chap1 - Chapter 1 Solutions Exercise 1.1 f ( x, y ) = x : a...

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Unformatted text preview: Chapter 1 Solutions Exercise 1.1 f ( x, y ) = x : a level curve f ( x, y ) = constant is a vertical line x = c . Curves for c = 1 , , 1 / 2 , 3 and 4 are graphed below:-2-1.5-1-0.5 0.5 1 1.5 2-1 1 2 3 4 y x Level sets x=c for c=-1,0,1/2,3, and 4 c=-1 c=0 c=1/2 c=3 c=4 Here is a surface plot of graph( f ): z=x-2-1.5-1-0.5 0.5 1 1.5 2 x-2-1.5-1-0.5 0.5 1 1.5 2 y-2-1.5-1-0.5 0.5 1 1.5 2 z The Matlab m-file used for this problem is % level sets and graph(f) for f: (x,y)-->x clear clf x=-2:0.1:2; y=x; % level curves f(x,y)=c for c=-1.0,1/2,3,4 c=[-1,0,0.5,3,4]; % generate level curves all at once xx=c*ones(size(x)); plot(xx,y,k); % put in axes, curve labels and title xlabel(x), ylabel(y) text(-0.9,0.25,c=-1) text(-0.25,-0.25,c=0) text(0.2,0.25,c=1/2) text(2.75,-0.25,c=3) text(3.75,0.25,c=4) title(Level sets x=c for c=-1,0,1/2,3, and 4) print -deps contour1.eps % print plot to .eps file pause % hit a key to continue close % close plot % surface plot of the graph of f [X,Y]=meshgrid(x,y); mesh(X,Y,X) % plot labels and title xlabel(x), ylabel(y), zlabel(z) title(z=x) print -deps surface1.eps pause close 2 Exercise 1.3 f ( x, y ) = x 2 y 2 : a level curve f ( x, y ) = constant is a hyperbola x 2 y 2 = c if c negationslash = 0. Vertices are on the y-axis if c < 0 and on the x-axis if c > 0. If c = 0 the level curve is the graph of the equation | x | = | y | , that is, the set of all points for which y = x .-2-1 .5-1-0 .5 0 .5 1 1 .5 2-2-1 1 2 y x x 2- y 2 = c f o r c = - 1 , 0 , 1 / 2 , 3 , 4 c = - 1 c = - 1 c = 0 c = 0 c = 1 / 2 c = 1 / 2 c = 3 c = 3 c = 4 c = 4 A surface plot of graph( f ): z = x 2- y 2-2-1 .5-1-0 .5 0 .5 1 1 .5 2 x-2-1 .5-1-0 .5 0 .5 1 1 .5 2 y-4-3-2-1 1 2 3 4 z 3 The Matlab m-file used for this problem is: % level sets and graph(f) for f: (x,y)-->x^2-y^2 clear clf x=-2:0.1:2; y=-2:0.1:2; % level curves f(x,y)=c for c=-1.0,1/2,3,4 % superimpose level curves hold on % label x & y axes and give plot a title xlabel(x) ylabel(y) title(x^2-y^2=c for c=-1,0,1/2,3,4) % c=-1 plot(x,[sqrt(x.^2+1); -sqrt(x.^2+1)],k); text(-1.25,1.8,c=-1) text(-1.25,-1.8,c=-1) % c=0 plot(x,[x;-x],k) text(-0.35,0.55,c=0) text(-0.35,-0.55,c=0)text(-0....
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chap1 - Chapter 1 Solutions Exercise 1.1 f ( x, y ) = x : a...

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