Deck 7: Calculus of Several Variables

Question 1

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

Answer 1

28

Question 2

$f ( x , y ) = 4 x ^ { 2 } - y$ . Compute f (3, 0).

Answer 2

36

Question 3

Let f (x, y) = 7xy. Compute f (5, 0).

Answer 3

0

Question 4

Compute f (7, 1) if $f ( x , y ) = 7 x y ^ { 3 }$

A) 49
B) 2,401
C) 7
D) 343

Question 5

Let f (x, y) = 5xy. Compute f (9, 0).

Question 6

Compute f (ln 5, ln 6) if $f ( x , y ) = e ^ { 2 x + y }$ .

A) 150
B) 30
C) 16
D) None of the above

Question 7

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

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

Question 8

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

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

Question 9

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

Question 10

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

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

Question 11

What is the domain of $f ( x , y ) = \frac { x } { \left( x ^ { 2 } + y ^ { 2 } - 16 \right) ^ { 1 / 2 } }$ ?

Question 12

What is the domain of $f ( x , y ) = \frac { x } { \left( x ^ { 2 } + y ^ { 2 } - 64 \right) ^ { 1 / 2 } }$ ?

Question 13

$Q ( x , y ) = 3 x y ^ { 3 }$ is the daily production for x skilled and y unskilled workers. What is the exact change in daily production if x changes from 4 to 3 and y from 2 to 1?

Question 14

Compute $f _ { y }$ for $f ( x , y ) = 5 x y ^ { 3 }$

A) 5x
B) $15 x y ^ { 2 }$
C) $15 x y ^ { 2 } + 5 y ^ { 3 }$
D) $y ^ { 3 }$

Question 15

Compute $f _ { x }$ for $f ( x , y ) = 5 x y ^ { 3 }$

A) 5x
B) $15 x y ^ { 2 }$
C) $15 x y ^ { 2 } + 5 y ^ { 3 }$
D) $5 y ^ { 3 }$

Question 16

Compute f_x for f (x, y) = 3x⁸y - 4x +e^xy.

Question 17

Compute $f _ { x }$ for $f ( x , y ) = 4 x ^ { 6 } y - 3 x + e ^ { x y }$ .

Question 18

Compute $f _ { y }$ for $f ( x , y ) = 2 x y ^ { 8 }$ .

A) $16 x y ^ { 7 }$
B) 2x
C) $2 y ^ { 8 }$
D) $16 x y ^ { 7 } + 2 y ^ { 8 }$

Question 19

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

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

Question 20

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 5 x + 8 y ) ^ { 3 }$

A) $f _ { x } = 15 ( 5 x + 8 y ) ^ { 2 }$ $f _ { y } = 24 ( 5 x + 8 y ) ^ { 2 }$
B) $f _ { x } = 15 ( 5 x + 8 y ) ^ { 4 }$ $f _ { y } = 24 ( 5 x + 8 y ) ^ { 4 }$
C) $f _ { x } = 15 ( x + 8 y ) ^ { 2 }$ $f _ { y } = 24 ( 5 x + 8 ) ^ { 2 }$
D) $f _ { x } = 15 ( 8 y ) ^ { 2 }$ $f _ { y } = 24 ( 5 x ) ^ { 2 }$

Question 21

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

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

Question 22

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

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

Question 23

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

Question 24

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

Question 25

If $f ( x , y ) = x e ^ { xy }$ , find $f _ { y } ( 6,2 )$ .

Question 26

If f (y) = x ln y, find $f _ { y } ( 2,1 )$ .

Question 27

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

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

Question 28

Let f (x, y) = x ln(1 + 2x - 5y). Find $f _ { x x } ( x , y )$ .

Question 29

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

A) $36 x ^ { 2 } e ^ { 6 xy }$
B) $36 y ^ { 2 } e ^ { 6 x y }$
C) $36 e ^ { 6 x y }$
D) $36 x ^ { 2 } y ^ { 2 } e ^ { 6 xy }$

Question 30

Find $f _ { y y }$ for $f ( x , y ) = e ^ { xy }$ .

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

Question 31

Let $f ( x , y ) = y e ^ { - 5 x ^ { 2 } y }$ . Find $f _ { y y } ( x , y )$ .

Question 32

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.

Question 33

The daily output for a manufacturer is $Q ( K , L ) = 10 K ^ { 1 / 3 } L ^ { 1 / 2 }$ units. Use marginal analysis to estimate the change in daily output as a result of changing L from 625 to 626 while K remains constant at 216. Round your answer to one decimal place, if necessary.

