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What is the shape of the curve z = x^2 + 2y^2 and how can it be sketched?
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[QUOTE="Mark44, post: 5915559, member: 147785"] The coefficient of 2 on the y[SUP]2[/SUP] term is [U]why[/U] this surface is an elliptic paraboloid. If that coefficient had been 1, all the horizontal sections would have been circular. All of the horizontal cross-sections (i.e., in planes parallel to the x-y plane) are ellipses, except for when both x and y are zero. Sketching traces is a good start, but you should also include cross sections or level curves for various values of z. For example, when z = 0, you get a single point. When z= 1, the cross section is the ellipse ##x^2 + 2y^2 = 1##. The same thinkiing applies for other values of ##z \ge 0##. Edit: Cross-sections are especially helpful when you have the sum of squares of two of the variables (cross sections are circles) or expressions like ##ax^2 + by^2## (elliptical cross sections). This advice applies not just for expressions involving x and y, but any two of the variables. [/QUOTE]
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Calculus and Beyond Homework Help
What is the shape of the curve z = x^2 + 2y^2 and how can it be sketched?
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