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Magazine Chapter 12: Q. 10 (page 943)
Compare the graphs in Exercises 8 and 9. Which of these surfaces do you think has a well-defined tangent plane at $$(0, 0, 0)$$?
Short Answer
The surface with graph, $$f(x,y)=\left | xy \right |$$ have a well-defined tangent plane at $$(0,0,0)$$
Step by step solution
Step 1. Given Information
Graphs in Exercises 8 and 9
Step 2. Explanation
We have the first graph of the expression, $$f(x,y)=\left | xy \right |$$
Differentiating it with respect to $$x$$ and $$y$$, we get
$$f_{x}(x,y)=\left | y \right |$$
$$f_{y}(x,y)=\left | x \right |$$
We have the second graph of the expression, $$g(x,y)=x^{2}-y^{2}$$
Differentiating it with respect to $$x$$ and $$y$$, we get
$$g_{x}(x,y)=2x$$
$$g_{y}(x,y)=2x$$
At point $$(0,0,0)$$, the functions $$g_{x}(x,y)$$ and $$g_{y}(x,y)$$ are equal to zero while the functions $$f_{x}(x,y)$$ and $$f_{y}(x,y)$$ will be in the positive direction.
Thus, the graph, $$f(x,y)=\left | xy \right |$$ have a well-defined tangent plane at the point, $$(0,0,0)$$.
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