Question 1

Sketch the graph of the function.   $f ( x , y ) = 3 x + y - 3$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 2

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   $q _ { 1 } \left( p _ { 1 } , p _ { 2 } \right) = 200,000 - 100 p _ { 1 } + 10 p _ { 2 }$  
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.

Accepted Answer

A) $874 for Ultra Mini and $1,033 for Big Stack 
B)  $1,033 for Ultra Mini and $874 for Big Stack 
C)  $1,186 for Ultra Mini and $859 for Big Stack 
D)  $1,086 for Ultra Mini and $859 for Big Stack 
E)  $859 for Ultra Mini and $1,086 for Big Stack 
A) $874 for Ultra Mini and $1,033 for Big Stack 
B)  $1,033 for Ultra Mini and $874 for Big Stack 
C)  $1,186 for Ultra Mini and $859 for Big Stack 
D)  $1,086 for Ultra Mini and $859 for Big Stack 
E)  $859 for Ultra Mini and $1,086 for Big Stack

Question 3

Locate the local minimum of the function.
​ $f ( x , y ) = x y + \frac { 3 } { x } + \frac { 3 } { y }$ ​
Express your answer as an ordered triple.

Accepted Answer

The answer of Locate the local minimum of the function.
​...

Question 4

Use Lagrange Multipliers to solve the problem.  
Find the maximum value of $f ( x , y ) = x y$ subject to $x + 2 y = 80$ .

Accepted Answer

A)  $f _ { \max } = 800$ 
B)  $f _ { \max } = 40$ 
C)  $f _ { \max } = 400$ 
D)  $f _ { \max } = 2$ 
E)  $f _ { \max } = 10$ 
A)  $f _ { \max } = 800$ 
B)  $f _ { \max } = 40$ 
C)  $f _ { \max } = 400$ 
D)  $f _ { \max } = 2$ 
E)  $f _ { \max } = 10$

Question 5

For the function   $f ( x , y ) = 4 + \sqrt { x ^ { 2 } + y ^ { 2 } }$  
Show the cross section at $z = 6$ .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 6

The US Postal Service (USPS) will accept only packages with length plus girth no more than 120 inches. (See the figure.)      
What are the dimensions of the largest volume package the USPS will accept

Accepted Answer

A)  $24 \times 24 \times 24$ inches 
B)  $24 \times 24 \times 48$ inches 
C)  $20 \times 20 \times 40$ inches 
D)  $24 \times 48 \times 48$ inches 
E)  $40 \times 40 \times 40$ inches 
A)  $24 \times 24 \times 24$ inches 
B)  $24 \times 24 \times 48$ inches 
C)  $20 \times 20 \times 40$ inches 
D)  $24 \times 48 \times 48$ inches 
E)  $40 \times 40 \times 40$ inches

Question 7

Graph the function.   $f ( x , y ) = 1.5 e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) }$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 8

Find the point on the given plane closest to $( 1,1,0 )$ .
​ $2 x - 2 y - 5 z + 66 = 0$ ​
NOTE: Please enter your answer in the form $( x , y , z )$ .

Accepted Answer

The answer of Find the point on the given plane...

Question 9

Locate the local minimum of the function.   $f ( x , y ) = x y + \frac { 4 } { x } + \frac { 4 } { y }$

Accepted Answer

A)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 64 ^ { \frac { 2 } { 3 } } \right)$ 
B)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 12 ^ { \frac { 2 } { 3 } } \right)$ 
C)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 2 } { 3 } } \right)$ 
D)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 4 ^ { \frac { 2 } { 3 } } \right)$ 
E)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 4 ^ { \frac { 1 } { 3 } } \right)$ 
A)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 64 ^ { \frac { 2 } { 3 } } \right)$ 
B)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 12 ^ { \frac { 2 } { 3 } } \right)$ 
C)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 2 } { 3 } } \right)$ 
D)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 4 ^ { \frac { 2 } { 3 } } \right)$ 
E)  $\left( 4 ^ { \frac { 1 } { 3 } } , 4 ^ { \frac { 1 } { 3 } } , 3 \cdot 4 ^ { \frac { 1 } { 3 } } \right)$

Question 10

Cobb-Douglas productivity functions have the form   $p ( x , y ) = k x ^ { a } y ^ { 1 - a }$  
Where P stands for the number of items produced per year, x is the number of employees, and y is the annual operating budget. The numbers k and a are constants that depend on the situation we are looking at, with $0 \leq a \leq 1$ .
 
