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Chapter 11: Problem 21
Describe the domain and range of the function. $$ f(x, y)=\ln (4-x-y) $$
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Differentiate implicitly to find the first partial derivatives of \(w\). \(x^{2}+y^{2}+z^{2}-5 y w+10 w^{2}=2\)
In Exercises 33 and \(34,\) find \(d^{2} w / d t^{2}\) using the appropriate Chain Rule. Evaluate \(d^{2} w / d t^{2}\) at the given value of \(t\) \(w=\arctan (2 x y), \quad x=\cos t, \quad y=\sin t, \quad t=0\)
Find the gradient of the function and the maximum value of the directional derivative at the given point. $$ \frac{\text { Function }}{g(x, y)=\ln \sqrt[3]{x^{2}+y^{2}}} \frac{\text { Point }}{(1,2)} $$
Resistance \(\quad\) The total resistance \(R\) of two resistors connected in parallel is \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) Approximate the change in \(R\) as \(R_{1}\) is increased from 10 ohms to 10.5 ohms and \(R_{2}\) is decreased from 15 ohms to 13 ohms.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(D_{\mathbf{u}} f(x, y)\) exists, then \(D_{\mathbf{u}} f(x, y)=-D_{-\mathbf{u}} f(x, y)\)
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