Exam 15: Functions of Several Variables

Your on-line bookstore is in direct competition with Amazon.com, BN.com, and Borders.com. Your company's daily revenue in dollars is given by $R ( x , y , z ) = 25,000 - 0.02 x - 0.02 y - 0.01 z + 0.00005 y z$ Where x, y, and z are the online daily revenues of Amazon.com, BN.com, and Borders.com, respectively. If Amazon.com and BN.com each show a daily revenue of $1,000, give an equation showing how your daily revenue depends on that of Borders.com.

(Multiple Choice)

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Question 21

Locate the maximum of the function. $f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 8 x \right) }$ Express your answer as an ordered triple. If there is more than one answer, separate them by commas.

(Essay)

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Question 22

Compute the integral. $\int _ { - 10 } ^ { 5 } \int _ { y - 5 } ^ { y + 5 } e ^ { x + y } \mathrm {~d} x \mathrm {~d} y$

(Multiple Choice)

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Question 23

Classify the highlighted value in the table.

(Multiple Choice)

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Question 24

Your latest CD - ROM drive is expected to sell between $q = 190,000 - p ^ { 2 }$ and $q = 210,000 - p ^ { 2 }$ units if priced at p. You plan to set the price between $200 and $300. What are the maximum and minimum possible revenues you can make What is the average of all the possible revenues you can make Please enter your answer as three numbers, separated by commas.

(Short Answer)

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Question 25

A productivity model at the Handy Gadget Company is $P = 13,000 x ^ { 0.1 } y ^ { 0.9 }$ Where P is the number of gadgets the company turns out per month, x is the number of employees at the company, and y is the monthly operating budget in thousands of dollars. Because the company hires part - time workers, it uses anywhere between 44 and 54 workers each month, and its operating budget varies from $12,000 to $14,000 per month. What is the average of the possible numbers gadgets it can turn out per month (Round the answer to the nearest 1,000 gadgets.)

(Multiple Choice)

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Question 26

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

(Multiple Choice)

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Question 27

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by $C ( x , y ) = 210,000 + 5,200 x + 3,400 y - 20 x y$ Compute the marginal cost of manufacturing cars at a production level of 14 cars and 13 trucks. $ __________

(Short Answer)

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Question 28

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 6 x ^ { 3 } - y ^ { 3 } + x - 12$

(Multiple Choice)

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Question 29

Calculate $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 8 x ^ { 2 } y$

(Multiple Choice)

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Question 30

Classify each labeled point on the graph. Choose the correct letter for each question. -a relative minimum

(Multiple Choice)

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Question 31

Calculate $\frac { \partial f } { \partial x }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 2,3 ) }$ when defined. $f ( x , y ) = 1,140 - 6 x + 7 y$ Enter your answers, separated by a comma.

(Short Answer)

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Question 32

Choose the correct letter for each question. -linear

(Multiple Choice)

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Question 33

Solve the given problem by using substitution. Find the minimum value of $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4$ subject to $y = 2 x$ .

(Multiple Choice)

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Question 34

Graph the function. $f ( x , y ) = 3 - \sqrt { x ^ { 2 } - y ^ { 2 } }$

(Multiple Choice)

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Question 35

Suppose your newspaper is trying to decide between two competing desktop publishing software packages, Macro Publish and Turbo Publish. You estimate that if you purchase x copies of Macro Publish and y copies of Turbo Publish, your company's daily productivity will be $U ( x , y ) = 8 x ^ { 0.9 } y ^ { 0.1 } + x$ Where U (x, y) is measured in pages per day (U is called a utility function.) If $x = y = 9$ , by approximately how many pages per day will productivity increase if x is increased by one unit

(Multiple Choice)

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Question 36

Evaluate $g ( - 2 , - 2,1 )$ for $g ( x , y , z ) = \frac { 9 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

(Multiple Choice)

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Question 37

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by $C ( x , y ) = 280,000 + 5,800 x + 4,000 y - 10 x y$ Compute the marginal cost of manufacturing cars at a production level of 20 cars and 10 trucks.

(Multiple Choice)

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Question 38

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

(Multiple Choice)

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Question 39

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

(Multiple Choice)

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Question 40

Locate the maximum of the function. $f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 8 x \right) }$ Express your answer as an ordered triple. If there is more than one answer, separate them by commas.

Compute the integral. $\int _ { - 10 } ^ { 5 } \int _ { y - 5 } ^ { y + 5 } e ^ { x + y } \mathrm {~d} x \mathrm {~d} y$

Classify the highlighted value in the table.

