Chapter 4

Evaluate the following limits, if they exist. If they do not exist, prove it. \(\lim _{(x, y) \rightarrow(0,0)} \frac{4 x y}{x-2 y^{2}}\)

For the following exercises, find the largest interval of continuity for the function. \(f(x, y)=x^{3} \sin ^{-1}(y)\)

For the following exercises, find the largest interval of continuity for the function. \(g(x, y)=\ln \left(4-x^{2}-y^{2}\right)\)

For the following exercises, find all first partial derivatives. \(f(x, y)=\sqrt{x^{2}-y^{2}}\)

For the following exercises, find all first partial derivatives. \(u(x, y)=x^{4}-3 x y+1, x=2 t, y=t^{3}\)

For the following exercises, find all second partial derivatives. \(g(t, x)=3 t^{2}-\sin (x+t)\)

For the following exercises, find all second partial derivatives. \(h(x, y, z)=\frac{x^{3} e^{2 y}}{z}\)

Find the equation of the tangent plane to the specified surface at the given point. \(z=x^{3}-2 y^{2}+y-1\) at point (1,1,-1)

Find the equation of the tangent plane to the specified surface at the given point. \(3 z^{3}=e^{x}+\frac{2}{y}\) at point (0,1,3)

Approximate \(f(x, y)=e^{x^{2}}+\sqrt{y}\) at (0.1,9.1) . Write down your linear approximation function \(L(x, y)\). How accurate is the approximation to the exact answer, rounded to four digits?

Find the differential \(d z\) of \(h(x, y)=4 x^{2}+2 x y-3 y\) and approximate \(\Delta z\) at the point \((1,-2) .\) Let \(\Delta x=0.1\) and \(\Delta y=0.01\).

Find the directional derivative of \(f(x, y)=x^{2}+6 x y-y^{2}\) in the direction \(v=\mathbf{i}+4 \mathbf{j}\).

Find the maximal directional derivative magnitude and direction for the function \(f(x, y)=x^{3}+2 x y-\cos (\pi y)\) at point (3,0).

Find the gradient. \(\quad f(x, y)=\frac{\sqrt{x}+y^{2}}{x y}\)

Find and classify the critical points. \(\quad z=x^{3}-x y+y^{2}-1\)

Use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints. \(f(x, y)=x^{2} y, x^{2}+y^{2}=4\)

Use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints. . \(f(x, y)=x^{2}-y^{2}, x+6 y=4\)

A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of \(5 \%\) in height and \(2 \%\) in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height \(6 \mathrm{~cm}\) and radius \(2 \mathrm{~cm} .\)

A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of 2 \(\mathrm{ft} / \mathrm{sec}\) and \(3 \mathrm{ft} / \mathrm{sec}\), respectively. Find the rate at which the liquid level is rising when the length is \(14 \mathrm{ft}\), the width is \(10 \mathrm{ft},\) and the height is \(4 \mathrm{ft}\)