91Ó°ÊÓ

Compute the directional derivative of \(f\) at the given point in the direction of the indicated vector. $$f(w, x, y, z)=\cos \left(w^{2} x y\right)+3 z-\tan 2 z,(2,-1,1,0), \mathbf{u} \text { in }$$ the direction of $$\langle-2,0,1,4\rangle$$

Show that the indicated limit exists. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{2}+y^{2}}$$

Use a graphing utility to sketch graphs of \(z=f(x, y)\) from two different viewpoints, showing different features of the graphs. $$f(x, y)=\frac{x^{2}}{x^{2}+y^{2}+1}$$

For a differentiable function \(f(x, y)\) with \(x=r \cos \theta\) and \(y=r \sin \theta,\) show that \(f_{\theta}=-f_{x} r \sin \theta+f_{y} r \cos \theta\).

The following data show the height and weight of a small number of people. Use the linear model to predict the weight of a \(6^{\prime} 8^{\prime \prime}\) person and a \(5^{\prime} 0^{\prime \prime}\) person. Comment on how accurate you think the model is. $$\begin{array}{|c|c|c|c|c|}\hline \text { Height (inches) } & 68 & 70 & 70 & 71 \\\\\hline \text { Weight (pounds) } & 160 & 172 & 184 & 180 \\\\\hline\end{array}$$

Use a graphing utility to sketch graphs of \(z=f(x, y)\) from two different viewpoints, showing different features of the graphs. $$f(x, y)=\cos \sqrt{x^{2}+y^{2}}$$

Compute the directional derivative of \(f\) at the given point in the direction of the indicated vector. $$f\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)=\frac{x_{1}^{2}}{x_{2}}-\sin ^{-1} 2 x_{3}+3 \sqrt{x_{4} x_{5}},(2,1,0,1,4)$$ \(\mathbf{u}\) in the direction of $$\langle 1,0,-2,4,-2\rangle$$

Show that the indicated limit exists. $$\lim _{(x, y) \rightarrow(0,0)} \frac{2 x^{2} \sin y}{2 x^{2}+y^{2}}$$

Suppose that the business in example 8.4 has profit function \(P(x, y, z)=3 x+6 y+6 z\) and manufacturing constraint \(2 x^{2}+y^{2}+4 z^{2} \leq 8800 .\) Maximize the profits.

Show that the indicated limit exists. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y+x^{2} y^{3}}{x^{2}+y^{2}}$$