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Chapter 13: Problem 13
Evaluate the following limits. $$\lim _{(x, y) \rightarrow(-3,3)}\left(4 x^{2}-y^{2}\right)$$
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The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1.\) \(u(x, t)=A e^{-a^{2} t} \cos a x,\) for any real numbers \(a\) and \(A\)
Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(-1,0)} \frac{x y e^{-y}}{x^{2}+y^{2}}$$
Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(1,1)} \frac{x^{2}+x y-2 y^{2}}{2 x^{2}-x y-y^{2}}$$
Use the Second Derivative Test to prove that if \((a, b)\) is a critical point
of \(f\) at which \(f_{x}(a, b)=f_{y}(a, b)=0\) and \(f_{x x}(a, b)<0
Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid \(36 x^{2}+4 y^{2}+9 z^{2}=36\).
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