Question 1

Find the maximum value of $f ( x , y ) = x y ^ { 2 }$ subject to the constraint g(x, y) = x - y = 2.

Accepted Answer

The answer of Find the maximum value of \(f (...

Question 2

Let $f ( x , y ) = \frac { x } { e ^ { y } }$ . Compute f (3, 0).

Accepted Answer

A)  0 
B)  3 
C)  Undefined 
D)  4 
A)  0 
B)  3 
C)  Undefined 
D)  4

Question 3

Use a double integral to find the area of R. R is the triangle with vertices (-4, 6), (4, 6), and (0, 2).

Accepted Answer

A)  $\int _ { x = - 4 } ^ { x - 4 } \int _ { y = - x + 2 } ^ { y - x + 2 } 1 d y d x = 32$ 
B)  $\int _ { y - 2 } ^ { y - 6 } \int _ { x - - y + 2 } ^ { x - y - 2 } 1 d x d y = 32$ 
C)  $\int _ { y - 2 } ^ { y - 6 } \int _ { x - - y + 2 } ^ { x - y - 2 } 1 d x d y = 16$ 
D)  $\int _ { x = - 4 } ^ { x - 4 } \int _ { y = - x + 2 } ^ { y - x + 2 } 1 d y d x = 16$ 
A)  $\int _ { x = - 4 } ^ { x - 4 } \int _ { y = - x + 2 } ^ { y - x + 2 } 1 d y d x = 32$ 
B)  $\int _ { y - 2 } ^ { y - 6 } \int _ { x - - y + 2 } ^ { x - y - 2 } 1 d x d y = 32$ 
C)  $\int _ { y - 2 } ^ { y - 6 } \int _ { x - - y + 2 } ^ { x - y - 2 } 1 d x d y = 16$ 
D)  $\int _ { x = - 4 } ^ { x - 4 } \int _ { y = - x + 2 } ^ { y - x + 2 } 1 d y d x = 16$

Question 4

Find $f _ { x y }$ for $f ( x , y ) = x ^ { 3 } + y ^ { 3 }$ .

Accepted Answer

A)  6x 
B)  $3 x ^ { 2 } + 3 y ^ { 2 }$ 
C)  0 
D)  6y 
A)  6x 
B)  $3 x ^ { 2 } + 3 y ^ { 2 }$ 
C)  0 
D)  6y

Question 5

Let $f ( x , y ) = 7 x ^ { 2 } - y$ . Compute f (2, 0).

Accepted Answer

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

Question 6

Use double integration to find the average value of $f ( x , y ) = 18 x y ^ { 2 }$ over the triangle with vertices (0, 0), (0, 2), and (3, 2).

Accepted Answer

The answer of Use double integration to find the average...

Question 7

Find the minimum value (if any) of $f ( x , y ) = x y ^ { 2 }$ subject to the constraint g(x, y) = x - y = 2.

Accepted Answer

The answer of Find the minimum value (if any) of...

Question 8

Use a double integral to find the area of R. R is the region bounded by y = 9x, y = ln x, y = 0, and y = 1.

Accepted Answer

A)  $\int _ { x = 0 } ^ { x - 2 } \int _ { y - lnx } ^ { y - 9 x } 1 d y d x = \frac { 9 e ^ { 2 } } { 2 }$ 
B)  $\int _ { y = 0 } ^ { y - 1 } \int _ { x - \frac { y } { 9 } } ^ { x - e ^ { y } } 1 d x d y = e - \frac { 19 } { 18 }$ 
C)  $\int _ { y - 0 } ^ { y - 1 } \int _ { x - 0 } ^ { x - e ^ { y } } 1 d x d y = e - 1$ 
D)  $\int _ { x - 0 } ^ { x - 2 } \int _ { y - \ln x } ^ { y - 1 } 1 d y d x = e$ 
A)  $\int _ { x = 0 } ^ { x - 2 } \int _ { y - lnx } ^ { y - 9 x } 1 d y d x = \frac { 9 e ^ { 2 } } { 2 }$ 
B)  $\int _ { y = 0 } ^ { y - 1 } \int _ { x - \frac { y } { 9 } } ^ { x - e ^ { y } } 1 d x d y = e - \frac { 19 } { 18 }$ 
C)  $\int _ { y - 0 } ^ { y - 1 } \int _ { x - 0 } ^ { x - e ^ { y } } 1 d x d y = e - 1$ 
D)  $\int _ { x - 0 } ^ { x - 2 } \int _ { y - \ln x } ^ { y - 1 } 1 d y d x = e$

Question 9

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 3 x + 5 y ) ^ { 3 }$

Accepted Answer

The answer of Compute all first-order partial derivatives of the...

