Exam 11: Partial Derivatives

Find the points on the hyperboloid of one sheet $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 1$ where the tangent plane is parallel to the plane $2 x - y + z = 3$ .

Evaluate $\lim _ { ( x , y , x ) \rightarrow ( 0,0,0 ) } \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { \left( x ^ { 2 } - y ^ { 2 } + z ^ { 2 } \right) }$

Find $\lim _ { h \rightarrow 0 } \frac { \ln \left( ( x + h ) ^ { 2 } y \right) - \ln \left( x ^ { 2 } y \right) } { h }$

Find the local maximum and minimum values and saddle points of the function $f ( x , y ) = x ^ { 3 } + y ^ { 3 } - 3 x y + 5$ .

Let $z = x ^ { 2 } y + x y ^ { 2 }$ and $x = 2 u + v$ , and $y = u - 3 v$ . Show that $\frac { \delta ^ { 2 } z } { \delta u \delta v } = - 16 x - 6 y$ .

Find the point at which the function $f ( x , y , z ) = 2 x + y - 2 z$ has the minimum value subject to the constraint that $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .

Find the direction of maximum increase of the function $f ( x , y , z ) = x ^ { 2 } - 2 x y + z ^ { 2 }$ at the point $( 1,1,2 )$ .

Find an equation of the tangent plant to the surface $z = e ^ { x + y }$ at the point $( 0,0,1 )$ .

The quality $Q$ of a good produced by a company is given by $Q = 10 K ^ { - 0.6 } L ^ { 0.4 }$ , where $K$ is the quantity of capital and $L$ is the quantity of labor used. Capital costs are $20 per unit, labor costs are $10 per unit, and the company wants to keep costs for capital and labor combined to $150.(a) What combination of labor and capital should be used to produce maximum quantity? What is the maximum value? (b) Draw the level curves of $Q$ and the graph of the budget constraint on the same set of axes.(c) Complete the value of $\lambda$ . What does $\lambda$ represent?

Find the unit vectors $\mathbf { u }$ and $\mathbf { V }$ for $f ( x , y ) = x ^ { 3 } + 3 y ^ { 2 }$ which describes the direction of maximal and minimal increase at $( 2,1 )$ on the level curve $f ( x , y ) = 11$ .

Given $f ( x , y ) = x ^ { 2 } y$ , $P _ { 0 } = ( 3,2 )$ , let $\mathbf { u }$ be the unit vector for which the directional derivative $D _ { \mathbf { u } } f \left( P _ { 0 } \right)$ has maximum value. This maximum value is

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

Find an equation of the tangent plane to the surface with parametric equations $x = \cos u \sin v$ , $y = \sin u \sin v$ , $z = \cos v$ at the point $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$ .

Find three positive numbers $x$ , $y$ , and $Z$ whose sum is 48 and product is maximum.

Find a point on the surface $x y z = 8$ such that the normal vector at the point is parallel to the vector $\{ 4,2,1 \}$ .

Find the maximum value of the function $f ( x , y ) = x y$ subject to the constraint that $x ^ { 2 } + y ^ { 2 } = 2$ .

Let $f ( x , y ) = x ^ { 2 } + 2 x y + y ^ { 2 }$ . If $x = 2$ , find $f ( x , 2 x )$ .

Let $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ .(a) Sketch the intersection of $z = f ( x , y )$ and $z = 1$ in the $x y$ -plane.(b) Sketch the intersection of $z = f ( x , y )$ and $x = 1$ in the $x y$ -plane.(c) Sketch the intersection of $z = f ( x , y )$ and $y = 1$ in the $x y$ -plane.(d) Sketch the graph of $z = f ( x , y )$ in $\mathbb { R } ^ { 3 }$ .

Describe the level surfaces of the function $f ( x , y , z ) = z - 2 x - 3 y$ .

Find an equation of the tangent plane to the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ at the point $( 1,2,2 )$ .