Question 1

Evaluate the double integral $\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } x y e ^ { - \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) } d x d y$ .
A) $\frac { 1 } { 180 }$
B) $\frac { 1 } { 19 }$
C) $\frac { 1 } { 144 }$
D) $- \frac { 1 } { 72 }$
E) $- \frac { 1 } { 108 }$

Accepted Answer

The answer of Evaluate the double integral \(\int _ {...

Question 2

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Data that are modeled by $y = 3.29 x - 4.17$ have a negative correlation.
A)True
B)False; The data modeled by $y = 3.29 x - 4.17$ have a positive correlation.

Accepted Answer

The slope of the line $y = 3.29x - 4.17$ is positive (3.29), indicating that as $x$ increases, $y$ also increases. This is characteristic of a positive correlation, not a negative one.

Question 3

Evaluate the double integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } ( 3 x + 2 y ) d x d y$ . Round your answer to two decimal places, where applicable.
A)-8.83
B)11.17
C)2.00
D)1.17
E)5.00

Accepted Answer

First, integrate with respect to $x$: $\int_{0}^{y} (3x + 2y) \, dx = \left[\frac{3}{2}x^2 + 2yx\right]_{0}^{y} = \frac{3}{2}y^2 + 2y^2 = \frac{7}{2}y^2$. Then, integrate with respect to $y$: $\int_{0}^{1} \frac{7}{2}y^2 \, dy = \left[\frac{7}{6}y^3\right]_{0}^{1} = \frac{7}{6} \approx 1.17$.

Question 4

A store manager wants to know the demand y for an energy bar as a function of price x. The daily sales for three different prices of the energy bar are shown in the table. Price, x
$ 1.02
$ 1.23
$ 1.54
Demand, y
410
365
280
(i) Use the regression capabilities of a graphing utility to find the least squares regression line for the data.
(ii) Use the model to estimate the demand when the price is $1.38.
A)(i) $y = - 24.314583 x + 382.38409$ ; (ii)348.708392
B)(i) $y = - 24.314583 x + 382.38409$ ; (ii)-416.059787
C)(i) $y = 382.38409 x - 24.314583$ ; (ii)505.287381
D)(i) $y = - 24.314583 x - 382.38409$ ; (ii)-416.059787
E)none of the above

Accepted Answer

The answer of A store manager wants to know the...

Question 5

Evaluate the double integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x$ .
A)134.00
B)144.00
C)154.00
D)42.00
E)24.00

Accepted Answer

The given double integral can be solved by first integrating with respect to $y$ and then with respect to $x$. Integrating $3x + 4y$ with respect to $y$ from 0 to 3 gives $3xy + 2y^2$ evaluated from 0 to 3, which simplifies to $9x + 18$. Then, integrating $9x + 18$ with respect to $x$ from 0 to 4 gives $4.5x^2 + 18x$ evaluated from 0 to 4, which simplifies to $72$. Therefore, the value of the double integral is 72, but since the integration with respect to $y$ was not correctly evaluated in this explanation, the correct procedure is as follows:First, integrate $3x + 4y$ with respect to $y$: $\int_{0}^{3} (3x + 4y) dy = 3x*y + 2y^2 |_{0}^{3} = 9x + 18$.Then, integrate the result with respect to $x$: $\int_{0}^{4} (9x + 18) dx = 4.5x^2 + 18x |_{0}^{4} = 72 + 72 = 144$.So, the correct answer is 144.

Question 6

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points $( 10,2 ) , ( 1,8 ) , ( 6,9 ) , ( 4,10 ) , ( 16,4 ) , ( 17,4 )$ . Round your answer to three decimal places.
A) $y = - 0.383 x + 9.524$
B) $y = - 0.373 x + 9.514$
C) $y = - 0.373 x + 9.524$
D) $y = - 0.363 x + 9.524$
E) $y = - 0.363 x + 9.514$

Accepted Answer

The least squares regression line calculated using the given points is $y = - 0.373 x + 9.524$.

Question 7

Sketch the region $R$ whose area is given by the following double integral. $$\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } d y d x$$
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Sketch the region $R$ whose area is...

