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Chapter 13: Problem 2
What is the domain of \(f(x, y)=x^{2} y-x y^{2} ?\)
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Identify and briefly describe the surfaces defined by the following equations. $$9 x^{2}+y^{2}-4 z^{2}+2 y=0$$
Let \(h\) be continuous for all real numbers. a. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{x}^{y} h(s) d s\) b. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{1}^{x y} h(s) d s\)
Find the points (if they exist) at which the following planes and curves
intersect.
$$8 x+y+z=60 ; \quad \mathbf{r}(t)=\left\langle t, t^{2}, 3
t^{2}\right\rangle, \text { for }-\infty
Let $$f(x, y)=\left\\{\begin{array}{ll}\frac{\sin \left(x^{2}+y^{2}-1\right)}{x^{2}+y^{2}-1} & \text { if } x^{2}+y^{2} \neq 1 \\\b & \text { if } x^{2}+y^{2}=1\end{array}\right.$$ Find the value of \(b\) for which \(f\) is continuous at all points in \(\mathbb{R}^{2}\).
Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}}$$
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