Question 1

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = 5 x ^ { 2 } - 6 y ^ { 2 }$ .

Accepted Answer

Saddle poi

Question 2

Use double integration to find the average value of f (x, y) = y over the region bounded by $x = 9 - y ^ { 2 }$ and the y axis.

Accepted Answer

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

Question 3

Find the volume of the solid bounded above by the graph of the function $f ( x , y ) = y e ^ { x }$ and below by the rectangular region R define by: 0 $\le$ x $\le$ 4 and 0 $\le$ y $\le$ 2.

Accepted Answer

The answer of Find the volume of the solid bounded...

Question 4

Use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = ex, y = 2, and x = 0

Accepted Answer

A)  Vertical cross sections: $\begin{aligned}
e ^ { x } & \leq x \leq 2 \
0 & \leq y \leq \ln 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq \ln 2 \
1 \leq x \leq 2
\end{array}$ 
B)  Vertical cross sections: $\begin{array} { c } 
e ^ { x } \leq x \leq \ln 2 \
0 \leq y \leq 2
\end{array}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq 2 \
1 \leq x \leq \ln 2
\end{array}$ 
C)  Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln y \
e ^ { x } & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 
1 \leq y \leq 2 \
0 \leq x \leq \ln 2
\end{array}$ 
D)  Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln 2 \
e ^ { x } & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 
1 \leq y \leq 2 \
0 \leq x \leq \ln y
\end{array}$ 
A)  Vertical cross sections: $\begin{aligned}
e ^ { x } & \leq x \leq 2 \
0 & \leq y \leq \ln 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq \ln 2 \
1 \leq x \leq 2
\end{array}$ 
B)  Vertical cross sections: $\begin{array} { c } 
e ^ { x } \leq x \leq \ln 2 \
0 \leq y \leq 2
\end{array}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq 2 \
1 \leq x \leq \ln 2
\end{array}$ 
C)  Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln y \
e ^ { x } & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 
1 \leq y \leq 2 \
0 \leq x \leq \ln 2
\end{array}$ 
D)  Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln 2 \
e ^ { x } & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 
1 \leq y \leq 2 \
0 \leq x \leq \ln y
\end{array}$

Question 5

Compute $f _ { y }$ for $f ( x , y ) = 9 x y ^ { 5 }$

Accepted Answer

The answer of Compute $f _ { y }$ for...

Question 6

Compute $f _ { y }$ for $f ( x , y ) = 7 x y ^ { 6 }$

Accepted Answer

A)  $7 y ^ { 6 }$ 
B)  $42 x y ^ { 5 } + 7 y ^ { 6 }$ 
C)  7x 
D)  $42 x y ^ { 5 }$ 
A)  $7 y ^ { 6 }$ 
B)  $42 x y ^ { 5 } + 7 y ^ { 6 }$ 
C)  7x 
D)  $42 x y ^ { 5 }$

Question 7

Use a double integral to find the area of R.R is the triangle with vertices (-5, 6), (5, 6), and (0, 1)

Accepted Answer

The answer of Use a double integral to find the...

Question 8

Let $f ( x , y ) = 5 x ^ { 2 } - e ^ { y }$ . Compute f (3, 0).

Accepted Answer

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

Question 9

Compute $f _ { y y }$ for $f ( x , y ) = e ^ { 6 x y}$

Accepted Answer

The answer of Compute $f _ { y y }$...

Question 10

Given the following points in the plane, find the slope of the least squares line: (1, 2), (2, 1), (3, 3), and (5, 13) round your answer to two decimal places, if necessary.

Accepted Answer

A)  2.94 
B)  2.89 
C)  2.83 
D)  2.77 
A)  2.94 
B)  2.89 
C)  2.83 
D)  2.77

Question 11

The following data shows the age and income for a small number of people. $$\begin{array} { c l l l l l } 
\text { Age (years) } & 24 & 32 & 39 & 54 & 60 \
\text { Incomes (\$) } & 30,000 & 34,000 & 53,000 & 72,000 & 92,000
\end{array}$$ 
Find the best fit straight line of this data, rounding coefficients and constants to the nearest whole number. Let x represent age and y represent income.

