Exam 14: Partial Derivatives

The radius of a right circular cone is increasing at a rate of 5 in/s while its height is decreasing at a rate of 3.6 in/s. At what rate is the volume of the cone changing when the radius is $108$ in. and the height is $132$ in.?

Find the differential of the function $z = \frac { x + 5 y } { x - 6 y }$

Find the direction in which the function $f ( x , y ) = x ^ { 4 } y - x ^ { 3 } y ^ { 2 }$ decreases fastest at the point $( 2 , - 1 )$ .

Find the domain and range of the function $g ( x , y ) = x ^ { 2 } + 2 y ^ { 2 } + 3$ .

Find $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } }$ for $z = y \tan 7 x$ .

If $z = x ^ { 2 } - x y + 7 y ^ { 2 }$ and $( x , y )$ changes from (2, 1) to $( 1.94,1.13 )$ find dz.

Find the direction in which the maximum rate of change of f at the given point occurs. $f ( x , y ) = 2 \sin ( x y ) , ( 1,0 )$

Find $f _ { x }$ for $f ( x , y ) = \int _ { y } ^ { x } \cos \left( t ^ { 8 } \right) d t$ .

Use the Chain Rule to find $\frac { \partial u } { \partial p }$ . $u = \frac { x + y } { y + z }$ $x = p + 5 r + 7 t , y = p - 5 r + 7 t , z = p + 5 r - 7 t$

Use the Chain Rule to find $\frac { d w } { d t }$ $w = 9 x ^ { 4 } y ^ { 3 } z , \quad x = 6 t , \quad y = \cos 7 t , \quad z = t \sin t$

Find all the saddle points of the function. $f ( x , y ) = x \sin \frac { y } { 2 }$

Describe the level survaces of the function $f ( x , y , z ) = 3 x + 8 y - 4 z + 2$ .

Determine the largest set on which the function is continuous. $F ( x , y ) = \arctan ( 10 x + 8 \sqrt { y - 5 } )$

The ellipsoid $4 x ^ { 2 } + 5 y ^ { 2 } + z ^ { 2 } = 28$ intersects the plane $y = 2$ in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1, 2, 2).

At what point is the following function a local minimum? $f ( x , y ) = 8 x ^ { 2 } + 5 y ^ { 2 }$

Find the second partial derivatives of the function $g ( x , y ) = 2 x ^ { 3 } y ^ { 2 } + 9 x y ^ { 3 } - 6 x + 8 y + 5$

Use the equation $\frac { d y } { d x } = - \frac { \frac { \partial F } { \partial x } } { \frac { \partial F } { \partial y } } = - \frac { F _ { x } } { F _ { y } }$ to find $\frac { d y } { d x }$ . $\cos ( x - 7 y ) = x e ^ { 4 y }$

Use a table of numerical values of f (x, y) for (x, y) near the origin to make a conjecture about the value of the limit of f (x, y) as $( x , y ) \rightarrow ( 0,0 )$ . $f ( x , y ) = \frac { x ^ { 2 } y ^ { 3 } + x ^ { 3 } y ^ { 2 } - 7 } { 1 - x y }$

Use Lagrange multipliers to find the maximum value of the function subject to the given constraint. $f ( x , y , z ) = 14 x + 8 y + 12 z , x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 101$

Find $h ( x , y ) = g ( f ( x , y ) )$ , if $g ( t ) = t ^ { 2 } + \sqrt { t }$ and $f ( x , y ) = 7 x + 6 y - 6$ .