How many items will be produced per year by a company that has 150 employees and an annual operating budget of $80,000 if $k = 500$ and $a = 0.4$   (Round your answer to one significant digit.)

Accepted Answer

A) 3245342.2 
B)  4,000,000 
C)  4245342.2 
D)  1,500,000 
E)  3,000,000 
A) 3245342.2 
B)  4,000,000 
C)  4245342.2 
D)  1,500,000 
E)  3,000,000

Question 11

Calculate $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ when defined.   $f ( x , y , z ) = - \frac { 2 } { x + y + z ^ { 2 } }$

Accepted Answer

A)  $\frac { \partial f } { \partial x } = - \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial y } = - \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 2 } { 2 z \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ 
B)  $\frac { \partial f } { \partial x } = \frac { 2 } { x + y + z ^ { 2 } }$ , $\frac { \partial f } { \partial y } = \frac { 2 } { x + y + z ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 4 z } { x + y + z ^ { 2 } }$ 
C)  $\frac { \partial f } { \partial x } = \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial y } = \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 4 z } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ 
D)  $\frac { \partial f } { \partial x } = \frac { 2 x } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial y } = \frac { 2 y } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 4 z } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ 
E)  $\frac { \partial f } { \partial x } = \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial y } = \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 4 z } { ( x + y + z ) ^ { 2 } }$ 
A)  $\frac { \partial f } { \partial x } = - \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial y } = - \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 2 } { 2 z \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ 
B)  $\frac { \partial f } { \partial x } = \frac { 2 } { x + y + z ^ { 2 } }$ , $\frac { \partial f } { \partial y } = \frac { 2 } { x + y + z ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 4 z } { x + y + z ^ { 2 } }$ 
C)  $\frac { \partial f } { \partial x } = \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial y } = \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 4 z } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ 
D)  $\frac { \partial f } { \partial x } = \frac { 2 x } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial y } = \frac { 2 y } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 4 z } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ 
E)  $\frac { \partial f } { \partial x } = \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial y } = \frac { 2 } { \left( x + y + z ^ { 2 } \right) ^ { 2 } }$ , $\frac { \partial f } { \partial z } = \frac { 4 z } { ( x + y + z ) ^ { 2 } }$

Question 12

Graph the function.   $f ( x , y ) = 4 - \left( x ^ { 2 } - y ^ { 2 } \right)$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 13

Let $f ( x , y , z ) = 27 - 1.4 x + 6.2 y + 10 z$ .
 
By what value is $f$ increased if $y$ is increased by 1

Accepted Answer

The answer of Let \(f ( x , y ,...

Question 14

For the function   $x ^ { 2 } \sqrt { 4 + x y }$  
Find $p ( 3,13 )$ .
 
Express the answer as a decimal rounded to four decimal places.

