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Chapter 11: Problem 19
Describe the domain and range of the function. $$ f(x, y)=\arcsin (x+y) $$
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In Exercises 59-62, differentiate implicitly to find the first partial derivatives of \(w\). \(x y z+x z w-y z w+w^{2}=5\)
Let \(w=f(x, y)\) be a function where \(x\) and \(y\) are functions of a single variable \(t\). Give the Chain Rule for finding \(d w / d t\)
In Exercises 21-26, find the gradient of the function and the maximum value of the directional derivative at the given point. $$ \frac{\text { Function }}{h(x, y)=x \tan y} \frac{\text { Point }}{\left(2, \frac{\pi}{4}\right)} $$
Heat-Seeking Path In Exercises 57 and \(58,\) find the path of a heat-seeking particle placed at point \(P\) on a metal plate with a temperature field \(T(x, y)\). $$ T(x, y)=400-2 x^{2}-y^{2}, \quad P(10,10) $$
The function \(f\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) Determine the degree of the homogeneous function, and show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\) \(f(x, y)=\frac{x^{2}}{\sqrt{x^{2}+y^{2}}}\)
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