Match the function to its contour diagram
Intermediate level — Class exercise, Section 12.3, Contour Diagrams
Match each function (a)–(i) with its contour diagram (I)–(IX). The numbers on the curves are the values of the function. Useful questions: What shape are the level curves? In which direction do the values increase? Are the curves equally spaced, or do they bunch up?
- (a) \(f(x,y) = x^2 + y^2\)
- (b) \(f(x,y) = 5 - x^2 - y^2\)
- (c) \(f(x,y) = \sqrt{x^2 + y^2}\)
- (d) \(f(x,y) = x + y\)
- (e) \(f(x,y) = 2x - y\)
- (f) \(f(x,y) = y^2\)
- (g) \(f(x,y) = x^2 - y^2\)
- (h) \(f(x,y) = xy\)
- (i) \(f(x,y) = |x| + |y|\)
Answer key
- (a) \(f(x,y) = x^2 + y^2\) → (V). Circles centered at the origin, with the values increasing outward; the circles get closer together as you move out, because the bowl gets steeper.
- (b) \(f(x,y) = 5 - x^2 - y^2\) → (VII). The same circles as (a), but the values decrease outward from 5 at the center.
- (c) \(f(x,y) = \sqrt{x^2 + y^2}\) → (II). Circles centered at the origin that are equally spaced: the cone has constant steepness, unlike the bowl in (a).
- (d) \(f(x,y) = x + y\) → (IX). A linear function: the contours x + y = c are parallel lines of slope −1, equally spaced, with values increasing toward the upper right.
- (e) \(f(x,y) = 2x - y\) → (III). Also a plane, so parallel equally spaced lines, but now 2x − y = c gives lines of slope 2, with values increasing toward the lower right.
- (f) \(f(x,y) = y^2\) → (VIII). No x: the contours y2 = c are the horizontal lines y = ±√c, symmetric about the x-axis and bunching up away from it.
- (g) \(f(x,y) = x^2 - y^2\) → (I). Hyperbolas: positive values open left and right around the x-axis, negative values open up and down; the zero contour is the pair of diagonal lines y = ±x.
- (h) \(f(x,y) = xy\) → (VI). Also hyperbolas, but with the axes as asymptotes: positive values in the first and third quadrants, negative in the second and fourth. The zero contour is the two axes.
- (i) \(f(x,y) = |x| + |y|\) → (IV). Diamonds (squares rotated 45°) centered at the origin, equally spaced, values increasing outward: a pyramid rather than a cone.