Accepted Answer

The answer of The following data shows the age and...

Question 12

Use Lagrange multipliers to find the maximum value of f (x, y) = 9xy subject to the constraint 9x + 5y = 45.

Accepted Answer

A)  
$f \left( \frac { 5 } { 4 } , \frac { 9 } { 4 } \right) = \frac { 405 } { 16 }$ 
B)  
$f \left( \frac { 5 } { 2 } , \frac { 9 } { 2 } \right) = \frac { 405 } { 4 }$ 
C)  f (5, 9) = 405 
D)  f (0, 0) = 0 
A)  
$f \left( \frac { 5 } { 4 } , \frac { 9 } { 4 } \right) = \frac { 405 } { 16 }$ 
B)  
$f \left( \frac { 5 } { 2 } , \frac { 9 } { 2 } \right) = \frac { 405 } { 4 }$ 
C)  f (5, 9) = 405 
D)  f (0, 0) = 0

Question 13

A manufacturer is planning to sell a new product at the price of $260 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is spent on promotion, approximately $\frac { 360 y } { y + 7 } + \frac { 180 x } { x + 14 }$ units of the product will be sold. The cost of manufacturing the product is $200 per unit. If the manufacturer has a total of $360,000 to spend on development and promotion, how should this money be allocated to generate the largest possible profit? [Hint: Profit equals (number of units)(price per unit minus cost per unit) minus total amount spent on development and promotion.]

Accepted Answer

A)  $177,000 on development, $183,000 on promotion 
B)  $183,000 on development, $177,000 on promotion 
C)  $183,500 on development, $177,000 on promotion 
D)  $176,500 on development, $183,500 on promotion 
A)  $177,000 on development, $183,000 on promotion 
B)  $183,000 on development, $177,000 on promotion 
C)  $183,500 on development, $177,000 on promotion 
D)  $176,500 on development, $183,500 on promotion

Question 14

If $f ( x , y ) = x e ^ {x y }$ , find $f _ { y } ( 2,3 )$ .

Accepted Answer

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

Question 15

Evaluate the following double integral: $\int _ { - 2 } ^ { 3 } \int _ { - 2 } ^ { 2 } x ^ { 2 } y ^ { 3 } d y d x$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  3 
C)  0 
D)  The integral can't be evaluated. 
A)  $\frac { 1 } { 2 }$ 
B)  3 
C)  0 
D)  The integral can't be evaluated.

Question 16

Evaluate the given double integral for the specified region R. $\iint _ { R } 2 x y d A$ , where R is the rectangle bounded by the lines x = -1, x = 2, y = -1, and y = 0.

Accepted Answer

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

Question 17

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 8 xy }$

Accepted Answer

A)  $8 x y e ^ { 8 x y }$ 
B)  $8 y e ^ { 8 x y }$ 
C)  $8 x e ^ { 8 x y }$ 
D)  $8 e ^ { 8 x y }$ 
A)  $8 x y e ^ { 8 x y }$ 
B)  $8 y e ^ { 8 x y }$ 
C)  $8 x e ^ { 8 x y }$ 
D)  $8 e ^ { 8 x y }$

Question 18

A mall kiosk sells two different models of pagers, the Elite and the Diamond. Their monthly profit from pager sales is $P ( x , y ) = ( x - 40 ) ( 20 - 5 x + 6 y ) + ( y - 50 ) ( 30 + 3 x - 4 y )$ where x and y are the prices of the Elite and the Diamond respectively, in dollars. At the moment, the Elite sells for $32 and the Diamond sells for $40. Use calculus to estimate the change in monthly profit if the kiosk operator raises the price of the Elite to $33 and lowers the price of the Diamond to $38.

