63y x Given as a linear equation with threevariable. 14

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12 How to sketch of the given functions 1) Determine the variables ) Sketch the trace in coordinate planes (based on the variables exist) 3) Make the projection onto the traceplane which is parallel to the (variables which is not exists)-axis 16

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24 How to sketch curved surfaces? domain and range Level curves Level Curves To sketch the graph of two variables, we need to familiar with the contour maps. Notice that when the plane z C intersects with the surface z f ( x, y), the result is the space curve with the equation so we called these as the level curves. f (, xy) C, Sketch of the surface z f ( x, y) * the set of point ( x, y) in xy-plane that satisfy f ( x, y) is called level curves /contour curves 8

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27 Sketching surfaces with level curves Let z f ( x, y) Plane z f z C (x, y) is a function of two variables intersects with the surface f ( xy, ) C The set of point ( xy, ) in the xy-plane that satisfy f ( x, y) Cis called the level curve of f at C An entire family of level curves is generated as C varies over the range of f The graph of z f ( x, y) is a surface which can be obtained by sketching the contour map (set of level curves) on xyplane 31

28 Example Sketch the contour lines/level curves and the graphs (i) (ii) z x y, c0,1,,3,4,9 z x y, c0,1,4, 9 (iii) z 6 x y, c0,,4,6 3

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30 Solution (i) z x y, c0,1,,3,4,9 Sketching the level curves - first, replace z with the value of c - second, plot the graph on the xy-plane 0 : 0 c x y 1 : 1 c x y c : x y c3 : x y c4 : x y c9 : x y The traces in the coordinate planes: yz-plane, z y x 0: the quadratic curve, xz-plane, z x y 0: the quadratic curve, 34

31 xy-plane, z 0: a point (the origin) (ii) z x y, c0,1,4, 9. 35

32 0 : 0 c x y 1 : 1 c x y 4 : 4 c x y 9 : 9 c x y The traces in the coordinate planes: yz-plane, xz-plane, x y 0: the straight line, 0: the straight line, z xy-plane, z 0: a point (the origin) parallel to xy-plane, z x y 4 z 4: the circle y x 36

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35 (ii) z x y c 6, 0,,4,6. Sketching the level curves - first, replace z with the value of c - second, plot the graph on the xy-plane c0 :6x y 0 y x c :6x y y x c4 :6x y 4 y x c6 :6x y 6 y x

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37 1.1.5 Domain and Range of z f ( x, y) Domain :{( xy, ) x, y,???} any constraint??? f ( x, y ) may consist: *Sometimes we need to sketch the domain of the function given. 41

38 Range z-values that results when (x,y) varies over the domain (i) z positive? (ii) z negative? (iii) z zero? (iv) z has maximum value? (v) z has minimum value? Range :{ zz,???} put the limitation of z here!! Example Describe the domain and the range of z x y

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40 Example Find the domain and range of Solution z x y 1. 44

41 Example Find the domain and the range of z ln( x y). Solution 45

42 Example Find the domain and the range of z 4 x y. Solution Domai n :{( xy, ) x, y} Rang e :{ zz, z4} 46

43 1. Functions of Three Variables 1..1 Domain and Range Definition A function f of three variables is a rule that assigns to each ordered triple ( x, y, z) in some domain D in space a unique real number w f ( x, y, z). The range consists of the output values for w. Example 1 Identify the domain and range for the following functions. a). w x y z x y z 0 for all points in space. Domain : entire space Domain :{( xyz,,) x, y, z, x y z 0} 47

44 Range : [ 0, ) Rang e :{ ww, w0} b) w 1 x y z 1 x y z 0 We must have order to have a real value for f ( x, y, z). Rewriting the condition, we obtained x y z 1 Thus the domain consists of all points on or within the sphere x y z 1, or Domain in :{( xyz,,) x, y, z, x y z 1} Range Rang e : [0, 1] or :{ ww, 0w1} c) w x 1 y z 48

45 Domain : ( x, y, z) : ( x, y, z) or ( 0, 0, 0 ) Domain :{( xyz,,) x, y, z, x y z 0} Range : ( 0, ) or Rang e :{ ww, w0} d) w xy ln z Domain :{( xyz,, ) x, y, z, z 0} Range : (, ) or Ran ge :{ ww, w} 49

46 1.. Level Surfaces The graphs of functions of three variables ( xyzf,,, ( xyz,, )) consist of points in four-dimensional space. lying Graphs cannot be sketch effectively in three-dimensional frame of reference. Can obtain insight of how function behaves by looking at its threedimensional level surfaces. The graph of the equation f ( x, y, z) will generally be a surface in 3-space which we call the level surface with constant k. Remark The term level surface is standard. It need not be level in the sense being horizontal; it is simply a surface on which all values of f are the same. k 50

47 Example Describe the level surfaces of (a) (b) f f Solution (a) ( x, y, z ) x y z ( x, y, z ) z x y f ( xyz,, ) x yz k 0 The level surfaces have equation of the form For x y, the graph of this equation is a sphere of radius k, centred at the origin. k 0 For ( 0, 0, 0 ). z, the graph is the single point k 51

48 For k 0, there is no level surface. Level surfaces of f ( xyz,, ) x y z 5

49 b) f ( x, y, z ) z x y The level surface have equation of the form For k 0 two sheets. z x y, the graph is a hyperboloid of k For k 0, the graph is a cone. For k 0 one sheet., the graph is a hyperboloid of 53

50 Level surfaces of f ( xyz,, ) z x y 54

51 Exersizes Describe the level surfaces of (i) f ( xyz,, ) x y for C 4, C 9. (ii) f xyz x y z (,, ) 4 4 w1, w4. for 55