Question 10

Given the function of three variables f (x, y, z) = xy + xz + yz, evaluate f (-9, 9, -3).

Accepted Answer

A)  -72 
B)  -78 
C)  -86 
D)  -81 
A)  -72 
B)  -78 
C)  -86 
D)  -81

Question 11

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + y ^ { 3 } - 6 x y$ .

Accepted Answer

Saddle point at (0,

Question 12

Evaluate the given double integral for the specified region R. $$\iint _ { R } ( 8 x + 2 y ) d A$$ , where R is the triangle with vertices (0, 0), (1, 0), and (0, 5)

Accepted Answer

The answer of Evaluate the given double integral for the...

Question 13

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = x2 and y = 7x.

Accepted Answer

A)  Vertical cross sections: 
$\begin{array} { c } 
0 \leq y \leq 49 \
\frac { y } { 7 } \leq x \leq \sqrt { y }
\end{array}$

Horizontal cross sections: 
$\begin{array} { c } 
0 \leq x \leq 7 \
x ^ { 2 } \leq y \leq 7 x
\end{array}$ 
B)  Vertical cross sections: 
$\begin{array} { c } 
0 \leq x \leq 7 \
x ^ { 2 } \leq y \leq 7 x
\end{array}$

Horizontal cross sections: 
$\begin{array} { c } 
0 \leq y \leq 49 \
\frac { y } { 7 } \leq x \leq \sqrt { y }
\end{array}$ 
C)  Vertical cross sections: 
$\begin{array} { c } 
0 \leq x \leq 49 \
x ^ { 2 } \leq y \leq 7 x
\end{array}$

Horizontal cross sections: 
$\begin{array} { c } 
0 \leq y \leq 7 \
\frac { y } { 7 } \leq x \leq \sqrt { y }
\end{array}$ 
D)  Vertical cross sections: 
$\begin{array} { c } 
0 \leq y \leq 7 \
\frac { y } { 7 } \leq x \leq \sqrt { y }
\end{array}$
 Horizontal cross sections: 
$\begin{array} { c } 
0 \leq x \leq 49 \
x ^ { 2 } \leq y \leq 7 x
\end{array}$ 
A)  Vertical cross sections: 
$\begin{array} { c } 
0 \leq y \leq 49 \
\frac { y } { 7 } \leq x \leq \sqrt { y }
\end{array}$

Horizontal cross sections: 
$\begin{array} { c } 
0 \leq x \leq 7 \
x ^ { 2 } \leq y \leq 7 x
\end{array}$ 
B)  Vertical cross sections: 
$\begin{array} { c } 
0 \leq x \leq 7 \
x ^ { 2 } \leq y \leq 7 x
\end{array}$

Horizontal cross sections: 
$\begin{array} { c } 
0 \leq y \leq 49 \
\frac { y } { 7 } \leq x \leq \sqrt { y }
\end{array}$ 
C)  Vertical cross sections: 
$\begin{array} { c } 
0 \leq x \leq 49 \
x ^ { 2 } \leq y \leq 7 x
\end{array}$

Horizontal cross sections: 
$\begin{array} { c } 
0 \leq y \leq 7 \
\frac { y } { 7 } \leq x \leq \sqrt { y }
\end{array}$ 
D)  Vertical cross sections: 
$\begin{array} { c } 
0 \leq y \leq 7 \
\frac { y } { 7 } \leq x \leq \sqrt { y }
\end{array}$
 Horizontal cross sections: 
$\begin{array} { c } 
0 \leq x \leq 49 \
x ^ { 2 } \leq y \leq 7 x
\end{array}$

Question 14

A computer-designed abstract glass sculpture is in the shape of the solid region under the graph of $f ( x , y ) = x ^ { 2 } y$ above the semi-circular region in the xy-plane bounded by the x-axis and $y = \sqrt { 4 - x ^ { 2 } }$ . If all measurements are in feet, what volume of glass will be needed for the sculpture?

Accepted Answer

The answer of A computer-designed abstract glass sculpture is in...

Question 15

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 20 x - 20 y$ .

Accepted Answer

Saddle poi

Question 16

Given the following points in the plane, find the corresponding least squares line: (1, 2), (2, 1), (4, 2), and (5, 1)

Accepted Answer

The answer of Given the following points in the plane,...

