91Ó°ÊÓ

Use the total differential dz to approximate the change in \(z\) as \((x, y)\) moves from \(P\) to \(Q\). Then use a calculator to find the corresponding exact change \(\Delta z\) (to the accuracy of your calculator). See Example \(3 .\) $$ z=x^{2}-5 x y+y ; P(2,3), Q(2.03,2.98) $$

In Problems 1-16, find all first partial derivatives of each function. \(f(s, t)=\ln \left(s^{2}-t^{2}\right)\)

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y, z)=x z \ln (x+y+z) $$

In Problems 9-12, find a unit vector in the direction in which \(f\) increases most rapidly at p. What is the rate of change in this direction? \(f(x, y)=e^{y} \sin x ; \mathbf{p}=(5 \pi / 6,0)\)

In Problems 7-12, find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\). $$ w=\ln (x+y)-\ln (x-y) ; x=t e^{s}, y=e^{s t} $$

Find the minimum distance between the origin and the surface \(x^{2} y-z^{2}+9=0\).

In Problems 7-16, sketch the graph of \(f\). $$ f(x, y)=\sqrt{16-x^{2}-y^{2}} $$

\(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}\)

In Problems 9-12, find a unit vector in the direction in which \(f\) increases most rapidly at p. What is the rate of change in this direction? \(f(x, y, z)=x^{2} y z ; \mathbf{p}=(1,-1,2)\)

In Problems 11-14, find the gradient vector of the given function at the given point \(\mathbf{p}\). Then find the equation of the tangent plane at \(\mathbf{p}\) (see Example 1). f(x, y)=x^{2} y-x y^{2}, \mathbf{p}=(-2,3)