Accepted Answer

A)  Profit will increase by about $26. 
B)  Profit will decrease by about $310. 
C)  Profit will increase by about $194. 
D)  Profit will stay the same. 
A)  Profit will increase by about $26. 
B)  Profit will decrease by about $310. 
C)  Profit will increase by about $194. 
D)  Profit will stay the same.

Question 19

Compute $f _ { x }$ for $f ( x , y ) = 2 x y ^ { 9 }$ .

Accepted Answer

A)  $2 y ^ { 9 }$ 
B)  $18 x y ^ { 8 }$ 
C)  2x 
D)  $18 x y ^ { 8 } + 2 y ^ { 9 }$ 
A)  $2 y ^ { 9 }$ 
B)  $18 x y ^ { 8 }$ 
C)  2x 
D)  $18 x y ^ { 8 } + 2 y ^ { 9 }$

Question 20

Evaluate the following double integral: $$\int _ { - 4 } ^ { 7 } \int _ { - 4 } ^ { 4 } x ^ { 4 } y ^ { 7 } d y d x$$

Accepted Answer

A)  4 
B)  The integral can't be evaluated. 
C)  $\frac { 1 } { 4 }$ 
D)  0 
A)  4 
B)  The integral can't be evaluated. 
C)  $\frac { 1 } { 4 }$ 
D)  0

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = 5 x ^ { 2 } - 6 y ^ { 2 }$ .

Use double integration to find the average value of f (x, y) = y over the region bounded by $x = 9 - y ^ { 2 }$ and the y axis.

Find the volume of the solid bounded above by the graph of the function $f ( x , y ) = y e ^ { x }$ and below by the rectangular region R define by: 0 $\le$ x $\le$ 4 and 0 $\le$ y $\le$ 2.

Use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = e^x, y = 2, and x = 0

Compute $f _ { y }$ for $f ( x , y ) = 9 x y ^ { 5 }$

Compute $f _ { y }$ for $f ( x , y ) = 7 x y ^ { 6 }$

Use a double integral to find the area of R.R is the triangle with vertices (-5, 6), (5, 6), and (0, 1)

Let $f ( x , y ) = 5 x ^ { 2 } - e ^ { y }$ . Compute f (3, 0).

Compute $f _ { y y }$ for $f ( x , y ) = e ^ { 6 x y}$

Given the following points in the plane, find the slope of the least squares line: (1, 2), (2, 1), (3, 3), and (5, 13) round your answer to two decimal places, if necessary.

Use Lagrange multipliers to find the maximum value of f (x, y) = 9xy subject to the constraint 9x + 5y = 45.

If $f ( x , y ) = x e ^ {x y }$ , find $f _ { y } ( 2,3 )$ .

Evaluate the following double integral: $\int _ { - 2 } ^ { 3 } \int _ { - 2 } ^ { 2 } x ^ { 2 } y ^ { 3 } d y d x$

Evaluate the given double integral for the specified region R. $\iint _ { R } 2 x y d A$ , where R is the rectangle bounded by the lines x = -1, x = 2, y = -1, and y = 0.

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 8 xy }$

Compute $f _ { x }$ for $f ( x , y ) = 2 x y ^ { 9 }$ .

Evaluate the following double integral: $\int _ { - 4 } ^ { 7 } \int _ { - 4 } ^ { 4 } x ^ { 4 } y ^ { 7 } d y d x$

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 extrema (minima, maxima, saddle points), if any, for f(x,y)=5x2−6y2f ( x , y ) = 5 x ^ { 2 } - 6 y ^ { 2 }f(x,y)=5x2−6y2 .

Use double integration to find the average value of f (x, y) = y over the region bounded by x=9−y2x = 9 - y ^ { 2 }x=9−y2 and the y axis.

Find the volume of the solid bounded above by the graph of the function f(x,y)=yexf ( x , y ) = y e ^ { x }f(x,y)=yex and below by the rectangular region R define by: 0 ≤\le≤ x ≤\le≤ 4 and 0 ≤\le≤ y ≤\le≤ 2.