Question 34

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.

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 35

If $z = x ^ { 2 } + y ^ { 2 }$ , $x = 5 \sqrt { t }$ , and $y = t ^ { 2 } - t$ , find $\frac { d z } { d t }$ when t = 9.

A) -2,111
B) 10,618
C) -8,100
D) 2,473

Question 36

Find the extrema (minima, maxima, saddle points), if any, for f (x, y) = (x - 1)(y + 2).

Question 37

Find the extrema (minima, maxima, saddle points), if any, for f (x, y) = (x - 9)(y + 1).

Question 38

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = 8 x ^ { 2 } - 4 y ^ { 2 }$ .

Question 39

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

Question 40

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = x ^ { 2 } - 6 x + y ^ { 2 } - 8 y$ .

Question 41

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

Question 42

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = x ^ { 2 } - 14 x + y ^ { 2 } - 6 y$ .

Question 43

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 4 } + y ^ { 4 } - 2 x ^ { 2 } - 2 y ^ { 2 }$ .

Question 44

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

Question 45

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 5 x - 5 y$ .

Question 46

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

Question 47

A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms. The price the manufacturer can expect to receive for the machines will depend on the number of machines made available. (For example, if only a few of the machines are placed on the market, competitive bidding among prospective purchasers will tend to drive the price up.) It is estimated that if the manufacturer supplies x machines to the domestic market and y machines to the foreign market, the machines will sell for $60 - \frac { x } { 5 } + \frac { y } { 20 }$ thousand dollars apiece at home and for $50 - \frac { y } { 10 } + \frac { x } { 20 }$ thousand dollars apiece abroad. If the manufacturer can produce the machines at a cost of $31,000 apiece, how many should be supplied to each market to generate the largest possible profit?

A) x = 150, y = 11
B) x = 38, y = 593
C) x = 110, y = 150
D) x = 55, y = 750

Question 48

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

Question 49

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.

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

Question 50

The following data shows the age and income for a small number of people. $$\begin{array} { c l l l l l } \text { Age (years) } & 24 & 32 & 39 & 54 & 60 \\\text { Incomes (\$) } & 30,000 & 34,000 & 53,000 & 72,000 & 90,000\end{array}$$ 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 = 1,666x - 13,842
B) y = 4,012x - 13,837
C) y = 3,288x - 13,849
D) None of the above

Question 51

The accompanying table lists the high-school GPA and college GPA for a number of students $$\begin{array} { l l l l l l l } \text { High school GPA } & 2.0 & 2.5 & 3.1 & 3.7 & 3.7 & 4.0 \\\text { College GPA } & 3.2 & 2.6 & 3.0 & 3.6 & 3.8 & 3.7\end{array}$$
Using the best fit straight line, predict the college GPA (to one decimal place) for a student whose high school GPA was 3.5.

A) 3.4
B) 3.3
C) 3.6
D) 3.5

Question 52

The accompanying table gives the Dow Jones Industrial Average (DJIA) at the close of the first trading day of the indicated years $$\begin{array} { l l l l l l l } \text { Year } & 1990 & 1992 & 1996 & 1998 & 2001 & 2002 \\\text { DJLA } & 2,810 & 3,172 & 5,177 & 7,965 & 10,646 & 10,073\end{array}$$
Find the least squares line for the DJIA, D, as a function of the year after 1990, t. Round numbers to two decimal places.

Question 53

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?

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

Question 54

A military radar is measuring the distance to a jet fighter. The radar has received the following measuremens: $$\begin{array} { l l l l l l l } \text { time } t \text { (minutes) } & 1 & 2 & 3 & 4 & 5 & 6 \\\text { distance (miles) } & 280 & 291 & 296 & 301 & 308 & 310\end{array}$$
Using a least squares fit to the data, extrapolate to the nearest tenth of a minute when the jet will be 388 miles away?

A) 18.8 minutes
B) 17.7 minutes
C) 21 minutes
D) 20.2 minutes

Question 55

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

Question 56

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

Question 57

Find the minimum value of f (x, y) = x + 2y subject to the constraint $g ( x , y ) = x y ^ { 2 } = 8$ .

Question 58

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

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 59

Find the maximum value of $f ( x , y ) = x y ^ { 2 }$ subject to the constraint g(x, y) = x - y = 4.