Accepted Answer

A) 11.3578 
B)  19.6723 
C)  27.3578 
D)  56.2050 
E)  59.0169 
A) 11.3578 
B)  19.6723 
C)  27.3578 
D)  56.2050 
E)  59.0169

Question 15

Calculate $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ when defined.   $f ( x , y , z ) = x ^ { 0.6 } y ^ { 0.1 } z ^ { 0.6 }$

Accepted Answer

A)  $\frac { \partial f } { \partial x } = 0.6 x ^ { - 0.4 } y ^ { 0.1 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.9 } z ^ { 0.6 }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.1 } z ^ { - 0.4 }$ 
B)  $\frac { \partial f } { \partial x } = 0.6 x ^ { - 0.4 } y ^ { 0.1 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.1 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.1 } z ^ { - 0.6 }$ 
C)  $\frac { \partial f } { \partial x } = 0.6 x ^ { - 0.4 } y ^ { 0.1 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.4 } z ^ { 0.1 }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.6 } z ^ { - 0.9 }$ 
D)  $\frac { \partial f } { \partial x } = 0.1 x ^ { - 0.9 } y ^ { 0.1 } z ^ { 0.6 }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.4 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.1 } z ^ { - 0.4 }$ 
E)  $\frac { \partial f } { \partial x } = 0.6 x ^ { - 0.4 } y ^ { 0.9 } z ^ { 0.4 }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.9 } z ^ { 0.6 }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.1 } z ^ { - 0.4 }$ 
A)  $\frac { \partial f } { \partial x } = 0.6 x ^ { - 0.4 } y ^ { 0.1 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.9 } z ^ { 0.6 }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.1 } z ^ { - 0.4 }$ 
B)  $\frac { \partial f } { \partial x } = 0.6 x ^ { - 0.4 } y ^ { 0.1 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.1 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.1 } z ^ { - 0.6 }$ 
C)  $\frac { \partial f } { \partial x } = 0.6 x ^ { - 0.4 } y ^ { 0.1 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.4 } z ^ { 0.1 }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.6 } z ^ { - 0.9 }$ 
D)  $\frac { \partial f } { \partial x } = 0.1 x ^ { - 0.9 } y ^ { 0.1 } z ^ { 0.6 }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.4 z ^ { 0.6 } }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.1 } z ^ { - 0.4 }$ 
E)  $\frac { \partial f } { \partial x } = 0.6 x ^ { - 0.4 } y ^ { 0.9 } z ^ { 0.4 }$ , $\frac { \partial f } { \partial x } = 0.1 x ^ { 0.6 } y ^ { - 0.9 } z ^ { 0.6 }$ , $\frac { \partial f } { \partial z } = 0.6 x ^ { 0.6 } y ^ { 0.1 } z ^ { - 0.4 }$

Question 16

The town of West Podunk is shaped like a rectangle 80 miles from west to east and 90 miles from north to south (see the figure). It has a population density of
  $P ( x , y ) = e ^ { - 0.1 ( x + y ) }$  
hundred people per square mile x miles east and y miles north of the southwest corner of town. What is the total population of the town

Please enter your answer as a number without the units.

Accepted Answer

The answer of The town of West Podunk is shaped...

Question 17

Sketch the graph of the function.   $f ( x , y ) = 7 - y$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 18

Cobb-Douglas productivity functions have the form
  $p ( x , y ) = k x ^ { a } y ^ { 1 - a }$  
where P stands for the number of items produced per year, x is the number of employees, and y is the annual operating budget. The numbers k and a are constants that depend on the situation we are looking at, with $0 \leq a \leq 1$ .
 
How many items will be produced per year by a company that has 250 employees and an annual operating budget of $100,000 if $k = 1,000$ and $a = 0.5$   (Round your answer to one significant digit.)

Accepted Answer

The answer of Cobb-Douglas productivity functions have the form
 ...

Question 19

Find the average value of the given function over the indicated domain.   $f ( x , y ) = y$

Accepted Answer

A)  $- 2$ 
B)  $\frac { 4 } { 3 }$ 
C)  $- \frac { 2 } { 3 }$ 
D)  2 
E)  $\frac { 2 } { 3 }$ 
A)  $- 2$ 
B)  $\frac { 4 } { 3 }$ 
C)  $- \frac { 2 } { 3 }$ 
D)  2 
E)  $\frac { 2 } { 3 }$

Question 20

Sketch the graph of the function.   $f ( x , y ) = 4 ( x - y )$

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Sketch the graph of the function. $f ( x , y ) = 3 x + y - 3$

Locate the local minimum of the function. $f ( x , y ) = x y + \frac { 3 } { x } + \frac { 3 } { y }$ Express your answer as an ordered triple.

