Question 1

Graph the function.   $f ( x , y ) = 1.5 e ^ { - y }$

Accepted Answer

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

Question 2

For the function   $f ( x , y ) = 3 - \sqrt { x ^ { 2 } + y ^ { 2 } }$  
Show the cross section at $y = 0$ .

Accepted Answer

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

Question 3

Trans World Airlines (TWA) has a rule for checked baggage, "The total dimensions (length + width + height) may not exceed 57 inches for each bag". What is the volume of the largest volume bag you can check on a TWA flight 
  Enter your answer as a number without the units.

Accepted Answer

The answer of Trans World Airlines (TWA) has a rule...

Question 4

In the table below, classify each highlighted value.

-saddle point

Accepted Answer

A) 2 
B) 8 
C) 9 
D) 12 
A) 2 
B) 8 
C) 9 
D) 12

Question 5

Graph the function.   $f ( x , y ) = 3 + \left( x ^ { 2 } + y ^ { 2 } \right)$

Accepted Answer

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

Question 6

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

Accepted Answer

The answer of Evaluate $g ( 0,1,0 )$ for
​ \(g...

Question 7

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

Accepted Answer

A) 13 
B)  2,197 
C)    $\frac { 2,197 } { 6 }$ 
D)  113 
E)  $\frac { 169 } { 6 }$ 
A) 13 
B)  2,197 
C)    $\frac { 2,197 } { 6 }$ 
D)  113 
E)  $\frac { 169 } { 6 }$

Question 8

Classify each labeled point on the graph.   Choose the correct letter for each question.
-saddle point

Accepted Answer

A) Q 
B) P 
C) R 
A) Q 
B) P 
C) R

Question 9

Locate the critical point of the function.   $f ( x , y ) = 5 x ^ { 2 } + 2 y ^ { 2 } + 4$

Accepted Answer

A)  $( 5,2,137 )$ 
B)  $( 5,2,4 )$ 
C)  $( 0,0,4 )$ 
D)  $( 0,4,4 )$ 
E)  $( 0,0,0 )$ 
A)  $( 5,2,137 )$ 
B)  $( 5,2,4 )$ 
C)  $( 0,0,4 )$ 
D)  $( 0,4,4 )$ 
E)  $( 0,0,0 )$

Question 10

How many critical points does the following function have    $f ( x , y ) = 4 - \left( y ^ { 2 } + x ^ { 2 } \right)$

Accepted Answer

A) four 
B)  one 
C)  two 
D)  three 
E)  none 
A) four 
B)  one 
C)  two 
D)  three 
E)  none

Question 11

At what point on the given surface is the quantity $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ a minimum  (The method of Lagrange multipliers can be used here.)   $z = \left( 4 x ^ { 2 } + 10 x + y ^ { 2 } + 22 \right) ^ { \frac { 1 } { 2 } }$

Accepted Answer

A)  $( - 1,0,5 )$ 
B)  $( 1,0,4 )$ 
C)  $( - 1,0,4 )$ 
D)  $( - 1,0,6 )$ 
E)  $( - 3,0,3 )$ 
A)  $( - 1,0,5 )$ 
B)  $( 1,0,4 )$ 
C)  $( - 1,0,4 )$ 
D)  $( - 1,0,6 )$ 
E)  $( - 3,0,3 )$

Question 12

In the table below, classify each highlighted value.
 
-relative minimum

Accepted Answer

A) 2 
B) 8 
C) 9 
D) 12 
A) 2 
B) 8 
C) 9 
D) 12

Question 13

Trans World Airlines (TWA) has a rule for checked baggage, "The total dimensions (length + width + height) may not exceed 66 inches for each bag". What is the volume of the largest volume bag you can check on a TWA flight

Accepted Answer

A) 10,648 in.3 
B)  66 in.3 
C)  287,496 in.3 
D)  132 in.3 
E)  22 in.3 
A) 10,648 in.3 
B)  66 in.3 
C)  287,496 in.3 
D)  132 in.3 
E)  22 in.3

Question 14

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) }$ when defined.   $f ( x , y ) = x ^ { 2 } y ^ { 3 } - x ^ { 3 } y ^ { 2 } - x y$

Accepted Answer

A)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = - 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2048$ 
B)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = - 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = - 2048$ 
C)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 2044$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 1016$ 
D)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 1024$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2048$ 
E)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2044$ 
A)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = - 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2048$ 
B)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = - 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = - 2048$ 
C)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 2044$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 1016$ 
D)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 1024$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2048$ 
E)  $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) } = 1016$ , $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) } = 2044$

Question 15

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by   $C ( x , y ) = 270,000 + 6,800 x + 3,000 y - 200,000 e ^ { - 0.03 ( x + y ) }$  
What is the marginal cost of a car ? Of a truck ?