Question 17

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + y ^ { 3 } - 6 x y$ .

Accepted Answer

Saddle poi

Question 18

The accompanying table lists the Gross Domestic Product (GNP) figures for China (in billions of yuan) for the period from 1996 to 2000 $$\begin{array} { c l l l l l } 
\text { Year } & 1996 & 1997 & 1998 & 1999 & 2000 \
\text { GNP } & 6,788 & 7,446 & 7,835 & 8,191 & 8,940
\end{array}$$
 Use the best fit straight line to extrapolate the GNP to the nearest yuan in 1990?

Accepted Answer

A)  3,801 billion yuan 
B)  3,876 billion yuan 
C)  3,942 billion yuan 
D)  3,989 billion yuan 
A)  3,801 billion yuan 
B)  3,876 billion yuan 
C)  3,942 billion yuan 
D)  3,989 billion yuan

Question 19

Compute f (6, 0) if $f ( x , y ) = \frac { x } { e ^ { y } }$

Accepted Answer

A)  Undefined 
B)  6 
C)  0 
D)  $\frac { 1 } { 6 }$ 
A)  Undefined 
B)  6 
C)  0 
D)  $\frac { 1 } { 6 }$

Question 20

The weekly output for a manufacturer is $Q ( x , y ) = 12 x + 50 y - 2 x ^ { 2 } + y ^ { 2 }$ units. Use marginal analysis to estimate the change in weekly output as a result of changing x from 20 to 21 while y remains constant at 10.

Accepted Answer

The answer of The weekly output for a manufacturer is...

Find the maximum value of $f ( x , y ) = x y ^ { 2 }$ subject to the constraint g(x, y) = x - y = 2.

Let $f ( x , y ) = \frac { x } { e ^ { y } }$ . Compute f (3, 0).

Use a double integral to find the area of R. R is the triangle with vertices (-4, 6), (4, 6), and (0, 2).

Find $f _ { x y }$ for $f ( x , y ) = x ^ { 3 } + y ^ { 3 }$ .

Let $f ( x , y ) = 7 x ^ { 2 } - y$ . Compute f (2, 0).

Use double integration to find the average value of $f ( x , y ) = 18 x y ^ { 2 }$ over the triangle with vertices (0, 0), (0, 2), and (3, 2).

Find the minimum value (if any) of $f ( x , y ) = x y ^ { 2 }$ subject to the constraint g(x, y) = x - y = 2.

Use a double integral to find the area of R. R is the region bounded by y = 9x, y = ln x, y = 0, and y = 1.

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 3 x + 5 y ) ^ { 3 }$

Given the function of three variables f (x, y, z) = xy + xz + yz, evaluate f (-9, 9, -3).

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + y ^ { 3 } - 6 x y$ .

Evaluate the given double integral for the specified region R. $\iint _ { R } ( 8 x + 2 y ) d A$ , where R is the triangle with vertices (0, 0), (1, 0), and (0, 5)

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = x² and y = 7x.

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 20 x - 20 y$ .

Given the following points in the plane, find the corresponding least squares line: (1, 2), (2, 1), (4, 2), and (5, 1)

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + y ^ { 3 } - 6 x y$ .

The accompanying table lists the Gross Domestic Product (GNP) figures for China (in billions of yuan) for the period from 1996 to 2000 Year 1996 1997 1998 1999 2000 GNP 6,788 7,446 7,835 8,191 8,940 Use the best fit straight line to extrapolate the GNP to the nearest yuan in 1990?

Compute f (6, 0) if $f ( x , y ) = \frac { x } { e ^ { y } }$

The weekly output for a manufacturer is $Q ( x , y ) = 12 x + 50 y - 2 x ^ { 2 } + y ^ { 2 }$ units. Use marginal analysis to estimate the change in weekly output as a result of changing x from 20 to 21 while y remains constant at 10.

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Appendix: Algebra Review

Filters

Exam 7: Calculus of Several Variables

Find the maximum value of f(x,y)=xy2f ( x , y ) = x y ^ { 2 }f(x,y)=xy2 subject to the constraint g(x, y) = x - y = 2.

Let f(x,y)=xeyf ( x , y ) = \frac { x } { e ^ { y } }f(x,y)=eyx​ . Compute f (3, 0).

Use a double integral to find the area of R. R is the triangle with vertices (-4, 6), (4, 6), and (0, 2).

