The plot of a function looks like a hill on a flat plane: h(x,y) = exp[xy - x2 – 2yz -
2x + y + 5] (a) where is the top of the hill location? (b) How high is the hill ? and (c)
How Steep is the slope at a point (1,1). In what direction is the slope steepest at that
point?.
Answers
Answer:
One way to visualize functions is through their graphs. If f(x,y) is a scalar-valued function of two variables, f:R2→R (confused?), then its graph is the surface formed by the set of all the points (x,y,z) where z=f(x,y), i.e., the set of points (x,y,f(x,y)). By graphing this surface, we can visualize the behavior of the function.
As an example, we graph the function f(x,y)=−x2−2y2 using the domain defined by −2≤x≤2 and −2≤y≤2. The graph of all points (x,y,f(x,y)) with (x,y) in this domain is an elliptic paraboloid, as shown in the following figure.
Graph of elliptic paraboloid. A graph of the function f(x,y)=−x2−2y2 over the domain −2≤x≤2 and −2≤y≤2.
Three-dimensional plots, such as the above figure, are more difficult to draw and visualize than two-dimensional plots. Moreover, the graph of a function f(x,y,z) of three variables would be the set of points (x,y,z,f(x,y,z)) in four dimensions, and it would be difficult to imagine what such a graph would look like.
Another way of visualizing a function is through level sets, i.e., the set of points in the domain of a function where the function is constant. The nice part of of level sets is that they live in the same dimensions as the domain of the function. A level set of a function of two variables f(x,y) is a curve in the two-dimensional xy-plane, called a level curve. A level set of a function of three variables f(x,y,z) is a surface in three-dimensional space, called a level surface.
Explanation:
Hope it will help you....
Answer:
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