Question 1

Find the limit.

Accepted Answer

The answer of Find the limit. 
...

Question 2

For  find all values of x and y such that  and  simultaneously. ​

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The answer of For  find all values of x...

Question 3

Find the limit  for the function  . ​

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The answer of Find the limit  for the function...

Question 4

Find the least squares regression line for the points  . Round numerical values in your answer to two decimal places.

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Question 5

Use Lagrange multipliers to find the minimum distance from the plane  to the point  . Round your answer to two decimal places.

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Question 6

For  , evaluate  at the point  . ​

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Question 7

Use the gradient to find the directional derivative of the function at P in the direction of Q.

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Question 8

Find  using the appropriate Chain Rule for  where  and  , and evaluate the partial derivative at  and  . Round your answer to two decimal places. ​

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Question 9

Find and simplify the function  at the given value  .

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Question 10

Find three positive numbers x, y, and z whose sum is 32 and  is a maximum. ​

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Question 11

Suppose 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  . Find the marginal costs  when  and  . Round your answer to the nearest integer. ​

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Question 12

The two radii of the frustum of a right circular cone are increasing at a rate of 5 centimeters per minute, and the height is increasing at a rate of 12 centimeters per minute (see figure). Find the rate at which the volume is changing when the two radii are 18 centimeters and 30 centimeters, and the height is 14 centimeters.

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Question 13

Assume in 2009, an investment of $2200 was made in a bond earning 9% compounded annually. Assume that the buyer pays at rate R and the annual rate of inflation is I. Suppose in the year 2019, the value V of the investment in constant 2009 dollars is  . Calculate the function  for  and  . Round your answer to the nearest cent.

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Question 14

Suppose a trough with trapezoidal cross sections is formed by turning up the edges of a 30-inch-wide sheet of aluminum (see figure). Find the cross section of maximum area. ​  ​

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Question 15

Use Lagrange multipliers to maximize the function  subject to the following constraint.  Assume that x and y are positive.

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Question 16

Suppose the utility function  is a measure of the utility (or satisfaction) derived by a person from the consumption of two products x and y. Determine the marginal utility of product y if the utility function is  . ​

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Question 17

Find the limit. ​  ​

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...

Question 18

Find and simplify the function  at the given value  .

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Question 19

Find symmetric equations of the normal line to the surface  at the point  .

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Question 20

For  , evaluate  at the point  . ​

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Quiz 13: Functions of Several Variables

Find the limit.

For find all values of x and y such that and simultaneously.

Find the limit for the function .

Find the least squares regression line for the points . Round numerical values in your answer to two decimal places.

Use Lagrange multipliers to find the minimum distance from the plane to the point . Round your answer to two decimal places.

For , evaluate at the point .

Use the gradient to find the directional derivative of the function at P in the direction of Q.

Find using the appropriate Chain Rule for where and , and evaluate the partial derivative at and . Round your answer to two decimal places.

Find and simplify the function at the given value .

Find three positive numbers x, y, and z whose sum is 32 and is a maximum.

Suppose 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 . Find the marginal costs when and . Round your answer to the nearest integer.

Suppose a trough with trapezoidal cross sections is formed by turning up the edges of a 30-inch-wide sheet of aluminum (see figure) . Find the cross section of maximum area.

Use Lagrange multipliers to maximize the function subject to the following constraint. Assume that x and y are positive.

Suppose the utility function is a measure of the utility (or satisfaction) derived by a person from the consumption of two products x and y. Determine the marginal utility of product y if the utility function is .

Find the limit.

Find and simplify the function at the given value .

Find symmetric equations of the normal line to the surface at the point .

For , evaluate at the point .

Quiz 13: Functions of Several Variables

Find the limit.

For find all values of x and y such that and simultaneously. ​

Find the limit for the function . ​

Find the least squares regression line for the points . Round numerical values in your answer to two decimal places.

Use Lagrange multipliers to find the minimum distance from the plane to the point . Round your answer to two decimal places.

For , evaluate at the point . ​

Use the gradient to find the directional derivative of the function at P in the direction of Q.

Find using the appropriate Chain Rule for where and , and evaluate the partial derivative at and . Round your answer to two decimal places. ​

Find and simplify the function at the given value .

Find three positive numbers x, y, and z whose sum is 32 and is a maximum. ​

Suppose 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 . Find the marginal costs when and . Round your answer to the nearest integer. ​

Suppose a trough with trapezoidal cross sections is formed by turning up the edges of a 30-inch-wide sheet of aluminum (see figure) . Find the cross section of maximum area. ​ ​

Use Lagrange multipliers to maximize the function subject to the following constraint. Assume that x and y are positive.

Suppose the utility function is a measure of the utility (or satisfaction) derived by a person from the consumption of two products x and y. Determine the marginal utility of product y if the utility function is . ​

Find the limit. ​ ​

Find and simplify the function at the given value .

Find symmetric equations of the normal line to the surface at the point .

For , evaluate at the point . ​

For find all values of x and y such that and simultaneously.

Find the limit for the function .

For , evaluate at the point .

Find using the appropriate Chain Rule for where and , and evaluate the partial derivative at and . Round your answer to two decimal places.

Find three positive numbers x, y, and z whose sum is 32 and is a maximum.

Suppose 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 . Find the marginal costs when and . Round your answer to the nearest integer.

Suppose a trough with trapezoidal cross sections is formed by turning up the edges of a 30-inch-wide sheet of aluminum (see figure) . Find the cross section of maximum area.

Suppose the utility function is a measure of the utility (or satisfaction) derived by a person from the consumption of two products x and y. Determine the marginal utility of product y if the utility function is .

Find the limit.

For , evaluate at the point .