Use the contours of the linear function z=f(x, y) in Figure...

Name: Problem 35, Chapter 12: Calculus Multivariable
Uploaded: 2021-07-10T19:56:53-08:00
Duration: 2 min 45 s
Channel: Kayla Robinson
Description: Use the contours of the linear function z=f(x, y) in Figure 12.77 to create possible contour labels for the linear function z=g(x, y) satisfying the given condition. (Figure cant copy) The graph of g is parallel but different from the graph of f.

Use the contours of the linear function $z=f(x, y)$ in Figure 12.77 to create possible contour labels for the linear function $z=g(x, y)$ satisfying the given condition.
(Figure cant copy)

The graph of $g$ is parallel but different from the graph of $f.$

Key Concepts

Linear Functions

A linear function in two variables typically has the form z = ax + by + c, where a, b, and c are constants. This form generates a plane in three-dimensional space. The coefficients a and b determine the slope of the plane, while c is the vertical intercept. The key characteristic of linear functions is that their graph (the plane) is flat, and any cross sections or level sets taken from it are represented by lines in the xy-plane.

Level Curves (Contours)

Level curves or contours are the set of points in the domain at which the function takes on a constant value, usually written as z = constant. For linear functions, these level curves are straight lines in the xy-plane. They provide a visual impression of how the function behaves by showing where it reaches specific output levels, and the spacing and orientation of these lines can indicate the rate and direction of change of the function.

Parallel Planes

When two linear functions have the same coefficients for x and y (i.e. the same slopes) yet differ by a constant term, the resulting graphs are parallel planes. This means that although the surfaces never intersect, the corresponding level curves in their respective domains will also be parallel, differing only by a constant shift. Recognizing this property is critical when comparing different linear functions that share similar directional properties in their gradients.

Intercept Shifts

Intercept shifts refer to the differences in the constant term in linear functions, which result in vertical translations of their graphs. Since the slope remains unchanged, the orientation of the level curves remains constant, and the graphs (or surfaces) remain parallel. Adjusting the constant term essentially shifts all contour labels by a fixed amount, which is key when converting the contours of one linear function to those of a parallel one.

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