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Calculus IIChapter 6: 3-Dimensional SpaceSection 6-5

Functions of Several Variables — Question 3

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Question 3

A contour map consists of the curves x−y2=cx-y^2=c for all real cc, labeled so values increase to the right. Reconstruct a simplest function producing the map. Find the level curve through (3,−1)(3,-1) and decide where the function is positive.

Original worksheet page 1: question and worked solution for 6-5-003
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Question 3 – Solution

Strategy Read the constant expression directly from the family of contours.

See the diagram in the original worksheet below.

Function A simplest choice is f(x,y)=x−y2\boxed{f(x,y)=x-y^2}. Any strictly increasing relabeling g(f)g(f) would preserve the geometric contours but change their numerical labels.

Requested contour At (3,−1)(3,-1), c=3−1=2c=3-1=2, so the curve is x−y2=2\boxed{x-y^2=2}.

Sign region The function is positive precisely when x>y2x>y^2, the region to the right of the parabola x=y2x=y^2.

Original worksheet page 2: question and worked solution for 6-5-003

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