Use Lagrange Multipliers to solve the problem. Find the maximum value of $f ( x , y ) = x y$ subject to $x + 2 y = 80$ .

For the function $f ( x , y ) = 4 + \sqrt { x ^ { 2 } + y ^ { 2 } }$ Show the cross section at $z = 6$ .

The US Postal Service (USPS) will accept only packages with length plus girth no more than 120 inches. (See the figure.) What are the dimensions of the largest volume package the USPS will accept

Graph the function. $f ( x , y ) = 1.5 e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) }$

Find the point on the given plane closest to $( 1,1,0 )$ . $2 x - 2 y - 5 z + 66 = 0$ NOTE: Please enter your answer in the form $( x , y , z )$ .

Locate the local minimum of the function. $f ( x , y ) = x y + \frac { 4 } { x } + \frac { 4 } { y }$

Calculate $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ when defined. $f ( x , y , z ) = - \frac { 2 } { x + y + z ^ { 2 } }$

Graph the function. $f ( x , y ) = 4 - \left( x ^ { 2 } - y ^ { 2 } \right)$

Let $f ( x , y , z ) = 27 - 1.4 x + 6.2 y + 10 z$ . By what value is $f$ increased if $y$ is increased by 1

For the function $x ^ { 2 } \sqrt { 4 + x y }$ Find $p ( 3,13 )$ . Express the answer as a decimal rounded to four decimal places.

Calculate $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ when defined. $f ( x , y , z ) = x ^ { 0.6 } y ^ { 0.1 } z ^ { 0.6 }$

Sketch the graph of the function. $f ( x , y ) = 7 - y$

Find the average value of the given function over the indicated domain. $f ( x , y ) = y$

Sketch the graph of the function. $f ( x , y ) = 4 ( x - y )$

Functions and Applications

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Exam 15: Functions of Several Variables

Sketch the graph of the function. f(x,y)=3x+y−3f ( x , y ) = 3 x + y - 3f(x,y)=3x+y−3

Locate the local minimum of the function. ​ f(x,y)=xy+3x+3yf ( x , y ) = x y + \frac { 3 } { x } + \frac { 3 } { y }f(x,y)=xy+x3​+y3​ ​ Express your answer as an ordered triple.

Use Lagrange Multipliers to solve the problem. Find the maximum value of f(x,y)=xyf ( x , y ) = x yf(x,y)=xy subject to x+2y=80x + 2 y = 80x+2y=80 .

For the function f(x,y)=4+x2+y2f ( x , y ) = 4 + \sqrt { x ^ { 2 } + y ^ { 2 } }f(x,y)=4+x2+y2​ Show the cross section at z=6z = 6z=6 .

The US Postal Service (USPS) will accept only packages with length plus girth no more than 120 inches. (See the figure.) What are the dimensions of the largest volume package the USPS will accept

Graph the function. f(x,y)=1.5e−(x2+y2)f ( x , y ) = 1.5 e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) }f(x,y)=1.5e−(x2+y2)

Find the point on the given plane closest to (1,1,0)( 1,1,0 )(1,1,0) . ​ 2x−2y−5z+66=02 x - 2 y - 5 z + 66 = 02x−2y−5z+66=0 ​ NOTE: Please enter your answer in the form (x,y,z)( x , y , z )(x,y,z) .

