91Ó°ÊÓ

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?

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

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

Use Lagrange multipliers to find the maximum and minimum values of \(f\) (when they exist) subject to the given constraint. $$f(x, y, z)=x-z \text { subject to } x^{2}+y^{2}+z^{2}-y=2$$

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

Find the first partial derivatives of the following functions. $$h(x, y)=\left(y^{2}+1\right) e^{x}$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(6,2)} \frac{x^{2}-3 x y}{x-3 y}$$

Find the following derivatives. $$z_{s} \text { and } z_{t}, \text { where } z=x^{2} \sin y, x=s-t, \text { and } y=t^{2}$$

$$\text { Find an equation of the following planes.}$$ The plane passing through the points \((2,-1,4),(1,1,-1),\) and (-4,1,1)

Tangent planes for \(z=f(x, y)\) Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). $$z=e^{x y} ;(1,0,1) \text { and }(0,1,1)$$