91Ó°ÊÓ

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$f(x, y)=y^{2}-2 y \cos x, \quad 1 \leqslant x \leqslant 7$$

\(17-18 \text { Verify the linear approximation at }(0,0)\) $$ \frac{2 x+3}{4 y+1} \approx 3+2 x-12 y $$

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$g(x, y, z)=(x+2 y+3 z)^{3 / 2}, \quad(1,1,2), \quad \mathbf{v}=2 \mathbf{j}-\mathbf{k}$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}$$

Find the first partial derivatives of the function. $$f(x, t)=e^{-t} \cos \pi x$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=y z+x y ; \quad x y=1, \quad y^{2}+z^{2}=1\)

Find the first partial derivatives of the function. $$f(x, t)=\sqrt{x} \ln t$$

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$f(x, y)=\sin x \sin y$$ $$-\pi< x <\pi, \quad-\pi< y <\pi$$

Find the extreme values of \(f\) on the region described by the inequality. \(f(x, y)=2 x^{2}+3 y^{2}-4 x-5, \quad x^{2}+y^{2} \leqslant 16\)

Find and sketch the domain of the function. $$f(x, y)=\arcsin \left(x^{2}+y^{2}-2\right)$$