Exam 13: Introduction to Functions of Several Variables

The average length of time that a customer waits in line for service is $W ( x , y ) = \frac { 1 } { x - y } , x > y$ where $y$ is the average arrival rate, written as the number of customers per unit of time, and $x$ is the average service rate, written in the same units. Evaluate $W ( 18,6 ) .$ Note: $x$ and $y$ are given as the number of customers per hour.

Find least squares regression line for the points shown in the graph is $y = \frac { 7 } { 10 } x + 2$ Calculate S, the sum of the squared errors.

Describe the level curves of the function. Sketch the level curves for the given c-values. $z = 6 - 2 x - 3 y , \quad c = 0,2,4,6$

Find and simplify $\frac { f ( x + \Delta x , y ) - f ( x , y ) } { \Delta x } \text { for the given function } f ( x , y ) = 7 x + 2 y ^ { 2 } .$

Find the first partial derivative for the function $f ( x , y , z ) = 5 x ^ { 2 } y - 3 x y z + 6 y z ^ { 2 } \text { with }$ respect to z.

Find the directional derivative of the function $g ( x , y , z ) = x y e ^ { z } \text { at } P ( 3,8,0 ) \text { in the }$ direction of $Q ( 0,0,0 )$ . Round your answer to two decimal places.

Suppose the temperature at any point $( x , y ) \text { in a steel plate is }$ $T = 500 - 0.6 x ^ { 2 } - 1.5 y ^ { 2 }$ where $x$ and $y$ are measured in meters. At the point $( 8,7 )$ find the rate of change of the temperature with respect to the distance moved along the plate in the direction of the $x$ -axis. Round your answer to one decimal place.

Find the partial derivative $f _ { y } \text { for the function } f ( x , y ) = x ^ { 2 } - 2 y ^ { 2 } + 7$

Suppose electrical power P is given by $P = \frac { E ^ { 2 } } { R } \text {, where } E \text { is voltage and } R \text { is }$ resistance. Approximate the maximum percent error in calculating the power if 140 volts are applied to a 3000-ohm resistor and the possible percent errors in measuring E and R are 5% and 8%, Respectively.

Find the limit. $\lim _ { ( x , y ) \rightarrow ( 0,0 ) } ( x + 5 y + 5 )$

Find the directional derivative of the function at P in the direction of $\overrightarrow { \mathbf { v } } \text {. }$ $f ( x , y , z ) = x y + y z + x z , P ( 1,1,1 ) , \quad \overrightarrow { \mathbf { v } } = 4 \hat { \mathbf { i } } + 5 \hat { \mathbf { j } } - 3 \hat { \mathbf { k } }$

The radius r and height h of a right circular cylinder are measured with possible errors of 5% and 1%, respectively. Approximate the maximum possible percent error in measuring The volume.

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function $f ( x , y ) \text { at the critical point }$ $\left( x _ { 0 } , y _ { 0 } \right) \text {, if } f _ { x x } \left( x _ { 0 } , y _ { 0 } \right) = 16 , f _ { y y } \left( x _ { 0 } , y _ { 0 } \right) = - 2 , f _ { x y } \left( x _ { 0 } , y _ { 0 } \right) = - 1$

Find and simplify the function $h ( x , y , z ) = \frac { x y } { z } \text { at the given value } ( 4,4,27 )$

Suppose a corporation manufactures candles at two locations. The cost of producing $x _ { 1 } \text { units at location } 1 \text { is } C _ { 1 } = 0.06 x _ { 1 } ^ { 2 } + 4 x _ { 1 } + 500 \text { and the cost of producing } x _ { 2 } \text { units at location } 2 \text { is }$ $C _ { 2 } = 0.05 x _ { 2 } ^ { 2 } + 4 x _ { 2 } + 400$ . The candles sell for $\$ 43$ per unit. Find the quantity that should be produced at each location to maximize the profit $P = 43 \left( x _ { 1 } + x _ { 2 } \right) - C _ { 1 } - C _ { 2 }$ .

Find three positive numbers x, y, and z whose sum is 24 and product is a maximum.

Determine the continuity of the function $f ( x , y ) = 11 \sin \frac { x } { y }$

Find the partial derivative $f _ { x } \text { for the function } f ( x , y ) = x ^ { 2 } - 8 y ^ { 2 } + 6 \text {. }$

Find the path of a heat-seeking particle placed at point $P ( 16,16 ) \text { on a metal plate }$ with a temperature field $T ( x , y ) = 401 - 6 x ^ { 2 } - 3 y ^ { 2 }$ .

Find the directional derivative of the function at P in the direction of $\overrightarrow { \mathbf { v } }$ $f ( x , y ) = 3 x - 2 x y + 7 y , P ( 1,9 ) , \quad \overrightarrow { \mathbf { v } } = \frac { 1 } { 2 } ( \hat { \mathbf { i } } + \sqrt { 3 } \hat { \mathbf { j } } )$