Find fxyf _ { x y }fxy​ for f(x,y)=x3+y3f ( x , y ) = x ^ { 3 } + y ^ { 3 }f(x,y)=x3+y3 .

Let f(x,y)=7x2−yf ( x , y ) = 7 x ^ { 2 } - yf(x,y)=7x2−y . Compute f (2, 0).

Use double integration to find the average value of f(x,y)=18xy2f ( x , y ) = 18 x y ^ { 2 }f(x,y)=18xy2 over the triangle with vertices (0, 0), (0, 2), and (3, 2).

Find the minimum value (if any) of f(x,y)=xy2f ( x , y ) = x y ^ { 2 }f(x,y)=xy2 subject to the constraint g(x, y) = x - y = 2.

Use a double integral to find the area of R. R is the region bounded by y = 9x, y = ln x, y = 0, and y = 1.

Compute all first-order partial derivatives of the given function. f(x,y)=(3x+5y)3f ( x , y ) = ( 3 x + 5 y ) ^ { 3 }f(x,y)=(3x+5y)3

Given the function of three variables f (x, y, z) = xy + xz + yz, evaluate f (-9, 9, -3).

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for f(x,y)=x2+y3−6xyf ( x , y ) = x ^ { 2 } + y ^ { 3 } - 6 x yf(x,y)=x2+y3−6xy .

Evaluate the given double integral for the specified region R. ∬R(8x+2y)dA\iint _ { R } ( 8 x + 2 y ) d A∬R​(8x+2y)dA , where R is the triangle with vertices (0, 0), (1, 0), and (0, 5)

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = x2 and y = 7x.

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for f(x,y)=x2+xy−y2+20x−20yf ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 20 x - 20 yf(x,y)=x2+xy−y2+20x−20y .

Given the following points in the plane, find the corresponding least squares line: (1, 2), (2, 1), (4, 2), and (5, 1)

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for f(x,y)=x2+y3−6xyf ( x , y ) = x ^ { 2 } + y ^ { 3 } - 6 x yf(x,y)=x2+y3−6xy .

The accompanying table lists the Gross Domestic Product (GNP) figures for China (in billions of yuan) for the period from 1996 to 2000 Year 1996 1997 1998 1999 2000 GNP 6,788 7,446 7,835 8,191 8,940 Use the best fit straight line to extrapolate the GNP to the nearest yuan in 1990?

Compute f (6, 0) if f(x,y)=xeyf ( x , y ) = \frac { x } { e ^ { y } }f(x,y)=eyx​

The weekly output for a manufacturer is Q(x,y)=12x+50y−2x2+y2Q ( x , y ) = 12 x + 50 y - 2 x ^ { 2 } + y ^ { 2 }Q(x,y)=12x+50y−2x2+y2 units. Use marginal analysis to estimate the change in weekly output as a result of changing x from 20 to 21 while y remains constant at 10.

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Appendix: Algebra Review

Filters

Find the maximum value of $f ( x , y ) = x y ^ { 2 }$ subject to the constraint g(x, y) = x - y = 2.

Let $f ( x , y ) = \frac { x } { e ^ { y } }$ . Compute f (3, 0).

Find $f _ { x y }$ for $f ( x , y ) = x ^ { 3 } + y ^ { 3 }$ .

Let $f ( x , y ) = 7 x ^ { 2 } - y$ . Compute f (2, 0).

Use double integration to find the average value of $f ( x , y ) = 18 x y ^ { 2 }$ over the triangle with vertices (0, 0), (0, 2), and (3, 2).

Find the minimum value (if any) of $f ( x , y ) = x y ^ { 2 }$ subject to the constraint g(x, y) = x - y = 2.

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 3 x + 5 y ) ^ { 3 }$

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + y ^ { 3 } - 6 x y$ .

Evaluate the given double integral for the specified region R. $\iint _ { R } ( 8 x + 2 y ) d A$ , where R is the triangle with vertices (0, 0), (1, 0), and (0, 5)

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = x² and y = 7x.

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 20 x - 20 y$ .

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + y ^ { 3 } - 6 x y$ .

Compute f (6, 0) if $f ( x , y ) = \frac { x } { e ^ { y } }$

The weekly output for a manufacturer is $Q ( x , y ) = 12 x + 50 y - 2 x ^ { 2 } + y ^ { 2 }$ units. Use marginal analysis to estimate the change in weekly output as a result of changing x from 20 to 21 while y remains constant at 10.