91Ó°ÊÓ

In Exercises \(1-10,\) find the total differential. \(z=3 x^{2} y^{3}\)

In Exercises 19-22, use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. [Hint: In Exercise 19, minimize \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(2 x+3 y=-1 .\) Plane: \(x+y+z=1, \quad(2,1,1)\)

Find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. $$ x^{2}-y^{2}+z^{2}=0, \quad(5,13,-12) $$

Maximum Volume Use Lagrange multipliers to find the dimensions of a rectangular box of maximum volume that can be inscribed (with edges parallel to the coordinate axes) in the ellipsoid \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\).

In Exercises 35-38, use the gradient to find a unit normal vector to the graph of the equation at the given point. Sketch your results $$ 4 x^{2}-y=6,(2,10) $$

Use the gradient to find a unit normal vector to the graph of the equation at the given point. Sketch your results $$ 9 x^{2}+4 y^{2}=40,(2,-1) $$

(a) find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection. $$ z=\sqrt{x^{2}+y^{2}}, \quad 5 x-2 y+3 z=22, \quad(3,4,5) $$

Find the angle of inclination \(\theta\) of the tangent plane to the surface at the given point. $$ 2 x y-z^{3}=0, \quad(2,2,2) $$

The temperature at the point \((x, y)\) on a metal plate is \(T=\frac{x}{x^{2}+y^{2}}\). Find the direction of greatest increase in heat from the point (3,4) .

Find the path of a heat-seeking particle placed at the given point in space with a temperature field \(T(x, y, z)\). $$ T(x, y, z)=100-3 x-y-z^{2}, \quad(2,2,5) $$