Exam 15: Partial Derivatives

Provide an appropriate response. - $\text { Determine the point on the paraboloid } z = 9 x ^ { 2 } + 2 y ^ { 2 } \text { that is closest to the point } ( 57 , - 10,88 ) \text {. }$

Find an upper bound for the magnitude |E| of the error in the approximation f(x, y) ≈ L(x, y) at the given point over the given region R. - $f ( x , y , z ) = \tan ^ { - 1 } x y z$ at $( 7,7,7 ) ; R : | x - 7 | \leq 0.2 , | y - 7 | \leq 0.2 , | z - 7 | \leq 0.2$

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. - $\text { Find } \frac { \partial f } { \partial y } \text { at the point } ( - 10,1 ) : f ( x , y ) = 9 - 10 x y + 3 x y ^ { 2 }$

Use the limit definition of the partial derivative to compute the indicated partial derivative of the function at the specified point. - $\text { Find } \frac { \partial f } { \partial x } \text { at the point } ( 4 , - 4 ) : f ( x , y ) = 3 - 8 x y + 8 x y ^ { 2 }$

Find the derivative of the function at P0 in the direction of u. - $f ( x , y , z ) = 4 x + 8 y - 7 z , \quad P _ { 0 } ( - 7,10 , - 10 ) , \quad \mathbf { u } = 3 \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }$

Provide an appropriate response. -Which order of differentiation will calculate $\mathrm { f } _ { \mathrm { xy } }$ faster, $x$ first or $\mathrm { y }$ first? $f ( x , y ) = \frac { 1 } { g ( y ) } , g ( y ) \neq 0$

Answer the question. -The graph below shows the level curves of a differentiable function $f ( x , y )$ (thin curves) as well as the constraint $g ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } } - \frac { 3 } { 2 } = 0$ (thick circle). Using the concepts of the orthogonal gradient theorem and the method of Lagrange multipliers, estimate the coordinates corresponding to the constrained extrema of $f ( x , y )$ .

Find the limit. - $\lim _ { ( x , y ) \rightarrow ( 11,13 ) } \sqrt { \frac { 1 } { x y } }$

Find two paths of approach from which one can conclude that the function has no limit as (x, y) approaches (0, 0). - $\text { Show that } f ( x , y , z ) = e ^ { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } \text { is continuous at the origin. }$

Solve the problem. - $\text { Find parametric equations for the normal line to the surface } z = 4 x ^ { 2 } - 2 y ^ { 2 } \text { at the point } ( 2,1,14 ) \text {. }$

Find the limit. -

Find the extreme values of the function subject to the given constraint. - $f ( x , y , z ) = ( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z - 4 ) ^ { 2 } , x - y + 3 z = 7$

Find the absolute maxima and minima of the function on the given domain. - $f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 } \text { on the square } - 8 \leq x , y \leq 8$

Solve the problem. - $\text { Evaluate } \frac { d w } { d t } \text { at } t = \frac { 1 } { 2 } \pi \text { for the function } w = x ^ { 2 } - y ^ { 2 } - 3 x ; x = \operatorname { cost } , y = \sin t$

Match the surface show below to the graph of its level curves. -

Find the limit. - $\lim _ { ( x , y ) \rightarrow ( 1,4 ) } \left( \frac { 5 } { x } - \frac { 4 } { y } \right)$

Solve the problem. -Write an equation for the tangent line to the curve $y ^ { 2 } - x = 4$ at the point $( 1 , \sqrt { 5 } )$ .

Use Taylor's formula to find the requested approximation of f(x, y) near the origin. -Quadratic approximation to $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \mathrm { e } ^ { \mathrm { x } + 3 \mathrm { y } }$

Find the extreme values of the function subject to the given constraint. - $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , x + 2 y + 3 z = 6$

Find the requested partial derivative. - $( \partial w / \partial y ) _ { x } \text { if } w = 4 x + 5 y + 6 z \text { and } x + y = z$