Deck 7: Calculus of Several Variables

Question 1

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

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

Answer 1

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

Question 2

Given the following points in the plane,find the slope of the least squares line: (1,2),(2,1),(3,3),and (5,6) round your answer to two decimal places,if necessary.

A)0.97
B)1.03
C)1.14
D)1.09

Answer 2

1.14

Question 3

Compute f (ln 2,ln 5)if $f ( x , y ) = e ^ { 2 x + y }$ .

A)10
B)9
C)20
D)None of the above

Answer 3

20

Question 4

Use a double integral to find the area of R. R is the triangle with vertices (-2,1),(2,1),and (0,-1).

A) $\int _ { x - 2 } ^ { x - 2 } \int _ { y - x - 1 } ^ { y - x - 1 } 1 d y d x = 4$
B) $\int _ { y - 1 } ^ { y - 1 } \int _ { x - - y - 1 } ^ { x - y + 1 } 1 d x d y = 4$
C) $\int _ { x = - 2 } ^ { x - 2 } \int _ { y - x - 1 } ^ { y - x - 1 } 1 d y d x = 8$
D) $\int _ { y - 1 } ^ { y - 1 } \int _ { x - - y - 1 } ^ { x - y + 1 } 1 d x d y = 8$

Question 5

Given the function of three variables f (x,y,z)= xy + xz + yz,evaluate f (3,7,7).

A)86
B)100
C)91
D)94

Question 6

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

A) $64 e ^ { 8 x }$
B) $64 x ^ { 2 } e ^ { 8 x }$
C) $64 x ^ { 2 } y ^ { 2 } e ^ { 8 x }$
D) $64 y ^ { 2 } e ^ { 8 x }$

Question 7

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }$

A) $f _ { x } = 16 ( 4 x + 2 y ) ^ { 5 }$ , $f _ { y } = 8 ( 4 x + 2 y ) ^ { 5 }$
B) $f _ { x } = 16 ( x + 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x + 2 ) ^ { 3 }$
C) $f _ { x } = 16 ( 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x ) ^ { 3 }$
D) $f _ { x } = 16 ( 4 x + 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x + 2 y ) ^ { 3 }$

Question 8

A military radar is measuring the distance to a jet fighter.The radar has received the following measurements:
$$\begin{array} { l l l l l l l } \text { time } t \text { (minutes) } & 1 & 2 & 3 & 4 & 5 & 6 \\\text { distance (miles) } & 280 & 289 & 294 & 299 & 308 & 310\end{array}$$
Using a least squares fit to the data,extrapolate to the nearest tenth of a minute when the jet will be 345 miles away?

A)10.4 minutes
B)13.7 minutes
C)12.9 minutes
D)11.5 minutes

Question 9

Given the following points in the plane,find the y-intercept of the least squares line to two decimal places: (3,3),(5,5),(7,8),and (9,8)

A)-0.90
B)-1.50
C)0.60
D)0.72

Question 10

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

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

Question 11

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

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

Question 12

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

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

Question 13

A manufacturer is planning to sell a new product at the price of $230 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is spent on promotion,approximately $\frac { 340 y } { y + 7 } + \frac { 170 x } { x + 14 }$ units of the product will be sold.The cost of manufacturing the product is $160 per unit.If the manufacturer has a total of $340,000 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.]

A)$173,500 on development,$167,000 on promotion
B)$167,000 on development,$173,000 on promotion
C)$166,500 on development,$173,500 on promotion
D)$173,000 on development,$167,000 on promotion

Question 14

Find the maximum value of the function f (x,y,z)= 7x + 8y + 6z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 596$ .

A)298
B)894
C)1,043.0
D)149

Question 15

At a certain factory,the output Q is related to inputs x and y by the expression $Q ( x , y ) = 2 x ^ { 3 } + 2 x ^ { 2 } y ^ { 4 } + 9 y ^ { 5 }$ .If 0 $\leq$ x $\leq$ 8 and 0 $\leq$ y $\leq$ 10,what is the average output of the factory? (Approximate to 2 decimal places.)

A)111,504.00 units
B)235,589.33 units
C)235,590.13 units
D)308,341.33 units

Question 16

Use Lagrange multipliers to find the maximum value of f (x,y)= 9xy subject to the constraint 9x + 3y = 27.

A)f (0,0)= 0
B)f (3,9)= 243
C) $f \left( \frac { 3 } { 4 } , \frac { 9 } { 4 } \right) = \frac { 243 } { 16 }$
D) $f \left( \frac { 3 } { 2 } , \frac { 9 } { 2 } \right) = \frac { 243 } { 4 }$

Question 17

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

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

Question 18

The following data shows the age and income for a small number of people.
$$\begin{array} { r l l l l l } \text { Age (years) } & 24 & 32 & 39 & 54 & 60 \\\text { Incomes (\$) } & 30,000 & 34,000 & 53,000 & 72,000 & 81,000\end{array}$$
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.

A)y = 3,830x - 8,036
B)y = 3,106x - 8,048
C)y = 1,484x - 8,041
D)None of the above

Question 19

The only grocery store in a small rural community carries two brands of frozen apple juice,a local brand that it obtains at the cost of 18 cents per can and a well-known national brand that it obtains at the cost of 40 cents per can.The grocer estimates that if the local brand is sold for x cents per can and the national brand for y cents per can,approximately 70 - 5x + 4y cans of the local brand and 80 + 6x - 7y cans of the national brand will be sold each day.How should the grocer price each brand to maximize the profit from the sale of the juice?

A)local brand (x)at 39 cents,national brand (y)at 104 cents
B)local brand (x)at 46 cents,national brand (y)at 52 cents
C)local brand (x)at 44 cents,national brand (y)at 52 cents
D)local brand (x)at 88 cents,national brand (y)at 50 cents

Question 20

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),(2,0),and (0,1)

A)18
B)12
C) $\frac { 68 } { 3 }$
D) $\frac { 94 } { 3 }$