Accepted Answer

A) Marginal cost of a car is $6,800 + 6,000 e ^ { - 0.03 x }$ marginal cost of a truck is $3,000 + 6,000 e ^ { - 0.03 y }$ 
B)  Marginal cost of a car is $6,800 - 200,000 e ^ { - 0.03 ( x + y ) }$
Marginal cost of a truck is $3,000 - 200,000 e ^ { - 0.03 ( x + y ) }$ 
C)  Marginal cost of a car is $3,000 + 200,000 e ^ { 0.03 ( x + y ) }$
Marginal cost of a truck is $6,800 + 200,000 e ^ { 0.03 ( x + y ) }$ 
D)  Marginal cost of a car is $3,000 - 6,000 e ^ { 0.03 ( x + y ) }$
Marginal cost of a truck is $6,800 - 6,000 e ^ { 0.03 \{ x + y \} }$ 
E)  Marginal cost of a car is $6,800 + 6,000 e ^ { - 0.03 ( x + y ) }$
Marginal cost of a truck is $3,000 + 6,000 e ^ { - 0.03 ( x + y ) }$ 
A) Marginal cost of a car is $6,800 + 6,000 e ^ { - 0.03 x }$ marginal cost of a truck is $3,000 + 6,000 e ^ { - 0.03 y }$ 
B)  Marginal cost of a car is $6,800 - 200,000 e ^ { - 0.03 ( x + y ) }$
Marginal cost of a truck is $3,000 - 200,000 e ^ { - 0.03 ( x + y ) }$ 
C)  Marginal cost of a car is $3,000 + 200,000 e ^ { 0.03 ( x + y ) }$
Marginal cost of a truck is $6,800 + 200,000 e ^ { 0.03 ( x + y ) }$ 
D)  Marginal cost of a car is $3,000 - 6,000 e ^ { 0.03 ( x + y ) }$
Marginal cost of a truck is $6,800 - 6,000 e ^ { 0.03 \{ x + y \} }$ 
E)  Marginal cost of a car is $6,800 + 6,000 e ^ { - 0.03 ( x + y ) }$
Marginal cost of a truck is $3,000 + 6,000 e ^ { - 0.03 ( x + y ) }$

Question 16

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

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)    $\frac { e ^ { 8 } } { 3 }$ 
C)  $2 e ^ { 4 }$ 
D)  $\frac { 1 } { 2 } \left( e ^ { 4 } - 1 \right)$ 
E)  $e ^ { 4 }$ 
A)  $\frac { 1 } { 2 }$ 
B)    $\frac { e ^ { 8 } } { 3 }$ 
C)  $2 e ^ { 4 }$ 
D)  $\frac { 1 } { 2 } \left( e ^ { 4 } - 1 \right)$ 
E)  $e ^ { 4 }$

Question 17

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

Accepted Answer

A) 243 
B)  729 
C)  324 
D)  972 
E)  81 
A) 243 
B)  729 
C)  324 
D)  972 
E)  81

Question 18

Classify each labeled point on the graph.   Choose the correct letter for each question.
-local maximum

Accepted Answer

A) Q 
B) P 
C) R 
A) Q 
B) P 
C) R

Question 19

In the table below, classify each highlighted value.

-none

Accepted Answer

A) 2 
B) 8 
C) 9 
D) 12 
A) 2 
B) 8 
C) 9 
D) 12

Question 20

Evaluate $f ( a , - 3 )$ for
​ $f ( x , y ) = x ^ { 2 } - y - x y + 6$

Accepted Answer

The answer of Evaluate \(f ( a , - 3...

Graph the function. $f ( x , y ) = 1.5 e ^ { - y }$

For the function $f ( x , y ) = 3 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ Show the cross section at $y = 0$ .

Trans World Airlines (TWA) has a rule for checked baggage, "The total dimensions (length + width + height) may not exceed 57 inches for each bag". What is the volume of the largest volume bag you can check on a TWA flight Enter your answer as a number without the units.

In the table below, classify each highlighted value. -saddle point

Graph the function. $f ( x , y ) = 3 + \left( x ^ { 2 } + y ^ { 2 } \right)$

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

Classify each labeled point on the graph. Choose the correct letter for each question. -saddle point

Locate the critical point of the function. $f ( x , y ) = 5 x ^ { 2 } + 2 y ^ { 2 } + 4$

How many critical points does the following function have $f ( x , y ) = 4 - \left( y ^ { 2 } + x ^ { 2 } \right)$

At what point on the given surface is the quantity $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ a minimum (The method of Lagrange multipliers can be used here.) $z = \left( 4 x ^ { 2 } + 10 x + y ^ { 2 } + 22 \right) ^ { \frac { 1 } { 2 } }$

In the table below, classify each highlighted value. -relative minimum

Trans World Airlines (TWA) has a rule for checked baggage, "The total dimensions (length + width + height) may not exceed 66 inches for each bag". What is the volume of the largest volume bag you can check on a TWA flight

Calculate $\left. \frac { \partial f } { \partial x } \right| _ { ( 4,8 ) }$ , and $\left. \frac { \partial f } { \partial y } \right| _ { ( 4,8 ) }$ when defined. $f ( x , y ) = x ^ { 2 } y ^ { 3 } - x ^ { 3 } y ^ { 2 } - x y$

Your weekly cost (in dollars) to manufacture x cars and y trucks is given by $C ( x , y ) = 270,000 + 6,800 x + 3,000 y - 200,000 e ^ { - 0.03 ( x + y ) }$ What is the marginal cost of a car ? Of a truck ?

