Question 1

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

Accepted Answer

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)  $f _ { x } = 15 ( 8 y ) ^ { 2 }$ $f _ { y } = 24 ( 5 x ) ^ { 2 }$ 
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 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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.

Accepted Answer

The answer of The daily output for a manufacturer is...

Question 5

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

Accepted Answer

The answer of Find the minimum value of \(f (...

Question 6

Given the following points in the plane, find the slope of the least squares line: (1, 2), (2, 1), (3, 3), and (5, 24)round your answer to two decimal places, if necessary.

Accepted Answer

The answer of Given the following points in the plane,...

Question 7

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

Accepted Answer

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)  $\int _ { y - - 1 } ^ { y - 1 } \int _ { x- - y - 1 } ^ { x - y + 1 } 1 d x d y = 8$ 
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 8

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

Accepted Answer

The answer of Let f (x, y) = 5xy. Compute...

Question 9

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

Accepted Answer

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

Question 10

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?

Accepted Answer

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

Question 11

A manufacturer is planning to sell a new product at the price of $80 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 $70 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

$176,500 o

Question 12

Given the function of three variables f (x, y, z) = xy + xz + yz, evaluate f (7, 3, 9).

Accepted Answer

The answer of Given the function of three variables f...

Question 13

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.

Accepted Answer

D = 693.75

Question 14

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

Accepted Answer

Saddle poi

Question 15

Use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = ex, y = 7, and x = 0

Accepted Answer

Vertical cross sections: \[\begin{array}

Question 16

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

Accepted Answer

The answer of Use Lagrange multipliers to find the maximum...

Question 17

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

Accepted Answer

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 
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 18

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

Accepted Answer

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 
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 19

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

Accepted Answer

The answer of Find the maximum value of \(f (...

Question 20

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

Accepted Answer

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

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

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

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

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.

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

Given the following points in the plane, find the slope of the least squares line: (1, 2), (2, 1), (3, 3), and (5, 24)round your answer to two decimal places, if necessary.

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

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

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

Given the function of three variables f (x, y, z) = xy + xz + yz, evaluate f (7, 3, 9).

Find the extrema (minima, maxima, saddle points), if any, for f (x, y) = (x - 9)(y + 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 = 7, and x = 0

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

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

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

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

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

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

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

Compute fyf _ { y }fy​ for f(x,y)=5xy3f ( x , y ) = 5 x y ^ { 3 }f(x,y)=5xy3

Find the maximum value of f (x, y, z) = xyz on the sphere x2+y2+z2=12x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 12x2+y2+z2=12 .

Find the minimum value of f(x,y)=xy2f ( x , y ) = x y ^ { 2 }f(x,y)=xy2 subject to the constraint g(x, y) = x - y = 2.

Given the following points in the plane, find the slope of the least squares line: (1, 2), (2, 1), (3, 3), and (5, 24)round your answer to two decimal places, if necessary.

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

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

Compute fxf _ { x }fx​ for f(x,y)=4x6y−3x+exyf ( x , y ) = 4 x ^ { 6 } y - 3 x + e ^ { x y }f(x,y)=4x6y−3x+exy .

Given the function of three variables f (x, y, z) = xy + xz + yz, evaluate f (7, 3, 9).

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

Use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = ex, y = 7, and x = 0

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

Find the minimum and maximum value of f(x,y)=x2+y2f ( x , y ) = x ^ { 2 } + y ^ { 2 }f(x,y)=x2+y2 on the ellipse 3x2+2y2=13 x ^ { 2 } + 2 y ^ { 2 } = 13x2+2y2=1 .

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

Find the maximum value of f(x,y)=xy2f ( x , y ) = x y ^ { 2 }f(x,y)=xy2 subject to the constraint g(x, y) = x - y = 4.

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

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

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

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

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

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

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

Use inequalities to describe R in terms of its vertical and horizontal cross sections.R is the region bounded by y = e^x, y = 7, and x = 0

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

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

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