Exam 15: Functions of Several Variables

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 8,7 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,7 ) }$ when defined. ​ $f ( x , y ) = 1,120 - 17 x + 19 y + 3 x y$ ​ Enter your answers, separated by a comma.

Graph the function. $f ( x , y ) = 3 x - 5 y + 3$

Graph the plane. $z = 5$

Calculate $\frac { \partial ^ { 2 } f } { \partial y ^ { 2 } }$ , $\frac { \partial ^ { 2 } f } { \partial x y }$ and evaluate each at $( 8,8 )$ . ​ $f ( x , y ) = 1,070 - 20 x + 11 y + 9 x y$ ​ Enter your answers, separated by a comma.

Your weekly cost (in dollars) to manufacture x cars and y trucks is $C ( x , y ) = 270,000 + 6,000 x + 5,000 y$ What is the marginal cost of a car Please enter your answer as a number without the units.

Classify each labeled point on the graph. ​ ​ Choose the correct letter for each question. ​ -none

If positive electric charges of Q and q coulombs are situated at positions $( a , b , c )$ and $( x , y , z )$ , respectively, then the force of repulsion they experience is given by $F = K \frac { Q q } { ( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } + ( z - c ) ^ { 2 } }$ Where $K \approx 9 \times 10 ^ { 9 }$ , F is given in newtons, and all positions are measured in meters. Assume that a charge of 6 coulombs is situated at the origin and that a second charge of 12 coulombs is situated $( 6,6,4 )$ and moving in the y direction at 1.5 meters per second. How fast is the electrostatic force it experiences decreasing Round your answer to the nearest trillion.

​Calculate $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } }$ , $\frac { \partial ^ { 2 } f } { \partial y x }$ when defined. ​ $f ( x , y ) = 4 x ^ { 0.8 } y ^ { 0.3 }$ ​ Enter your answers, separated by a comma.

For the function $f ( x , y ) = 5 x ^ { 2 } + 5 y ^ { 2 }$ Show cross section at $z = 1$ .

Evaluate $g ( 0,0,1 )$ for $g ( x , y , z ) = \ln ( x + y + z )$

Evaluate $g ( - 2,1 , - 1 )$ for ​ $g ( x , y , z ) = \frac { 6 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 9 x ^ { 0.7 } y ^ { 0.9 }$

Calculate $\frac { \partial ^ { 2 } f } { \partial y ^ { 2 } }$ , $\frac { \partial ^ { 2 } f } { \partial x y }$ and evaluate each at $( 5,7 )$ . $f ( x , y ) = 950 - 16 x + 14 y + 6 x y$

The temperature at the point $( x , y )$ on the square with vertices (0, 0), (0, 2), (2, 0) and (2, 2) is given by ​ $T ( x , y ) = 2 x ^ { 2 } + 4 y ^ { 2 }$ ​ Find the hottest and coldest points on the square. ​ NOTE: Please enter your answers in the form $( x , y )$ , separated by commas.

Compute the integral. $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 2 - y } x \mathrm {~d} x \mathrm {~d} y$

Let $f ( x , y , z ) = 48.4 - 3.1 x + 7.7 y + 8.5 z$ . By what value is $f$ increased if $y$ is increased by 1

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by $C ( x , y ) = 300,000 + 6,500 x + 4,500 y$ Find $\frac { \partial C } { \partial x }$ , and $\frac { \partial C } { \partial y }$ .

Compute the integral. $\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 5 } \left( y e ^ { x } - x - y \right) \mathrm { d } x \mathrm {~d} y$

Find x and y values of the relative extrema of the function. $f ( x , y ) = x ^ { 4 } + 8 x y ^ { 2 } + 2 y ^ { 4 }$

Calculate $\frac { \partial f } { \partial x }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 8,2 ) }$ when defined. $f ( x , y ) = 1100 - 16 x + 11 y$