Question 8

A manufacturer has an order for 1100 units of fine paper that can be produced at two locations. Let $x _ { 1 }$ and $x _ { 2 }$ be the numbers of units produced at the two plants. Find the number of units that should be produced at each plant to minimize the cost if the cost function is given by $C = 0.2 x _ { 1 } ^ { 2 } + 25 x _ { 1 } + 0.05 x _ { 2 } ^ { 2 } + 12 x _ { 2 }$ .
A) $x _ { 1 } = 388$ units and $x _ { 2 } = 906$ units
B) $x _ { 1 } = 906$ units and $x _ { 2 } = 194$ units
C) $x _ { 1 } = 194$ units and $x _ { 2 } = 906$ units
D) $x _ { 1 } = 194$ units and $x _ { 2 } = 1812$ units
E) $x _ { 1 } = 1812$ units and $x _ { 2 } = 388$ units

Accepted Answer

To minimize the cost, we need to set up the problem as a constrained optimization problem. The constraint is $x_1 + x_2 = 1100$, since the total production must equal 1100 units. The objective function to minimize is $C = 0.2x_1^2 + 25x_1 + 0.05x_2^2 + 12x_2$. To find the minimum cost, we would take the partial derivatives of C with respect to $x_1$ and $x_2$, set them equal to zero, and solve the system of equations along with the constraint. Without going through all the algebraic steps here, the correct answer reflects the optimal production levels at each plant to minimize the cost while meeting the production requirement.

Question 9

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $( - 4 , - 1 ) , ( - 2,0 ) , ( 2,4 ) , ( 4,5 )$
A) $y = 1.8 x + 5$
B) $y = 1.2 x + 2$
C) $y = 0.5 x + 1$
D) $y = 0.8 x + 2$
E) $y = 0.2 x + 1$

Accepted Answer

Using a graphing utility or spreadsheet to perform a linear regression analysis on the given points, the least squares regression line is found to be $y = 0.8x + 2$, which matches option D. This line best fits the data points provided, minimizing the sum of the squares of the vertical distances of the points from the line.

Question 10

Evaluate the double integral $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { x } \frac { 2 } { x ^ { 2 } + 8 } d y d x$ .
A) $\ln 43$
B) $\ln 43 - \ln 8$
C) $\ln 44$
D) $\ln 44 + \ln 8$
E) $\ln 44 - \ln 8$

Accepted Answer

The inner integral with respect to $y$ is straightforward because the integrand does not depend on $y$, so the integral becomes $\int_0^x \frac{2}{x^2+8} dy = \frac{2y}{x^2+8}\Big|_0^x = \frac{2x}{x^2+8}$. Then, integrating with respect to $x$ from 0 to 6 gives $\int_0^6 \frac{2x}{x^2+8} dx$, which can be solved by a substitution method (let $u = x^2 + 8$, then $du = 2x dx$). The integral becomes $\int \frac{1}{u} du$ evaluated from $x=0$ to $x=6$, which is $\ln|u|\Big|_{8}^{44} = \ln 44 - \ln 8$.

Question 11

Find the least squares regression line for the points (1,0) , (6,6) , (11,12).
A) $y = \frac { 6 } { 5 } x - \frac { 6 } { 5 }$
B) $y = \frac { 6 } { 5 } x - \frac { 30 } { 133 }$
C) $y = \frac { 30 } { 133 } x - \frac { 6 } { 5 }$
D) $y = \frac { 30 } { 133 } x - \frac { 30 } { 133 }$
E)none of the above

Accepted Answer

The least squares regression line is given by the equation y = mx + b, where m is the slope and b is the y-intercept. The slope is calculated as the average change in y divided by the average change in x, and the y-intercept is calculated as the average value of y minus the product of the slope and the average value of x.For the given points, the average change in y is (6 - 0) / (6 - 1) = 1, and the average change in x is (11 - 6) / (11 - 1) = 1. Therefore, the slope is m = 1.The average value of y is (0 + 6 + 12) / 3 = 6, and the average value of x is (1 + 6 + 11) / 3 = 6. Therefore, the y-intercept is b = 6 - 1 * 6 = 0.Therefore, the least squares regression line is y = 1x + 0, which is equivalent to y = x.

Question 12

Evaluate the following integral. $\int _ { 3 x } ^ { x ^ { 3 } } \frac { 4 y } { x } d y$
A) $2 \left( x ^ { 5 } - 9 x \right)$
B) $4 \left( x ^ { 5 } - 9 x \right)$
C) $2 \left( x ^ { 5 } - 3 x \right)$
D) $2 \left( 9 x - x ^ { 5 } \right)$
E)none of the above

Accepted Answer

The integral evaluates to $ \frac{4}{x} \left( \frac{(x^3)^2}{2} - \frac{(3x)^2}{2} \right) = 2(x^5 - 9x) $.

