A company sells two products whose demand functions are given by x1 = 500 - 3p1 and x2 = 750 - 2.4p2 . So, the total revenue is given by R = x1p1 + x2p2 . Estimate the average revenue if the price p1 varies between $35 and $60 and the price p2 varies between $35 and $60.

A) $ 45,895 B) $ 48,045 C) $ 45,393 D) $ 46,910 E) $ 48,455 A) $ 45,895 B) $ 48,045 C) $ 45,393 D) $ 46,910 E) $ 48,455

Exam 14: Functions of Several Variables

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

(Multiple Choice)

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Question 81

Find the four second partial derivatives. Observe that the second mixed partials are equal. $z = x ^ { 2 } + 8 x y + 6 y ^ { 2 }$

(Multiple Choice)

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Question 82

Use Lagrange multipliers to minimize the function $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the following constraint. $x + y + z - 21 = 0$ Assume that x, y, and z are positive.

(Multiple Choice)

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Question 83

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.

(Multiple Choice)

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Question 84

Examine the function $f ( x , y ) = x ^ { 3 } - 12 x y + y ^ { 3 } + 4$ for relative extrema and saddle points.

(Multiple Choice)

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Question 85

A company sells two products whose demand functions are given by x₁ = 500 - 3p₁ and x₂ = 750 - 2.4p₂ . So, the total revenue is given by R = x₁p₁ + x₂p₂ . Estimate the average revenue if the price p₁ varies between $35 and $60 and the price p₂ varies between $35 and $60.

(Multiple Choice)

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Question 86

A rectangular box is resting on the $x y$ -plane with one vertex at the origin. The opposite lies in the plane Find the dimensions that maximize the volume. (Hint: Maximize $V = x y z$ subject to the constraint $2 x + 3 y + 5 z - 90 = 0$ ).

(Multiple Choice)

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Question 87

A company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing x freestanding and y fireplace-insert stoves is $C = 30 \sqrt { x y } + 160 x + 195 y + 1050$ . Find the marginal costs ( $\partial C / \partial x$ and $\partial c / \partial y$ ) when $x = 60$ and $y = 10$ . Round your answers to two decimal places.

(Multiple Choice)

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Question 88

The value $A ( t , r )$ of an investment of $4,000 after t years in an account for which the interest rate 100r% is compounded continuously is given by the function $A ( t , r ) = 4,000 e ^ { r t }$ dollars. Write the partial derivative $\frac { \partial A } { \partial t }$

(Multiple Choice)

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Question 89

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.

(Multiple Choice)

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Question 90

Examine the function given below for relative extrema and saddle points. $f ( x , y ) =$ $( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 }$

(Multiple Choice)

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Question 91

The two planes $x - 8 y - 9 z = 4$ and $4 x - 21 y - 7 z = - 4$ are parallel.

(Multiple Choice)

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Question 92

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

Find the four second partial derivatives. Observe that the second mixed partials are equal. $z = x ^ { 2 } + 8 x y + 6 y ^ { 2 }$

Use Lagrange multipliers to minimize the function $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the following constraint. $x + y + z - 21 = 0$ Assume that x, y, and z are positive.

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.

Examine the function $f ( x , y ) = x ^ { 3 } - 12 x y + y ^ { 3 } + 4$ for relative extrema and saddle points.

A company sells two products whose demand functions are given by x₁ = 500 - 3p₁ and x₂ = 750 - 2.4p₂ . So, the total revenue is given by R = x₁p₁ + x₂p₂ . Estimate the average revenue if the price p₁ varies between $35 and $60 and the price p₂ varies between $35 and $60.

A rectangular box is resting on the $x y$ -plane with one vertex at the origin. The opposite lies in the plane Find the dimensions that maximize the volume. (Hint: Maximize $V = x y z$ subject to the constraint $2 x + 3 y + 5 z - 90 = 0$ ).

The value $A ( t , r )$ of an investment of $4,000 after t years in an account for which the interest rate 100r% is compounded continuously is given by the function $A ( t , r ) = 4,000 e ^ { r t }$ dollars. Write the partial derivative $\frac { \partial A } { \partial t }$

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.

Examine the function given below for relative extrema and saddle points. $f ( x , y ) =$ $( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 }$

The two planes $x - 8 y - 9 z = 4$ and $4 x - 21 y - 7 z = - 4$ are parallel.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

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Exam 14: Functions of Several Variables

Find the sum of the squared errors for the linear model f(x)f ( x )f(x) and the quadratic model g(x)g ( x )g(x) using the given points. f(x)=1.6x+6,g(x)=0.29+2.2x+6 (-3,2),(-2,2),(-1,4),(0,6),(18)

Find the four second partial derivatives. Observe that the second mixed partials are equal. z=x2+8xy+6y2z = x ^ { 2 } + 8 x y + 6 y ^ { 2 }z=x2+8xy+6y2

Use Lagrange multipliers to minimize the function f(x,y,z)=x2+y2+z2f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }f(x,y,z)=x2+y2+z2 subject to the following constraint. x+y+z−21=0x + y + z - 21 = 0x+y+z−21=0 Assume that x, y, and z are positive.

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.

Examine the function f(x,y)=x3−12xy+y3+4f ( x , y ) = x ^ { 3 } - 12 x y + y ^ { 3 } + 4f(x,y)=x3−12xy+y3+4 for relative extrema and saddle points.

A company sells two products whose demand functions are given by x1 = 500 - 3p1 and x2 = 750 - 2.4p2 . So, the total revenue is given by R = x1p1 + x2p2 . Estimate the average revenue if the price p1 varies between $35 and $60 and the price p2 varies between $35 and $60.

A rectangular box is resting on the xyx yxy -plane with one vertex at the origin. The opposite lies in the plane Find the dimensions that maximize the volume. (Hint: Maximize V=xyzV = x y zV=xyz subject to the constraint 2x+3y+5z−90=02 x + 3 y + 5 z - 90 = 02x+3y+5z−90=0 ).

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.

Examine the function given below for relative extrema and saddle points. f(x,y)=f ( x , y ) =f(x,y)= (x−1)2+(y−3)2( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 }(x−1)2+(y−3)2

The two planes x−8y−9z=4x - 8 y - 9 z = 4x−8y−9z=4 and 4x−21y−7z=−44 x - 21 y - 7 z = - 44x−21y−7z=−4 are parallel.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

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

Find the four second partial derivatives. Observe that the second mixed partials are equal. $z = x ^ { 2 } + 8 x y + 6 y ^ { 2 }$

Use Lagrange multipliers to minimize the function $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the following constraint. $x + y + z - 21 = 0$ Assume that x, y, and z are positive.

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.

Examine the function $f ( x , y ) = x ^ { 3 } - 12 x y + y ^ { 3 } + 4$ for relative extrema and saddle points.

A company sells two products whose demand functions are given by x₁ = 500 - 3p₁ and x₂ = 750 - 2.4p₂ . So, the total revenue is given by R = x₁p₁ + x₂p₂ . Estimate the average revenue if the price p₁ varies between $35 and $60 and the price p₂ varies between $35 and $60.

A rectangular box is resting on the $x y$ -plane with one vertex at the origin. The opposite lies in the plane Find the dimensions that maximize the volume. (Hint: Maximize $V = x y z$ subject to the constraint $2 x + 3 y + 5 z - 90 = 0$ ).

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.

Examine the function given below for relative extrema and saddle points. $f ( x , y ) =$ $( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 }$

The two planes $x - 8 y - 9 z = 4$ and $4 x - 21 y - 7 z = - 4$ are parallel.