Question 60

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

Question 61

Find the minimum value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } - 4 y + 4$ on the hyperbola $x ^ { 2 } - y ^ { 2 } = 4$ .

A) $12 \sqrt { 3 }$
B) 43
C) 6
D) No minimum, unbounded below

Question 62

Find the minimum and maximum value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ on the ellipse $3 x ^ { 2 } + 2 y ^ { 2 } = 1$ .

A) Minimum $\frac { 1 } { 4 }$ , maximum $\frac { 5 } { 12 }$
B) Minimum $\frac { 1 } { 3 }$ , maximum $\frac { 1 } { 2 }$
C) Minimum $\frac { 1 } { 6 }$ , maximum $\frac { 5 } { 6 }$
D) No maximum or minimum, unbounded

Question 63

Find the maximum value of f (x, y) = xy on the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ .

Question 64

Find the maximum value of f (x, y, z) = xyz on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 12$ .

A) 8
B) $8 \sqrt { 3 }$
C) 16
D) No maximum, unbounded

Question 65

Use Lagrange multipliers to find the maximum value of f (x, y, z) = 6xyz subject to 4x + 3y + 4z = 144.

A) f (0, 0, 0) = 0
B) f (15, 17, 10) = 15,300
C) f (12, 16, 12) = 13,824
D) f (13, 14, 12) = 13,104

Question 66

Find the maximum value of the function f (x, y, z) = 4x + 5y + 7z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 360$ .

A) 630
B) 480.0
C) 90
D) 180

Question 67

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.]

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 68

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

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

Question 69

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.

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 70

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the rectangle with vertices (1, -3), (5, -3), (5, 2), (1, 2).

A) Vertical cross sections: $\begin{array} { r } - 3 \leq x \leq 2 \\1 \leq y \leq 5\end{array}$ Horizontal cross sections: $\begin{array} { r } - 3 \leq y \leq 2 \\1 \leq x \leq 5\end{array}$


B) Vertical cross sections: $\begin{array} { r } 1 \leq x \leq 5 \\- 3 \leq y \leq 2\end{array}$ Horizontal cross sections: $\begin{array} { r } 1 \leq y \leq 5 \\- 3 \leq x \leq 2\end{array}$


C) Vertical cross sections: $\begin{array} { r } - 3 \leq x \leq 2 \\1 \leq y \leq 5\end{array}$ Horizontal cross sections: $\begin{array} { r } 1 \leq y \leq 5 \\- 3 \leq x \leq 2\end{array}$


D) Vertical cross sections: $\begin{array} { r } 1 \leq x \leq 5 \\- 3 \leq y \leq 2\end{array}$ Horizontal cross sections: $\begin{array} { r } - 3 \leq y \leq 2 \\1 \leq x \leq 5\end{array}$

Question 71

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = e^x, y = 4, and x = 0.

A) Vertical cross sections: $\begin{aligned}0 & \leq x \leq \ln 4 \\e ^ { x } & \leq y \leq 4\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 1 \leq y \leq 4 \\0 \leq x \leq \ln y\end{array}$

B) Vertical cross sections: $\begin{aligned}0 & \leq x \leq \ln y \\e ^ { x } & \leq y \leq 4\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 1 \leq y \leq 4 \\0 \leq x \leq \ln 4\end{array}$

C) Vertical cross sections: $\begin{aligned}e ^ { x } & \leq x \leq 4 \\0 & \leq y \leq \ln 4\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 0 \leq y \leq \ln 4 \\1 \leq x \leq 4\end{array}$

D) Vertical cross sections: $\begin{array} { c } e ^ { x } \leq x \leq \ln 4 \\0 \leq y \leq 4\end{array}$ Horizontal cross sections: $\begin{array} { l } 0 \leq y \leq 4 \\1 \leq x \leq \ln 4\end{array}$

Question 72

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.

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

Question 73

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

Question 74

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

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 75

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.

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 76

Find the volume of the solid bounded above by the graph of the function f (x, y) = 4x - y + 8 and below by the rectangular region R defined by: 0 $\le$ x $\le$ 2 and 0 $\le$ y $\le$ 1.

Question 77

Find the volume of the solid bounded above by the graph of the function f (x, y) = xy and below by the rectangular region R defined by: 0 $\le$ x $\le$ 3 and 0 $\le$ y $\le$ 4.

Question 78

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.

Question 79

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).

Question 80

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.