Exam 15: Functions of Several Variables

What point on the surface z is closest to the origin $z = x ^ { 2 } + y - 2$ Hint : Minimize the square of the distance from $( x , y , z )$ to the origin. NOTE: Please enter your answer(s) in the form $( x , y , z )$ . If there is more than one answer, separate them with commas.

Compute the integral. $\int _ { 0 } ^ { 9 } \int _ { - x ^ { 2 } } ^ { x ^ { 2 } } x \mathrm {~d} y \mathrm {~d} x$

Let $H = f _ { x x } ( a , b ) f _ { y y } ( a , b ) - f ^ { 2 } x y ( a , b )$ What condition on H guarantees that f has a relative extremum at the point $( a , b )$

Use Lagrange Multipliers to solve the problem. Find the maximum value of $f ( x , y ) = 3 x y$ subject to $x ^ { 2 } + y ^ { 2 } = 8$ .

Write the integral with the order of integration reversed (changing the limits of integration as necessary). $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 + y } } f ( x , y ) \mathrm { d } x \mathrm {~d} y$

Locate the local maximum of the function. $f ( x , y ) = x ^ { 2 } y - 8 x ^ { 2 } - 4 y ^ { 2 }$

Solve the given problem by using substitution. Find the maximum value of $f ( x , y , z ) = 8 - x ^ { 2 } - y ^ { 2 } - z ^ { 2 }$ subject to $z = 3 y$ .

Use the given tabular representation of the function $f$ to compute $f ( 2,4 ) - f ( 4,2 )$ . -5 197 185 -5 -9 191 100 -4 -20 201 225 -19 -19 216 235 -18

Use Lagrange Multipliers to solve the given problem. Find the maximum value of $f ( x , y ) = x y$ subject to $2 x + y = 60$ .

For the function $p ( x , y ) = x - 0.2 y + 0.44 x y$ Find $p ( 40,20 )$ .

Locate the saddle point of the function. $f ( x , y ) = x ^ { 2 } - 6 x + y - e ^ { y }$

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 4 coulombs is situated at the origin and that a second charge of 14 coulombs is situated $( 1,4,3 )$ and moving in the y direction at 1.4 meters per second. How fast is the electrostatic force it experiences decreasing NOTE: Please enter your answer without the units. Round the answer to the nearest trillion.

Your company manufactures two models of stereo speakers, the Ultra Mini and the Big Stack. Demand for each depends partly on the price of the other. If one is expensive then more people will buy the other. If p₁ is the price per pair of the Ultra Mini and p₂ is the price of the Big Stack, demand for the Ultra Mini is given by $q _ { 1 } \left( p _ { 1 } , p _ { 2 } \right) = 50,000 - 100 p _ { 1 } + 10 p _ { 2 }$ Where q₁ represents the number of pairs of Ultra Minis that will be sold in a year. The demand for the Big Stack is given by $q _ { 2 } \left( p _ { 1 } , p _ { 2 } \right) = 150,000 + 10 p _ { 1 } - 100 p _ { 2 }$ Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue. Round your answer to the nearest dollar.

Solve the given problem by using substitution. Minimize $S = x y + 25 x z + 5 y z$ subject to $x y z = 1$ with $x > 0$ , $y > 0$ , $z > 0$ .

Evaluate $f ( a , 2 )$ for $f ( x , y ) = x ^ { 2 } - y - x y + 5$

Compute the integral. $\int _ { 0 } ^ { 10 } \int _ { 4 - x } ^ { 25 - x } ( x + y ) ^ { \frac { 1 } { 2 } } \mathrm {~d} y \mathrm {~d} x$

How many critical points does the function have $f ( x , y ) = 2 x e ^ { y }$

Find $\iint _ { R } f ( x , y ) \mathrm { d } x \mathrm {~d} y$ , where $f ( x , y ) = x y ^ { 2 }$ and R is the indicated domain. (Remember that you often have a choice as to the order of integration.) $f ( y ) = \sqrt { 9 - y ^ { 2 } }$

Compute the integral. $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 8 } e ^ { x + y } \mathrm {~d} x \mathrm {~d} y$

Locate the local maximum of the function. $f ( x , y ) = e ^ { - \left( x ^ { 6 } + y ^ { 2 } \right) }$