Locate the local minimum of the function. f(x,y)=xy+4x+4yf ( x , y ) = x y + \frac { 4 } { x } + \frac { 4 } { y }f(x,y)=xy+x4​+y4​

Calculate ∂f∂x\frac { \partial f } { \partial x }∂x∂f​ , ∂f∂y\frac { \partial f } { \partial y }∂y∂f​ , and ∂f∂z\frac { \partial f } { \partial z }∂z∂f​ when defined. f(x,y,z)=−2x+y+z2f ( x , y , z ) = - \frac { 2 } { x + y + z ^ { 2 } }f(x,y,z)=−x+y+z22​

Graph the function. f(x,y)=4−(x2−y2)f ( x , y ) = 4 - \left( x ^ { 2 } - y ^ { 2 } \right)f(x,y)=4−(x2−y2)

Let f(x,y,z)=27−1.4x+6.2y+10zf ( x , y , z ) = 27 - 1.4 x + 6.2 y + 10 zf(x,y,z)=27−1.4x+6.2y+10z . By what value is fff increased if yyy is increased by 1

For the function x24+xyx ^ { 2 } \sqrt { 4 + x y }x24+xy​ Find p(3,13)p ( 3,13 )p(3,13) . Express the answer as a decimal rounded to four decimal places.

Calculate ∂f∂x\frac { \partial f } { \partial x }∂x∂f​ , ∂f∂y\frac { \partial f } { \partial y }∂y∂f​ , and ∂f∂z\frac { \partial f } { \partial z }∂z∂f​ when defined. f(x,y,z)=x0.6y0.1z0.6f ( x , y , z ) = x ^ { 0.6 } y ^ { 0.1 } z ^ { 0.6 }f(x,y,z)=x0.6y0.1z0.6

Sketch the graph of the function. f(x,y)=7−yf ( x , y ) = 7 - yf(x,y)=7−y

Find the average value of the given function over the indicated domain. f(x,y)=yf ( x , y ) = yf(x,y)=y

Sketch the graph of the function. f(x,y)=4(x−y)f ( x , y ) = 4 ( x - y )f(x,y)=4(x−y)

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Techniques of Differentiation

Applications of the Derivative

The Integral

Further Integration Techniques and Applications of the Integral

Trigonometric Models

Filters

Sketch the graph of the function. $f ( x , y ) = 3 x + y - 3$

Locate the local minimum of the function. $f ( x , y ) = x y + \frac { 3 } { x } + \frac { 3 } { y }$ Express your answer as an ordered triple.

Use Lagrange Multipliers to solve the problem. Find the maximum value of $f ( x , y ) = x y$ subject to $x + 2 y = 80$ .

For the function $f ( x , y ) = 4 + \sqrt { x ^ { 2 } + y ^ { 2 } }$ Show the cross section at $z = 6$ .

Graph the function. $f ( x , y ) = 1.5 e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) }$

Find the point on the given plane closest to $( 1,1,0 )$ . $2 x - 2 y - 5 z + 66 = 0$ NOTE: Please enter your answer in the form $( x , y , z )$ .

Locate the local minimum of the function. $f ( x , y ) = x y + \frac { 4 } { x } + \frac { 4 } { y }$

Calculate $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ when defined. $f ( x , y , z ) = - \frac { 2 } { x + y + z ^ { 2 } }$

Graph the function. $f ( x , y ) = 4 - \left( x ^ { 2 } - y ^ { 2 } \right)$

Let $f ( x , y , z ) = 27 - 1.4 x + 6.2 y + 10 z$ . By what value is $f$ increased if $y$ is increased by 1

For the function $x ^ { 2 } \sqrt { 4 + x y }$ Find $p ( 3,13 )$ . Express the answer as a decimal rounded to four decimal places.

Calculate $\frac { \partial f } { \partial x }$ , $\frac { \partial f } { \partial y }$ , and $\frac { \partial f } { \partial z }$ when defined. $f ( x , y , z ) = x ^ { 0.6 } y ^ { 0.1 } z ^ { 0.6 }$

Sketch the graph of the function. $f ( x , y ) = 7 - y$

Find the average value of the given function over the indicated domain. $f ( x , y ) = y$

Sketch the graph of the function. $f ( x , y ) = 4 ( x - y )$