91Ó°ÊÓ

Does knowing that \(|\sin (1 / x)| \leq 1\) tell you anything about $$ \lim _{(x, y) \rightarrow(0,0)} y \sin \frac{1}{x} ? $$ Give reasons for your answer.

Use the limit definition of partial derivative to compute the partial derivatives of the functions at the specified points. \(f(x, y)=1-x+y-3 x^{2} y, \quad \frac{\partial f}{\partial x} \quad\) and \(\quad \frac{\partial f}{\partial y} \quad\) at (1,2)

Among all closed rectangular boxes of volume \(27 \mathrm{cm}^{3},\) what is the smallest surface area?

Use the limit definition of partial derivative to compute the partial derivatives of the functions at the specified points. \(f(x, y)=4+2 x-3 y-x y^{2}, \quad \frac{\partial f}{\partial x} \quad\) and \(\quad \frac{\partial f}{\partial y} \quad\) at (-2,1)

A smooth curve is normal to a surface \(f(x, y, z)=c\) at a point of intersection if the curve's velocity vector is a nonzero scalar multiple of \(\nabla f\) at the point. Show that the curve $$\mathbf{r}(t)=\sqrt{t} \mathbf{i}+\sqrt{t} \mathbf{j}-\frac{1}{4}(t+3) \mathbf{k}$$ is normal to the surface \(x^{2}+y^{2}-z=3\) when \(t=1\)

You are to construct an open rectangular box from \(12 \mathrm{m}^{2}\) of material. What dimensions will result in a box of maximum volume?

Sketch a typical level surface for the function. $$f(x, y, z)=z-x^{2}-y^{2}$$

Use the limit definition of partial derivative to compute the partial derivatives of the functions at the specified points. \(f(x, y)=\sqrt{2 x+3 y-1}, \quad \frac{\partial f}{\partial x} \quad\) and \(\quad \frac{\partial f}{\partial y} \quad\) at (-2,3)

Sketch a typical level surface for the function. $$f(x, y, z)=\left(x^{2} / 25\right)+\left(y^{2} / 16\right)+\left(z^{2} / 9\right)$$

Use the limit definition of partial derivative to compute the partial derivatives of the functions at the specified points. \(f(x, y)=\left\\{\begin{array}{ll}\frac{\sin \left(x^{3}+y^{4}\right)}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0),\end{array}\right.\) \(\frac{\partial f}{\partial x} \quad\) and \(\quad \frac{\partial f}{\partial y} \quad\) at (0,0)