Final answer:
To create a contour map of the function f(x, y) = x² + 6y², set the function to constant values to get ellipses with major axes along the y-axis, centered at the origin.
Step-by-step explanation:
To sketch a contour map of the function f(x, y) = x² + 6y², consider the function as a surface in a 3D space where each contour line represents a particular height or value of f(x, y). To plot the contours, we can set f(x, y) to constant values such as 1, 4, 9, etc., and solve for y in terms of x, resulting in ellipses with the major axis along the y-axis because of the higher coefficient of y². The contour lines on the xy-plane are ellipses centered at the origin, with the equation (x/a)² + (y/b)² = 1, where a and b are squared roots of the respective coefficients in f(x, y). The axes of the ellipses extend along the x and y axes, with the width of the ellipse determined by the value of a and the height determined by the value of b.