91Ó°ÊÓ

Minimum surface area with fixed volume Find the dimensions of the closed right circular cylindrical can of smallest surface area whose volume is 16\(\pi \mathrm{cm}^{3} .\)

Find the linearization \(L(x, y)\) of the function at each point. $$ f(x, y)=e^{2 y-x} \text { at } \quad \text { a. }(0,0), \quad \text { b. }(1,2) $$

Minimize the function \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) subject to the constraints \(x+2 y+3 z=6\) and \(x+3 y+9 z=9\) .

Display the values of the functions in Exercises \(37-48\) in two ways: (a) by sketching the surface \(z=f(x, y)\) and \((b)\) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value. $$ f(x, y)=1-|y| $$

Change along the involute of a circle Find the derivative of \(f(x, y)=x^{2}+y^{2}\) in the direction of the unit tangent vector of the curve $$ \mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}, \quad t>0 $$

Each of Exercises \(75-78\) gives a function \(f(x, y, z)\) and a positive number \(\epsilon .\) In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y, z)\) , \(\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\epsilon.\) $$f(x, y, z)=x^{2}+y^{2}+z^{2}, \quad \epsilon=0.015$$

If a function \(f(x, y)\) has continuous second partial derivatives throughout an open region \(R,\) must the first-order partial derivatives of \(f\) be continuous on \(R ?\) Give reasons for your answer.