91Ó°ÊÓ

Find the gradient of the function. Assume the variables are restricted to a domain on which the function s defined. $$f(m, n)=m^{2}+n^{2}$$

Find the partial derivatives. The variables are restricted to a domain on which the function is defined. $$f_{x}(1,2) \text { and } f_{y}(1,2) \text { if } f(x, y)=x^{3}+3 x^{2} y-2 y^{2}$$

List the points in the \(x y\) -plane, if any, at which the function \(z=f(x, y)\) is not differentiable. $$z=|x|+|y|$$

Find the gradient of the function. $$f(x, y, z)=\cos (x+y)+\sin (y+z)$$

find the equation of the tangent plane at the given point. \(z=e^{y}+x+x^{2}+6\) at the point (1,0,9)

List the points in the \(x y\) -plane, if any, at which the function \(z=f(x, y)\) is not differentiable. $$z=|x+2|-|y-3|$$

Find the gradient of the function. Assume the variables are restricted to a domain on which the function s defined. $$z=x e^{y}$$

Calculate all four second-order partial derivatives and check that \(f_{x y}=f_{y x} .\) Assume the variables are restricted to a domain on which the function is defined. $$f(x, y)=e^{2 x y}$$

The price \(P\) in dollars to purchase a used car is a function of its original cost, \(C,\) in dollars, and its age, \(A,\) in years. (a) What are the units of \(\partial P / \partial A ?\) (b) What is the sign of \(\partial P / \partial A\) and why? (c) What are the units of \(\partial P / \partial C ?\) (d) What is the sign of \(\partial P / \partial C\) and why?

Find the partial derivatives. The variables are restricted to a domain on which the function is defined. $$\frac{\partial}{\partial y}\left(3 x^{5} y^{7}-32 x^{4} y^{3}+5 x y\right)$$