Sketch the graph using matlab
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Draw the graph of f and its tangent plane at the given point. (Use your computer algebra system both to compute the partial derivatives and to graph the surface and its tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable. f(x, y)=[xy sin(x-y)]/[1+x^2+y^2], and the given point(1, 1, 0)
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Scott MacKenzie
2021-7-5
编辑：Scott MacKenzie
2021-7-5
I think this is what you are looking for. NOTE: My script is based on code in Find Tangent Plane to Surface which you should review for further details.
% function domain
x = -3:0.25:3;
y = -3:0.25:3;
% your function
f = @(x,y) (x .* y .* sin(x-y)) ./ (1 + x.^2 + y.^2);
% use gradient to find partial derivatives of f.
[xx, yy] = meshgrid(x,y);
[fx, fy] = gradient(f(xx,yy), 0.25);
% find tangent plane at query point of interest
xq = 1;
yq = 1;
t = (xx == xq) & (yy == yq);
indt = find(t);
fxq = fx(indt);
fyq = fy(indt);
% plot the function over domain
surf(xx,yy,f(xx,yy),'EdgeAlpha',0.7,'FaceAlpha',0.9)
hold on;
xlabel('X'); ylabel('Y'); zlabel('Z');
% tangent plane equation and points
z = @(x,y) f(xq,yq) + fxq*(x-xq) + fyq*(y-yq);
zz = z(xx,yy);
% plot tangent plain and point-of-intersection
surf(xx,yy,zz);
plot3(1,1,f(1,1), 'or', 'markerfacecolor', 'r', 'markersize', 5);
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