Question 1

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

Accepted Answer

The answer of If f (y) = x ln y,...

Question 2

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.

Accepted Answer

A)  y = 1,666x - 13,842 
B)  y = 4,012x - 13,837 
C)  y = 3,288x - 13,849 
D)  None of the above 
A)  y = 1,666x - 13,842 
B)  y = 4,012x - 13,837 
C)  y = 3,288x - 13,849 
D)  None of the above

Question 3

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

Accepted Answer

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

Question 4

$Q ( x , y ) = 3 x y ^ { 3 }$ 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?

Accepted Answer

The answer of \(Q ( x , y ) =...

Question 5

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

Accepted Answer

Saddle poi

Question 6

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.

Accepted Answer

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

Question 7

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.

Accepted Answer

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

Question 8

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

Accepted Answer

A)  $36 x ^ { 2 } e ^ { 6 xy }$ 
B)  $36 y ^ { 2 } e ^ { 6 x y }$ 
C)  $36 e ^ { 6 x y }$ 
D)  $36 x ^ { 2 } y ^ { 2 } e ^ { 6 xy }$ 
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 9

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?

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

At a certain factory, the output Q is related to inputs x and y by the expression $Q ( x , y ) = 9 x ^ { 2 } + 6 x ^ { 3 } y ^ { 4 } + 4 y ^ { 4 }$ . If 0 $\le$ x $\le$ 4 and 0 $\le$y $\le$3, what is the average output of the factory? (Approximate to 2 decimal places.)

Accepted Answer

A)  1,669.33 units 
B)  536.07 units 
C)  2,226.60 units 
D)  1,668.00 units 
A)  1,669.33 units 
B)  536.07 units 
C)  2,226.60 units 
D)  1,668.00 units

Question 12

Evaluate the following double integral: $$\int _ { - 1 } ^ { 9 } \int _ { - 5 } ^ { 5 } x ^ { 4 } y ^ { 9 } d y d x$$

Accepted Answer

The answer of Evaluate the following double integral: \[\int _...

Question 13

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).

Accepted Answer

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: $\begin{array} { r } 
- 3 \leq y \leq 2 \
1 \leq x \leq 5
\end{array}$ 
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: $\begin{array} { r } 
- 3 \leq y \leq 2 \
1 \leq x \leq 5
\end{array}$

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

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

$Q ( x , y ) = 3 x y ^ { 3 }$ 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?

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

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.

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

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

At a certain factory, the output Q is related to inputs x and y by the expression $Q ( x , y ) = 9 x ^ { 2 } + 6 x ^ { 3 } y ^ { 4 } + 4 y ^ { 4 }$ . If 0 $\le$ x $\le$ 4 and 0 $\le$ y $\le$ 3, what is the average output of the factory? (Approximate to 2 decimal places.)

Evaluate the following double integral: $\int _ { - 1 } ^ { 9 } \int _ { - 5 } ^ { 5 } x ^ { 4 } y ^ { 9 } d y d x$

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).

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

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

Compute f (ln 5, ln 6) if f(x,y)=e2x+yf ( x , y ) = e ^ { 2 x + y }f(x,y)=e2x+y .

Q(x,y)=3xy3Q ( x , y ) = 3 x y ^ { 3 }Q(x,y)=3xy3 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?

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

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.

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

Find fyyf _ { y y }fyy​ for f(x,y)=exyf ( x , y ) = e ^ { xy }f(x,y)=exy .

Evaluate the following double integral: ∫−19∫−55x4y9dydx\int _ { - 1 } ^ { 9 } \int _ { - 5 } ^ { 5 } x ^ { 4 } y ^ { 9 } d y d x∫−19​∫−55​x4y9dydx

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).

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

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

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

$Q ( x , y ) = 3 x y ^ { 3 }$ 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?

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.

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

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

Evaluate the following double integral: $\int _ { - 1 } ^ { 9 } \int _ { - 5 } ^ { 5 } x ^ { 4 } y ^ { 9 } d y d x$