Exam 15: Partial Derivatives

Solve the problem. -Find an equation for the level surface of the function $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ that passes through the point $( 3,12,4 )$ .

Sketch the surface z = f(x,y). - $f ( x , y ) = 1 - | x |$

Solve the problem. -Write parametric equations for the tangent line to the curve of intersection of the surfaces $x = y ^ { 2 }$ and $y = 5 z ^ { 2 }$ at the point $( 25,5,1 )$ .

Find the extreme values of the function subject to the given constraint. - $f ( x , y , z ) = x ^ { 3 } + y ^ { 3 } + z ^ { 3 } , x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$

Find the extreme values of the function subject to the given constraint. - $f ( x , y ) = x ^ { 2 } + 4 y ^ { 3 } , x ^ { 2 } + 2 y ^ { 2 } = 2$

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Cubic approximation to $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \frac { 1 } { 1 + 4 \mathrm { x } + \mathrm { y } }$

Solve the problem. -Find the derivative of the function $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \tan ^ { - 1 } \frac { \mathrm { y } } { \mathrm { x } }$ at the point $( - 3,3 )$ in the direction in which the function increases most rapidly.

Provide an appropriate response. -Let $\mathrm { T } ( \mathrm { x } , \mathrm { y } ) = 25 \mathrm { x } ^ { 2 } - 2 \mathrm { xy } + 9 \mathrm { y } ^ { 2 }$ be the temperature at the point $( \mathrm { x } , \mathrm { y } )$ on the ellipse $\mathrm { x } = 3 \cos \mathrm { t } , \mathrm { y } = 5$ sin $t$ , $0 \leq t \leq 2 \pi$ . Find the minimum and maximum temperatures, $\mathrm { T } _ { \mathrm { min } }$ and $\mathrm { T } _ { \mathrm { max } }$ , respectively, on the ellipse.

Provide an appropriate response. -Find any local extrema (maxima, minima, or saddle points) of $f ( x , y )$ given that $\mathrm { f } _ { \mathrm { X } } = 8 \mathrm { x } - 4 \mathrm { y }$ and $\mathrm { f } _ { \mathrm { y } } = - 8 \mathrm { x } + 6 \mathrm { y }$ .

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$ - $f ( x , y , z ) = z \left( e ^ { x } \right) ^ { y }$

Provide an appropriate response. -Determine whether the function $f ( x , y ) = 6 x ^ { 2 } y ^ { 2 } + 8 x ^ { 4 } y ^ { 4 }$ has a maximum, a minimum, or neither at the origin.

Find the extreme values of the function subject to the given constraint. - $f ( x , y , z ) = x ^ { 2 } + 2 y ^ { 2 } + 3 z ^ { 2 } , x - y - z = 1$

Find the requested partial derivative. - $\frac { \partial z } { \partial y }$ if $z ^ { 3 } = z + x y - 1$ and $y ^ { 3 } = x + y - 1$

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Quadratic approximation to $f ( x , y ) = \frac { 1 } { ( 1 + x + 6 y ) ^ { 2 } }$

Solve the problem. -Find the point on the paraboloid $z = 2 - x ^ { 2 } - y ^ { 2 }$ that is closest to the point $( 1,1,2 )$ .

Give an appropriate answer. -Given the function $f ( x , y , z )$ and the positive number $\varepsilon$ as in the formal definition of a limit, find a positive number $\delta$ as in the definition that insures $| f ( x , y , z ) - f ( 0,0,0 ) | < \varepsilon$ . $f ( x , y , z ) = \sin ^ { 2 } x + \sin ^ { 2 } y + \sin ^ { 2 } z ; \varepsilon = 0.12$

Solve the problem. -About how much will $f ( x , y , z ) = \ln ( - 7 x + 10 y + 9 z )$ change if the point $( x , y , z )$ moves from $( - 1 , - 8,3 )$ a distance of $\mathrm { ds } = \frac { 1 } { 10 }$ unit in the direction of $12 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$ ?

Solve the problem. -Find the distance from the point (1, -1, 2) to the plane x + y - z = 3.

Find the requested partial derivative. - $( \partial w / \partial x ) _ { y }$ if $w = x ^ { 3 } + y ^ { 3 } + z ^ { 3 } + 18 x y z$ and $z = x ^ { 2 } + y ^ { 2 }$

Find the linearization of the function at the given point. - $f ( x , y , z ) = \ln ( 6 x - 10 y - 4 z ) \text { at } \left( \frac { 1 } { 6 } , - \frac { 1 } { 10 } , - \frac { 1 } { 4 } \right)$