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

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by $C ( x , y ) = 210,000 + 5,200 x + 3,400 y - 20 x y$ Compute the marginal cost of manufacturing cars at a production level of 14 cars and 13 trucks. $ __________

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 6 x ^ { 3 } - y ^ { 3 } + x - 12$

Calculate $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 8 x ^ { 2 } y$

Classify each labeled point on the graph. Choose the correct letter for each question. -a relative minimum

Calculate $\frac { \partial f } { \partial x }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 2,3 ) }$ when defined. $f ( x , y ) = 1,140 - 6 x + 7 y$ Enter your answers, separated by a comma.

Choose the correct letter for each question. -linear

Solve the given problem by using substitution. Find the minimum value of $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4$ subject to $y = 2 x$ .

Graph the function. $f ( x , y ) = 3 - \sqrt { x ^ { 2 } - y ^ { 2 } }$

Evaluate $g ( - 2 , - 2,1 )$ for $g ( x , y , z ) = \frac { 9 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by $C ( x , y ) = 280,000 + 5,800 x + 4,000 y - 10 x y$ Compute the marginal cost of manufacturing cars at a production level of 20 cars and 10 trucks.

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

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

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

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Exam 15: Functions of Several Variables

Locate the maximum of the function. ​ f(x,y)=e−(x2+y2+8x)f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 8 x \right) }f(x,y)=e−(x2+y2+8x) ​ Express your answer as an ordered triple. If there is more than one answer, separate them by commas.

Compute the integral. ∫−105∫y−5y+5ex+y dx dy\int _ { - 10 } ^ { 5 } \int _ { y - 5 } ^ { y + 5 } e ^ { x + y } \mathrm {~d} x \mathrm {~d} y∫−105​∫y−5y+5​ex+y dx dy

Classify the highlighted value in the table. ​

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

Calculate ∂f∂x\frac { \partial f } { \partial x }∂x∂f​ , and ∂f∂y\frac { \partial f } { \partial y }∂y∂f​ when defined. f(x,y)=6x3−y3+x−12f ( x , y ) = 6 x ^ { 3 } - y ^ { 3 } + x - 12f(x,y)=6x3−y3+x−12

Calculate ∂f∂x\frac { \partial f } { \partial x }∂x∂f​ and ∂f∂y\frac { \partial f } { \partial y }∂y∂f​ when defined. f(x,y)=8x2yf ( x , y ) = 8 x ^ { 2 } yf(x,y)=8x2y

Classify each labeled point on the graph. ​ ​ Choose the correct letter for each question. ​ -a relative minimum

Choose the correct letter for each question. -linear

Solve the given problem by using substitution. Find the minimum value of f(x,y,z)=x2+y2+z2−4f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4f(x,y,z)=x2+y2+z2−4 subject to y=2xy = 2 xy=2x .

Graph the function. f(x,y)=3−x2−y2f ( x , y ) = 3 - \sqrt { x ^ { 2 } - y ^ { 2 } }f(x,y)=3−x2−y2​

Evaluate g(−2,−2,1)g ( - 2 , - 2,1 )g(−2,−2,1) for g(x,y,z)=9xyzx2+y2+z2g ( x , y , z ) = \frac { 9 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }g(x,y,z)=x2+y2+z29xyz​

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by C(x,y)=280,000+5,800x+4,000y−10xyC ( x , y ) = 280,000 + 5,800 x + 4,000 y - 10 x yC(x,y)=280,000+5,800x+4,000y−10xy Compute the marginal cost of manufacturing cars at a production level of 20 cars and 10 trucks.

Solve the given problem by using substitution. Find the maximum value of f(x,y,z)=3−x2−x−y2+y−z2+zf ( x , y , z ) = 3 - x ^ { 2 } - x - y ^ { 2 } + y - z ^ { 2 } + zf(x,y,z)=3−x2−x−y2+y−z2+z subject to y=3xy = 3 xy=3x .

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

Locate the maximum of the function. $f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 8 x \right) }$ Express your answer as an ordered triple. If there is more than one answer, separate them by commas.

Compute the integral. $\int _ { - 10 } ^ { 5 } \int _ { y - 5 } ^ { y + 5 } e ^ { x + y } \mathrm {~d} x \mathrm {~d} y$

Classify the highlighted value in the table.

Calculate $\frac { \partial f } { \partial x }$ , and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 6 x ^ { 3 } - y ^ { 3 } + x - 12$

Calculate $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ when defined. $f ( x , y ) = 8 x ^ { 2 } y$

Classify each labeled point on the graph. Choose the correct letter for each question. -a relative minimum

Solve the given problem by using substitution. Find the minimum value of $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4$ subject to $y = 2 x$ .

Graph the function. $f ( x , y ) = 3 - \sqrt { x ^ { 2 } - y ^ { 2 } }$

Evaluate $g ( - 2 , - 2,1 )$ for $g ( x , y , z ) = \frac { 9 x y z } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by $C ( x , y ) = 280,000 + 5,800 x + 4,000 y - 10 x y$ Compute the marginal cost of manufacturing cars at a production level of 20 cars and 10 trucks.

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

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