Level Curves B Level Curves and Contour Map One of the most useful and common methods...

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Level Curves B Level Curves and Contour Map One of the most useful and common methods for visualizing functions (or surfaces) of two varibles is a Contour Map in which points of constant elevation are joined in a 2D plane to form level curves (or contour curves). Definition: The level curves of a function f of two variables are the curves with equations f(x, y) =k, where k is a constant (in the range of f). A level curve f(x,y) = k is the set of all points in the domain off at which f takes on a given value k. In other words, it shows where the graph of f has height k. You can see from the picture below (Figure 1) the relation between level curves and horizontal traces. The level curves f(x, y) =k are just the traces of the graph off in the horizontal plane z k projected down to the xy-plane. Demonstration a = -1; b = 1; h = 0.1; [x, y] = meshgrid(a:h:b); z = 9-x.^2-2*y.^2 subplot (2,1,1); surf(x,y,z); title("Surface Plot"); subplot (2,2,2); mesh (x,y,z); title("Mesh Plot"); subplot (2,2,3); contour3(x, y, z); title("Level Curves"); grid on subplot (2,2,4); contour(x,y,z); title("Contour Plot"); axis equal grid on %Enter the lower limit of both x and y in the domain of the sketch %Enter the upper limit of both x and y in the domain of the sketch %Identify the limit %Create the domain of the graph %Encode the function in terms of x and y (Note that all operations between x's and y's are done with dotted operationF1 %Plot the given surface %Set the title as "Surface Plot" %Use the second plotting space %Mesh Plot the given surface % Set the title as "Mesh Plot" %Use the third plotting space %Plot the Level Curves of the given surface % Set the title as "Level Curves" %Use the fourth plotting space %Plot the Level Curvesof the given surface %Set the title as "Contour Plot" %Set the x axis and y axis to have the same scale My Solutions > % Start Using Rotate 3D Tools to rotate the solids Exercise: Perform the Surface, Mesh, Level Curves and Contour Plot of the surface over the indicated domain z = xye-²-², -2<x<2 and -2<y<2 at h=0.2 Exercise: Perform the Surface, Mesh, Level Curves and Contour Plot of the surface over the indicated domain z = xye-²-², -2<x<2 and -2<y<2 at h=0.2 Script %Set the Domain of both x and y by setting the lower limit 2 a = 3 b = 4 h = 5 6 %Setup the graphing properties for both x and y and z. 7 8 9 10 %First Plotting Space. Sketch the surface with the title "Surface Plot" 11 12 13 %Enter the lower limit of both x and y in the domain of the sketch %Enter the upper limit of both x and y in the domain of the sketch %Identify the increment 28 28 29 14 15 %Second Plotting Space. Sketch the Mesh with the title "Mesh Plot" 16 17 18 19 20 21 %Third Plotting Space. Sketch the Level Curves at various heights with the title "Level Curves" %Use the third plotting space 22 30 31 %Divide the plotting space into Four parts and use the first plotting area %Plot the given surface % Set the title as "Surface Plot" %Create the domain of the graph %Encode the function in terms of x and y (Note that all operations between x's and y's are done with dotted operat 23 24 23 25 26 27 %Fourth Plotting Space. Sketch the Contour Map with the title "Contour Plot" %Use the fourth plotting space %Plot the Level Curvesof the given surface %Set the title as "Contour Plot" %First Plotting Space. Sketch the surface with the title "Surface Plot" %Use the second plotting space %Mesh Plot the given surface % Set the title as "Mesh Plot" %Plot the Level Curves of the given surface % Set the title as "Level Curves" % Set Grid On Save %Set Grid On %Set the Axes to have the same scale 32 33 34 % Start Using Rotate 3D Tools to rotate the solids C Reset MATLAB Documentation ▶ Run Script ? Level Curves B Level Curves and Contour Map One of the most useful and common methods for visualizing functions (or surfaces) of two varibles is a Contour Map in which points of constant elevation are joined in a 2D plane to form level curves (or contour curves). Definition: The level curves of a function f of two variables are the curves with equations f(x, y) =k, where k is a constant (in the range of f). A level curve f(x,y) = k is the set of all points in the domain off at which f takes on a given value k. In other words, it shows where the graph of f has height k. You can see from the picture below (Figure 1) the relation between level curves and horizontal traces. The level curves f(x, y) =k are just the traces of the graph off in the horizontal plane z k projected down to the xy-plane. Demonstration a = -1; b = 1; h = 0.1; [x, y] = meshgrid(a:h:b); z = 9-x.^2-2*y.^2 subplot (2,1,1); surf(x,y,z); title("Surface Plot"); subplot (2,2,2); mesh (x,y,z); title("Mesh Plot"); subplot (2,2,3); contour3(x, y, z); title("Level Curves"); grid on subplot (2,2,4); contour(x,y,z); title("Contour Plot"); axis equal grid on %Enter the lower limit of both x and y in the domain of the sketch %Enter the upper limit of both x and y in the domain of the sketch %Identify the limit %Create the domain of the graph %Encode the function in terms of x and y (Note that all operations between x's and y's are done with dotted operationF1 %Plot the given surface %Set the title as "Surface Plot" %Use the second plotting space %Mesh Plot the given surface % Set the title as "Mesh Plot" %Use the third plotting space %Plot the Level Curves of the given surface % Set the title as "Level Curves" %Use the fourth plotting space %Plot the Level Curvesof the given surface %Set the title as "Contour Plot" %Set the x axis and y axis to have the same scale My Solutions > % Start Using Rotate 3D Tools to rotate the solids Exercise: Perform the Surface, Mesh, Level Curves and Contour Plot of the surface over the indicated domain z = xye-²-², -2<x<2 and -2<y<2 at h=0.2 Exercise: Perform the Surface, Mesh, Level Curves and Contour Plot of the surface over the indicated domain z = xye-²-², -2<x<2 and -2<y<2 at h=0.2 Script %Set the Domain of both x and y by setting the lower limit 2 a = 3 b = 4 h = 5 6 %Setup the graphing properties for both x and y and z. 7 8 9 10 %First Plotting Space. Sketch the surface with the title "Surface Plot" 11 12 13 %Enter the lower limit of both x and y in the domain of the sketch %Enter the upper limit of both x and y in the domain of the sketch %Identify the increment 28 28 29 14 15 %Second Plotting Space. Sketch the Mesh with the title "Mesh Plot" 16 17 18 19 20 21 %Third Plotting Space. Sketch the Level Curves at various heights with the title "Level Curves" %Use the third plotting space 22 30 31 %Divide the plotting space into Four parts and use the first plotting area %Plot the given surface % Set the title as "Surface Plot" %Create the domain of the graph %Encode the function in terms of x and y (Note that all operations between x's and y's are done with dotted operat 23 24 23 25 26 27 %Fourth Plotting Space. Sketch the Contour Map with the title "Contour Plot" %Use the fourth plotting space %Plot the Level Curvesof the given surface %Set the title as "Contour Plot" %First Plotting Space. Sketch the surface with the title "Surface Plot" %Use the second plotting space %Mesh Plot the given surface % Set the title as "Mesh Plot" %Plot the Level Curves of the given surface % Set the title as "Level Curves" % Set Grid On Save %Set Grid On %Set the Axes to have the same scale 32 33 34 % Start Using Rotate 3D Tools to rotate the solids C Reset MATLAB Documentation ▶ Run Script ?

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