Question 13

An agronomist used four test plots to determine the relationship between the wheat yield $y$ (in bushels per acre) and the amount of fertilizer $x$ (in pounds per acre). The results are shown in the table. $\begin{array} { | c | c | c | c | c | } 
\hline \text { Fertilizer, } x & 100 & 150 & 200 & 250 \
\hline \text { Yield, } y & 35 & 44 & 50 & 56 \
\hline
\end{array}$ (a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data. (b) Estimate the yield for a fertilizer application of 160 pounds per acre.
A) $y = 0.15 x + 15.1$
B) $y = 0.155 x + 21.1$
C) $y = 0.138 x + 22.1$
D) $y = 0.052 x + 34$
E) $y = 0.234 x + 17.5$

Accepted Answer

The answer of An agronomist used four test plots to...

Question 14

Evaluate the double integral $\int _ { 0 } ^ { 2 } \int _ { y } ^ { 2 y } \left( 1 + x ^ { 2 } + y ^ { 2 } \right) d x d y$ . Round your answer to two decimal places, where applicable.
A)7.33
B)14.33
C)16.33
D)15.33
E)14.83

Accepted Answer

The answer of Evaluate the double integral \(\int _ {...

Question 15

Find the sum of the squared errors for the linear model $f ( x )$ and the quadratic model $g ( x )$ using the given points. $\begin{array} { l } 
f ( x ) = 1.6 x + 6 , g ( x ) = 0.29 x ^ { 2 } + 2.2 x + 6 \
( - 3,2 ) , ( - 2,2 ) , ( - 1,4 ) , ( 0,6 ) , ( 18 )
\end{array}$
A) $S = 1.5$ ; $S = 0.7159$
B) $S = 1.6$ ; $S = 0.8259$
C) $S = 1.2$ ; $S = 0.8623$
D) $S = 1.3$ ; $S = 0.4160$
E) $S = 1.1$ ; $S = 0.7621$

Accepted Answer

The sum of squared errors for the linear model $f(x)$ is 1.6, and for the quadratic model $g(x)$ is 0.8259.

Question 16

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A linear regression model with a positive correlation will have a slope that is greater than 0.
A)True
B)False

Accepted Answer

A positive correlation between two variables means that as one variable increases, the other variable also increases. In a linear regression model, the slope measures the rate of change in the dependent variable for every one-unit increase in the independent variable. Therefore, if there is a positive correlation, the slope must be greater than 0 because as the independent variable increases, the dependent variable must also increase in a linear fashion.

Question 17

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. When the correlation coefficient is $r \approx - 0.98781$ , the model is a good fit.
A)False
B)True

Accepted Answer

The correlation coefficient $r \approx -0.98781$ indicates a very strong negative linear relationship between the variables, suggesting the model is a good fit.

Question 18

A rancher plans to use an existing stone wall and the side of a barn as a boundary for two adjacent rectangular corrals. Fencing for the perimeter costs 15 per foot. To separate the corrals, a fence that costs 6 per foot will divide the region. The total area of the two corrals is to be $6000$ square feet. Use Lagrange multipliers to find the dimensions that will minimize the cost of the fencing. 
A)dimensions: $50$ feet by $60$ feet
B)dimensions: $60$ feet by $50$ feet
C)dimensions: $25$ feet by $60$ feet
D)dimensions: $30$ feet by $50$ feet
E)dimensions: $25$ feet by $30$ feet

Accepted Answer

The answer of A rancher plans to use an existing...

Question 19

Use a double integral to find the area of the region bounded by the graphs of $y = x ^ { 13 / 2 }$ and $y = x$ .
A) $2$
B) $11$
C) $\frac { 13 } { 2 }$
D) $\frac { 13 } { 15 }$
E) $\frac { 11 } { 30 }$

Accepted Answer

The area between the curves $y = x^{13/2}$ and $y = x$ from $x = 0$ to $x = 1$ is given by the integral $\int_0^1 (x - x^{13/2}) \, dx$. Evaluating this integral gives $\frac{11}{30}$.

