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Chapter 11: Problem 26
Describe the domain and range of the function. $$ f(x, y)=x^{2}+y^{2} $$
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Find \(\partial w / \partial s\) and \(\partial w / \partial t\) by using the appropriate Chain Rule. \(w=x^{2}+y^{2}+z^{2}, \quad x=t \sin s, \quad y=t \cos s, \quad z=s t^{2}\)
Show that the function is differentiable by finding values for \(\varepsilon_{1}\) and \(\varepsilon_{2}\) as designated in the definition of differentiability, and verify that both \(\varepsilon_{1}\) and \(\varepsilon_{2} \rightarrow 0\) as \((\boldsymbol{\Delta x}, \boldsymbol{\Delta} \boldsymbol{y}) \rightarrow(\mathbf{0}, \mathbf{0})\) \(f(x, y)=x^{2}+y^{2}\)
Find \(\partial w / \partial s\) and \(\partial w / \partial t\) by using the appropriate Chain Rule. \(w=x \cos y z, \quad x=s^{2}, \quad y=t^{2}, \quad z=s-2 t\)
Find \(d w / d t\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(t\) before differentiating. \(w=x^{2}+y^{2}+z^{2}, \quad x=e^{t} \cos t, \quad y=e^{t} \sin t, \quad z=e^{t}\)
The temperature at the point \((x, y)\) on a metal plate is \(T=\frac{x}{x^{2}+y^{2}}\). Find the direction of greatest increase in heat from the point (3,4) .
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