Exam 14: Partial Derivatives

Use Lagrange multipliers to find the maximum and minimum values of the function $f ( x , y , z ) = 5 x - y - 5 z$ subject to the constraints $x + 2 y - z = 0$ and $x ^ { 2 } + 4 y ^ { 2 } = 1$ .

Find the differential of the function $z = e ^ { 3 x } \sin 5 y$

Let $g ( r , s , t ) = r e ^ { s / t }$ Find $g \left( 8 , \ln 3 , \frac { 1 } { 2 } \right)$

Find the domain and range of the function $f ( x , y , z ) = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 } }$ .

Evaluate the gradient of f at the point P. $f ( x , y , z ) = x y ^ { 2 } z ^ { 3 } , \quad P ( - 1,3 , - 1 )$

Find the absolute minimum value of the function $f ( x , y ) = 6 + 3 x y - 2 x - 4 y$ on the set D. D is the region bounded by the parabola $y = x ^ { 2 }$ and the line $y = 4$

Use partial derivatives to find the implicit derivative $\frac { d y } { d x }$ $8 x ^ { 2 } + 9 \sqrt { x y } - 7 y = 5$

Find the equation of the tangent plane to the given surface at the specified point. $3 x ^ { 2 } + 3 y ^ { 2 } + 8 z ^ { 2 } = 353 , \quad ( 3,6,5 )$

Find the indicated partial derivative. $u = x ^ { e } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$

Find the linearization L(x, y) of the function at the given point. $f ( x , y ) = x \sqrt { y } , ( - 5,25 )$ Round the answers to the nearest hundredth.

Suppose that over a certain region of space the electrical potential V is given by $V ( x , y , z ) = 8 x ^ { 2 } - 7 x y + 7 x y z$ . Find the rate of change of the potential at $( - 1,1 , - 1 )$ in the direction of the vector $\mathbf { v } = 8 \mathbf { i } + 10 \mathbf { j } - 8 \mathbf { k }$ .

Find $f _ { y } ( - 24,8 )$ for $f ( x , y ) = \sin ( 4 x + 12 y )$ .