91Ó°ÊÓ

Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. $$f(x, y)=x^{2}-y^{2} ; P(-1,-3) ;\left\langle\frac{3}{5},-\frac{4}{5}\right\rangle$$

Find the first partial derivatives of the following functions. $$f(w, z)=\frac{w}{w^{2}+z^{2}}$$

The volume of a pyramid with a square base \(x\) units on a side and a height of \(h\) is \(V=\frac{1}{3} x^{2} h\). a. Assume that \(x\) and \(h\) are functions of \(t\). Find \(V^{\prime}(t)\). b. Suppose that \(x=t /(t+1)\) and \(h=1 /(t+1),\) for \(t \geq 0\) Use part (a) to find \(V^{\prime}(t)\). c. Does the volume of the pyramid in part (b) increase or decrease as \(t\) increases?

Find the first partial derivatives of the following functions. $$g(x, z)=x \ln \left(z^{2}+x^{2}\right)$$

Find all critical points of the following functions. $$f(x, y)=e^{x^{2} y^{2}-2 x y^{2}+y^{2}}$$

Find an equation of the plane tangent to the following surfaces at the given points. $$z=2+2 x^{2}+\frac{y^{2}}{2} ;\left(-\frac{1}{2}, 1,3\right) \text { and }(3,-2,22)$$

Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. $$f(x, y)=3 x^{2}+y^{3} ; P(3,2) ;\left\langle\frac{5}{13}, \frac{12}{13}\right\rangle$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(1,-1)} \frac{10 x y-2 y^{2}}{x^{2}+y^{2}}$$

Find an equation of the following planes. The plane passing through the points \((-1,1,1),(0,0,2),\) and (3,-1,-2)

Find the domain of the following functions. $$f(x, y)=\sin ^{-1}\left(y-x^{2}\right).$$