91Ó°ÊÓ

Find the gradient at the point. $$f(x, y, z)=x y z, \text { at }(1,2,3)$$

assume points \(P\) and \(Q\) are close. Estimate \(\Delta f=f(Q)-f(\bar{P})\) using the differential \(d f\) $$d f=10 d x-5 d y, P=(200,400), Q=(202,405)$$

Find the gradient at the point. $$f(r, h)=2 \pi r h+\pi r^{2}, \text { at }(2,3)$$

Use the chain rule to find \(\partial w / \partial \rho\) and \(\partial w / \partial \theta,\) given that $$ w=x^{2}+y^{2}-z^{2} $$ and $$ x=\rho \sin \phi \cos \theta, \quad y=\rho \sin \phi \sin \theta, \quad z=\rho \cos \phi $$

Find the quadratic Taylor polynomials about (0,0) for the function. $$\cos (x+3 y)$$

Find the gradient at the point. $$f(x, y, z)=\sin (x y)+\sin (y z), \text { at }(1, \pi,-1)$$

Consider the function $$f(x, y)=\left\\{\begin{array}{ll}\frac{x y^{2}}{x^{2}+y^{4}}, & (x, y) \neq(0,0) \\\0, & (x, y)=(0,0)\end{array}\right.$$ (a) Use a computer to draw the contour diagram for \(f\) (b) Show that the directional derivative \(f_{\vec{u}}(0,0)\) exists for each unit vector \(\vec{u}\). (c) Is \(f\) continuous at (0,0)\(?\) Is \(f\) differentiable at (0,0)\(?\) Explain.

Find the partial derivatives. The variables are restricted to a domain on which the function is defined. $$\frac{\partial A}{\partial h} \text { if } A=\frac{1}{2}(a+b) h$$

Find the gradient at the point. $$f(x, y)=e^{\sin y}, \text { at }(0, \pi)$$

Find the gradient at the point. $$f(x, y, z)=x \ln (y z), \text { at }(2,1, e)$$