Use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = ex, y = 2, and x = 0

Compute fyf _ { y }fy​ for f(x,y)=9xy5f ( x , y ) = 9 x y ^ { 5 }f(x,y)=9xy5

Compute fyf _ { y }fy​ for f(x,y)=7xy6f ( x , y ) = 7 x y ^ { 6 }f(x,y)=7xy6

Use a double integral to find the area of R.R is the triangle with vertices (-5, 6), (5, 6), and (0, 1)

Let f(x,y)=5x2−eyf ( x , y ) = 5 x ^ { 2 } - e ^ { y }f(x,y)=5x2−ey . Compute f (3, 0).

Compute fyyf _ { y y }fyy​ for f(x,y)=e6xyf ( x , y ) = e ^ { 6 x y}f(x,y)=e6xy

Given the following points in the plane, find the slope of the least squares line: (1, 2), (2, 1), (3, 3), and (5, 13) round your answer to two decimal places, if necessary.

Use Lagrange multipliers to find the maximum value of f (x, y) = 9xy subject to the constraint 9x + 5y = 45.

If f(x,y)=xexyf ( x , y ) = x e ^ {x y }f(x,y)=xexy , find fy(2,3)f _ { y } ( 2,3 )fy​(2,3) .

Evaluate the following double integral: ∫−23∫−22x2y3dydx\int _ { - 2 } ^ { 3 } \int _ { - 2 } ^ { 2 } x ^ { 2 } y ^ { 3 } d y d x∫−23​∫−22​x2y3dydx

Evaluate the given double integral for the specified region R. ∬R2xydA\iint _ { R } 2 x y d A∬R​2xydA , where R is the rectangle bounded by the lines x = -1, x = 2, y = -1, and y = 0.

Compute fxf _ { x }fx​ for f(x,y)=e8xyf ( x , y ) = e ^ { 8 xy }f(x,y)=e8xy

Compute fxf _ { x }fx​ for f(x,y)=2xy9f ( x , y ) = 2 x y ^ { 9 }f(x,y)=2xy9 .

Evaluate the following double integral: ∫−47∫−44x4y7dydx\int _ { - 4 } ^ { 7 } \int _ { - 4 } ^ { 4 } x ^ { 4 } y ^ { 7 } d y d x∫−47​∫−44​x4y7dydx

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 extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = 5 x ^ { 2 } - 6 y ^ { 2 }$ .

Use double integration to find the average value of f (x, y) = y over the region bounded by $x = 9 - y ^ { 2 }$ and the y axis.

Find the volume of the solid bounded above by the graph of the function $f ( x , y ) = y e ^ { x }$ and below by the rectangular region R define by: 0 $\le$ x $\le$ 4 and 0 $\le$ y $\le$ 2.

Use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = e^x, y = 2, and x = 0

Compute $f _ { y }$ for $f ( x , y ) = 9 x y ^ { 5 }$

Compute $f _ { y }$ for $f ( x , y ) = 7 x y ^ { 6 }$

Let $f ( x , y ) = 5 x ^ { 2 } - e ^ { y }$ . Compute f (3, 0).

Compute $f _ { y y }$ for $f ( x , y ) = e ^ { 6 x y}$

If $f ( x , y ) = x e ^ {x y }$ , find $f _ { y } ( 2,3 )$ .

Evaluate the following double integral: $\int _ { - 2 } ^ { 3 } \int _ { - 2 } ^ { 2 } x ^ { 2 } y ^ { 3 } d y d x$

Evaluate the given double integral for the specified region R. $\iint _ { R } 2 x y d A$ , where R is the rectangle bounded by the lines x = -1, x = 2, y = -1, and y = 0.

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 8 xy }$

Compute $f _ { x }$ for $f ( x , y ) = 2 x y ^ { 9 }$ .

Evaluate the following double integral: $\int _ { - 4 } ^ { 7 } \int _ { - 4 } ^ { 4 } x ^ { 4 } y ^ { 7 } d y d x$