Exam 14: Partial Derivatives

The ellipsoid $8 x ^ { 2 } + 3 y ^ { 2 } + z ^ { 2 } = 14$ intersects the plane $y = 2$ in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1, 2, 2).

Find the equation of the normal line to the given surface at the specified point. $2 x ^ { 2 } + 8 y ^ { 2 } + 3 z ^ { 2 } = 235 , ( 4,4,5 )$

Let $z = 4 x ^ { 2 } + 2 y ^ { 2 }$ and suppose that $( x , y )$ changes from $( 2 , - 1 )$ to $( 2.01 , - 0.98 )$ (a) Compute $\Delta z .$ (b) Compute $d z$

The height of a hill (in feet) is given by $h ( x , y ) = 30 \left( 14 - 5 x ^ { 2 } - 2 y ^ { 2 } + 3 x y + 30 x - 18 y \right)$ where x is the distance (in miles) east and y is the distance (in miles) north of your cabin. If you are at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the height of the hill (a) in a northerly direction and (b) in an easterly direction?

Use implicit differentiation to find $\frac { \partial z } { \partial x }$ $x ^ { 4 } y + x z + y z ^ { 2 } = 7$

Find the directional derivative of $f ( x , y ) = 2 \sqrt { x } - y ^ { 3 }$ at the point (1, 3) in the direction toward the point (3, 1). Select the correct answer.

Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s). $f ( x , y ) = x y z ; \quad x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 15$

Find the first partial derivatives of the function $z = \ln \left( e ^ { 2 x } + y ^ { 4 } \right)$ .

Find three positive real numbers whose sum is 388 and whose product is as large as possible.

A cardboard box without a lid is to have a volume of $5324$ cm $2$ . Find the dimensions that minimize the amount of cardboard used.

Find the first partial derivatives of the function $f ( x , y , z ) = 2 x ^ { 3 } + 9 x y + 3 y z - z ^ { 5 }$

Find three positive numbers whose sum is $400$ and whose product is a maximum.

Use differentials to estimate the amount of metal in a closed cylindrical can that is 12 cm high and 8 cm in diameter if the metal in the top and bottom is 0.09 cm thick and the metal in the sides is 0.01 cm thick. (rounded to the nearest hundredth.)

Describe the level surfaces of the function $f ( x , y , z ) = 9 x + 7 y - 2 z + 7$ .

Find the absolute extrema of the function $f ( x , y ) = x ^ { 2 } + 3 y ^ { 2 } + 4 x + 10$ on the region bounded by the disk defined by $\left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 9 \right\}$ .

Find and classify the relative extrema and saddle points of the function $f ( x , y ) = x ^ { 2 } + 2 y ^ { 2 } + x ^ { 2 } y + 13$ .

Find the first partial derivatives of the function $f ( x , y ) = 9 x ^ { 2 } - 7 x y + 8 y ^ { 2 }$

A contour map for a function f is shown. Use it to estimate the value of $f ( - 1,2 )$ .

Use Lagrange multipliers to find the maximum value of the function subject to the given constraints. $f ( x , y , z ) = x + 2 y , x + y + z = 1 , y ^ { 2 } + z ^ { 2 } = 4$

Determine where the function $f ( x , y ) = \frac { 6 x y } { 9 x + 4 y - 1 }$ is continuous.