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

Classify each labeled point on the graph. Choose the correct letter for each question. -local maximum

In the table below, classify each highlighted value. -none

Evaluate $f ( a , - 3 )$ for $f ( x , y ) = x ^ { 2 } - y - x y + 6$

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

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Exam 15: Functions of Several Variables

Graph the function. f(x,y)=1.5e−yf ( x , y ) = 1.5 e ^ { - y }f(x,y)=1.5e−y

For the function f(x,y)=3−x2+y2f ( x , y ) = 3 - \sqrt { x ^ { 2 } + y ^ { 2 } }f(x,y)=3−x2+y2​ Show the cross section at y=0y = 0y=0 .

Trans World Airlines (TWA) has a rule for checked baggage, "The total dimensions (length + width + height) may not exceed 57 inches for each bag". What is the volume of the largest volume bag you can check on a TWA flight Enter your answer as a number without the units.

In the table below, classify each highlighted value. -saddle point

Graph the function. f(x,y)=3+(x2+y2)f ( x , y ) = 3 + \left( x ^ { 2 } + y ^ { 2 } \right)f(x,y)=3+(x2+y2)

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

Find ∬Rf(x,y)dxdy\iint _ { R } f ( x , y ) d x d y∬R​f(x,y)dxdy , where f(x,y)=xf ( x , y ) = xf(x,y)=x and R is the indicated domain. (Remember that you often have a choice as to the order of integration.)

Classify each labeled point on the graph. Choose the correct letter for each question. -saddle point

Locate the critical point of the function. f(x,y)=5x2+2y2+4f ( x , y ) = 5 x ^ { 2 } + 2 y ^ { 2 } + 4f(x,y)=5x2+2y2+4

How many critical points does the following function have f(x,y)=4−(y2+x2)f ( x , y ) = 4 - \left( y ^ { 2 } + x ^ { 2 } \right)f(x,y)=4−(y2+x2)

At what point on the given surface is the quantity x2+y2+z2x ^ { 2 } + y ^ { 2 } + z ^ { 2 }x2+y2+z2 a minimum (The method of Lagrange multipliers can be used here.) z=(4x2+10x+y2+22)12z = \left( 4 x ^ { 2 } + 10 x + y ^ { 2 } + 22 \right) ^ { \frac { 1 } { 2 } }z=(4x2+10x+y2+22)21​

In the table below, classify each highlighted value. -relative minimum

Trans World Airlines (TWA) has a rule for checked baggage, "The total dimensions (length + width + height) may not exceed 66 inches for each bag". What is the volume of the largest volume bag you can check on a TWA flight

Compute the integral. ∫02∫0xex2 dy dx\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x } e ^ { x ^ { 2 } } \mathrm {~d} y \mathrm {~d} x∫02​∫0x​ex2 dy dx

Find ∬Rf(x,y)dx dy\iint _ { R } f ( x , y ) \mathrm { d } x \mathrm {~d} y∬R​f(x,y)dx dy , where f(x,y)=9+yf ( x , y ) = 9 + yf(x,y)=9+y and R is the indicated domain. (Remember that you often have a choice as to the order of integration.) f(y)=9−y2f ( y ) = 9 - y ^ { 2 }f(y)=9−y2

Classify each labeled point on the graph. Choose the correct letter for each question. -local maximum

In the table below, classify each highlighted value. -none

Evaluate f(a,−3)f ( a , - 3 )f(a,−3) for ​ f(x,y)=x2−y−xy+6f ( x , y ) = x ^ { 2 } - y - x y + 6f(x,y)=x2−y−xy+6

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

Graph the function. $f ( x , y ) = 1.5 e ^ { - y }$

For the function $f ( x , y ) = 3 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ Show the cross section at $y = 0$ .

Graph the function. $f ( x , y ) = 3 + \left( x ^ { 2 } + y ^ { 2 } \right)$

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

Locate the critical point of the function. $f ( x , y ) = 5 x ^ { 2 } + 2 y ^ { 2 } + 4$

How many critical points does the following function have $f ( x , y ) = 4 - \left( y ^ { 2 } + x ^ { 2 } \right)$

At what point on the given surface is the quantity $x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ a minimum (The method of Lagrange multipliers can be used here.) $z = \left( 4 x ^ { 2 } + 10 x + y ^ { 2 } + 22 \right) ^ { \frac { 1 } { 2 } }$

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

Evaluate $f ( a , - 3 )$ for $f ( x , y ) = x ^ { 2 } - y - x y + 6$