Sketch a contour diagram of the function f(x,y)=x2+2y2. What is the predominant shape of the contour lines (that is, the graphs of f(x,y)=c where c∈R)?

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To understand the contour diagram of f(x,y) = x^2 + 2y^2, we examine the level curves where f(x,y) = c for constant c. 
Option 1: Hyperbolas. A hyperbola would arise from equations like x^2 - y^2 = c or similar with opposite signs in the squared terms. Here both squared terms have positive coefficients, so the level sets cannot be hyperbolas. 
Option 2: Ellipses. Setting x^2 + 2y^2 = c with c > 0 yields a closed, oval shape whose axes are aligned with the coordinate axes; the x-radius grows with sqrt(c), and the y-radius grows with sqrt(c/2), giving ellipses. This matches the standard form of an ellipse 
(x^2)/(c) + (y^2)/(c/2) = 1 after normalization, indicating elliptical contours. 
Option 3: Parabolas. Parabolas arise from equations like y = x^2 or x = y^2, which involve a single squared term without both terms balanced as here. The level sets do not produce parabolic shapes since both x^2 and y^2 terms are present, not a single squared term with a linear counterpart. 
Option 4: Lines. Lines would correspond to linear level sets such as ax + by = d, which would occur if the function were linear in x and y or if the level set collapsed to a straight line for certain c. Here the dependence is quadratic in both variables, so straight-line contours do not occur. 
In summary, all viable level curves for this function form closed, oval shapes in the plane rather than open curves, and the positive definite quadratic form confirms elliptical contours. The predominant shape of the contour lines is therefore ellipses.

MATH1062/1005/1023 (ND) MATH1062/1023 Calculus Quiz 6

Sketch a contour diagram of the function f(x,y)=x2+2y2. What is the predominant shape of the contour lines (that is, the graphs of f(x,y)=c where c∈R)?

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