Question 1

Examine the function   for relative extrema. ​

Accepted Answer

A)  relative minimum:   
B)  relative maximum:  
​ 
C)  relative maximum:   
D)  relative maximum:   
E)  relative minimum:   
A)  relative minimum:   
B)  relative maximum:  
​ 
C)  relative maximum:   
D)  relative maximum:   
E)  relative minimum:

Question 2

Find the point(s) on the hyperboloid   where the tangent plane is perpendicular to the line with parametric equations   ,   and   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 3

Given   , use the total differential to approximate   at   towards   . Round your answer to four decimal places. ​

Accepted Answer

A)  -32.5115 
B)  -0.0200 
C)  25.4885 
D)  2.9770 
E)  -9.5115 
A)  -32.5115 
B)  -0.0200 
C)  25.4885 
D)  2.9770 
E)  -9.5115

Question 4

Suppose a home improvement contractor is painting the walls and ceiling of a rectangular room. The volume of the room is 911.25 cubic feet. The cost of wall paint is $0.08 per square foot and the cost of ceiling paint is $0.19 per square foot. Let x, y, and z be the length, width, and height of a rectangular room respectively. Find the room dimensions that result in a minimum cost for the paint. Round your answers to two decimal places.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

Find the limit.

Accepted Answer

A)    
B)    
C)  0 
D)  1 
E)    
A)    
B)    
C)  0 
D)  1 
E)

Question 6

Find the partial derivative   for the function   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 7

Use spherical coordinates to find the limit   . [Hint: Let

Accepted Answer

A)  2 
B)  8 
C)  9 
D)  1 
E)  0 
A)  2 
B)  8 
C)  9 
D)  1 
E)  0

Question 8

Find the gradient of the function at the given point.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 9

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 10

Find the total differential of the function   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 11

Let   , where   and   . Find   . ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 12

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

Accepted Answer

A)  -15.00 
B)  -10.00 
C)  -12.00 
D)    7.39 
E)  -16.00 
A)  -15.00 
B)  -10.00 
C)  -12.00 
D)    7.39 
E)  -16.00

Question 13

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

Accepted Answer

A)  1,210.01 
B)  806.73 
C)  1,210.16 
D)  806.71 
E)  1,209.34 
A)  1,210.01 
B)  806.73 
C)  1,210.16 
D)  806.71 
E)  1,209.34

Question 14

Use Lagrange multipliers to find the minimum distance from the line   to the point   . ​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Question 15

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. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 16

Given   , calculate   by evaluating   and   . Round your answer to four decimal places. ​

Accepted Answer

A)  9.2459 
B)  11.4918 
C)  -14.7541 
D)  -3.7541 
E)  -24.5082 
A)  9.2459 
B)  11.4918 
C)  -14.7541 
D)  -3.7541 
E)  -24.5082

Question 17

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

Accepted Answer

A)  2.63 
B)  6.94 
C)  172.47 
D)  13.13 
E)  1.74 
A)  2.63 
B)  6.94 
C)  172.47 
D)  13.13 
E)  1.74

Question 18

Find an equation of the tangent plane to the surface   at the point   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 19

Find the total differential for the function   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 20

A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from   units of running shoes and   units of basketball shoes is: ​   ,
​
Where   and   are in thousands of units. Find   and   so as to maximize the revenue.
​

Accepted Answer

A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)    ​ 
C)    ​ 
D)    ​ 
E)    ​

Examine the function for relative extrema.

Find the point(s) on the hyperboloid where the tangent plane is perpendicular to the line with parametric equations , and .

Given , use the total differential to approximate at towards . Round your answer to four decimal places.

Find the limit.

Find the partial derivative for the function .

Use spherical coordinates to find the limit . [Hint: Let

Find the gradient of the function at the given point.

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

Find the total differential of the function .

Let , where and . Find .

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

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

Use Lagrange multipliers to find the minimum distance from the line to the point .

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.

Given , calculate by evaluating and . Round your answer to four decimal places.

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

Find an equation of the tangent plane to the surface at the point .

Find the total differential for the function .

A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from units of running shoes and units of basketball shoes is: , Where and are in thousands of units. Find and so as to maximize the revenue.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Anal

Filters

Exam 13: Functions of Several Variables

Examine the function for relative extrema. ​

Find the point(s) on the hyperboloid where the tangent plane is perpendicular to the line with parametric equations , and .

Given , use the total differential to approximate at towards . Round your answer to four decimal places. ​

Find the limit.

Find the partial derivative for the function .

Use spherical coordinates to find the limit . [Hint: Let

Find the gradient of the function at the given point.

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

Find the total differential of the function .

Let , where and . Find . ​

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

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

Use Lagrange multipliers to find the minimum distance from the line to the point . ​

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. ​ ​

Given , calculate by evaluating and . Round your answer to four decimal places. ​

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

Find an equation of the tangent plane to the surface at the point .

Find the total differential for the function .

A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from units of running shoes and units of basketball shoes is: ​ , ​ Where and are in thousands of units. Find and so as to maximize the revenue. ​

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Multiple Integration

Vector Anal

Filters

Examine the function for relative extrema.

Given , use the total differential to approximate at towards . Round your answer to four decimal places.

Let , where and . Find .

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

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

Use Lagrange multipliers to find the minimum distance from the line to the point .

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.

Given , calculate by evaluating and . Round your answer to four decimal places.

A company manufactures two types of sneakers: running shoes and basketball shoes. The total revenue from units of running shoes and units of basketball shoes is: , Where and are in thousands of units. Find and so as to maximize the revenue.