Question 20

Find the least squares regression line for the given points. Then plot the points and sketch the regression line. $( - 2 , - 1 ) , ( 0,0 ) , ( 2,3 )$
A) $y = x + \frac { 2 } { 3 }$
B) $y = x - \frac { 2 } { 3 }$
C) $y = x - \frac { 1 } { 3 }$
D) $y = x - \frac { 1 } { 2 }$
E) $y = x + \frac { 1 } { 3 }$

Accepted Answer

To find the least squares regression line, we first calculate the means of the x-values ($\bar{x}$) and y-values ($\bar{y}$). For the given points $(-2, -1)$, $(0, 0)$, and $(2, 3)$, $\bar{x} = 0$ and $\bar{y} = \frac{2}{3}$. Then, we calculate the slope ($m$) of the regression line using the formula $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$. Plugging in the values, we find $m = 1$. Thus, the equation of the regression line is $y = mx + b$, where $b$ is the y-intercept. Since $\bar{y} = m\bar{x} + b$ and $\bar{x} = 0$, $b = \bar{y} = \frac{2}{3}$. Therefore, the least squares regression line is $y = x + \frac{2}{3}$.

Quiz 14: Functions of Several Variables

Evaluate the double integral $\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } x y e ^ { - \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) } d x d y$ .

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Data that are modeled by $y = 3.29 x - 4.17$ have a negative correlation.

Evaluate the double integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } ( 3 x + 2 y ) d x d y$ . Round your answer to two decimal places, where applicable.

Evaluate the double integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x$ .

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points $( 10,2 ) , ( 1,8 ) , ( 6,9 ) , ( 4,10 ) , ( 16,4 ) , ( 17,4 )$ . Round your answer to three decimal places.

Sketch the region $R$ whose area is given by the following double integral. $\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } d y d x$

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. $( - 4 , - 1 ) , ( - 2,0 ) , ( 2,4 ) , ( 4,5 )$

Evaluate the double integral $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { x } \frac { 2 } { x ^ { 2 } + 8 } d y d x$ .

Find the least squares regression line for the points (1,0) , (6,6) , (11,12) .

Evaluate the following integral. $\int _ { 3 x } ^ { x ^ { 3 } } \frac { 4 y } { x } d y$

Evaluate the double integral $\int _ { 0 } ^ { 2 } \int _ { y } ^ { 2 y } \left( 1 + x ^ { 2 } + y ^ { 2 } \right) d x d y$ . Round your answer to two decimal places, where applicable.

Find the sum of the squared errors for the linear model $f ( x )$ and the quadratic model $g ( x )$ using the given points. $\begin{array} { l } f ( x ) = 1.6 x + 6 , g ( x ) = 0.29 x ^ { 2 } + 2.2 x + 6 \\( - 3,2 ) , ( - 2,2 ) , ( - 1,4 ) , ( 0,6 ) , ( 18 ) \end{array}$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A linear regression model with a positive correlation will have a slope that is greater than 0.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. When the correlation coefficient is $r \approx - 0.98781$ , the model is a good fit.

Use a double integral to find the area of the region bounded by the graphs of $y = x ^ { 13 / 2 }$ and $y = x$ .

Find the least squares regression line for the given points. Then plot the points and sketch the regression line. $( - 2 , - 1 ) , ( 0,0 ) , ( 2,3 )$

Quiz 14: Functions of Several Variables

Evaluate the double integral ∫0∞∫0∞xye−(4x2+9y2) dxdy\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } x y e ^ { - \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) } d x d y∫0∞​∫0∞​xye−(4x2+9y2) dxdy .

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Data that are modeled by y=3.29x−4.17y = 3.29 x - 4.17y=3.29x−4.17 have a negative correlation.

Evaluate the double integral ∫01∫0y(3x+2y) dxdy\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } ( 3 x + 2 y ) d x d y∫01​∫0y​(3x+2y) dxdy . Round your answer to two decimal places, where applicable.

Evaluate the double integral ∫04∫03(3x+4y) dydx\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x∫04​∫03​(3x+4y) dydx .

Sketch the region RRR whose area is given by the following double integral. ∫04∫x2dydx\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } d y d x∫04​∫x​2​dydx

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the given points. (−4,−1) ,(−2,0) ,(2,4) ,(4,5) ( - 4 , - 1 ) , ( - 2,0 ) , ( 2,4 ) , ( 4,5 ) (−4,−1) ,(−2,0) ,(2,4) ,(4,5)

Evaluate the double integral ∫06∫0x2x2+8dydx\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { x } \frac { 2 } { x ^ { 2 } + 8 } d y d x∫06​∫0x​x2+82​dydx .