Question 1

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

Accepted Answer

A)    
B)    
C)    
D)    
E)  no absolute maximum 
A)    
B)    
C)    
D)    
E)  no absolute maximum

Question 2

Suppose the centripetal acceleration of a particle moving in a circle is   , where v is the velocity and r is the radius of the circle. Approximate the maximum percent error in measuring the acceleration due to errors of 7% in v and 5% in r. ​

Accepted Answer

A)  23% 
B)  25% 
C)  19% 
D)  40% 
E)  24% 
A)  23% 
B)  25% 
C)  19% 
D)  40% 
E)  24%

Question 3

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

Accepted Answer

A) 1 
B) 0 
C) 2 
D)    
E)  limit does not exist. 
A) 1 
B) 0 
C) 2 
D)    
E)  limit does not exist.

Question 4

Find and simplify   for the given function   . ​

Accepted Answer

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

Question 5

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.

Accepted Answer

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

Question 6

A rectangular box with an open top has a length of x feet, a width of y feet, and a height of z feet. It costs $1.80 per square foot to build the base and $0.45 per square foot to build the sides. Write the cost C of constructing the box as a function of x, y, and z.

Accepted Answer

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

Question 7

Find the angle of inclination   of the tangent plane to the surface   at the point   . Round your answer to two decimal places.

Accepted Answer

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

Question 8

Find the least squares regression line for the points shown in the graph.

Accepted Answer

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

Question 9

Find the total differential for the function   .

Accepted Answer

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

Question 10

Find the path of a heat-seeking particle placed at point   on a metal plate with a temperature field   .

Accepted Answer

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

Question 11

Find the partial derivative   for the function   .

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

Find the limit   for the function   . ​

Accepted Answer

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

Question 14

Determine the continuity of the composite function   where   and   . ​

Accepted Answer

A)  continuous except at  
​ 
B)  continuous for all  
​ 
C)  continuous for all  
​ 
D)  continuous except at  
​ 
E)  continuous everywhere
​ 
A)  continuous except at  
​ 
B)  continuous for all  
​ 
C)  continuous for all  
​ 
D)  continuous except at  
​ 
E)  continuous everywhere
​

Question 15

Find the highest point on the curve of intersection of the following surfaces. Cone:   , Plane:

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)  -4.4276 
B)  -38.4276 
C)  -14.4276 
D)  0.1448 
E)  5.1448 
A)  -4.4276 
B)  -38.4276 
C)  -14.4276 
D)  0.1448 
E)  5.1448

Question 17

Assume a rule that is one of several methods used to determine the lumber yield of a log (in board-feet) in terms of its diameter d (in inches) and its length L (in feet). The number of board-feet is   . Find the number of board-feet of lumber in a log 46 inches in diameter and 20 feet in length.

Accepted Answer

A)  9,680 board-feet 
B)  2,205 board-feet 
C)  440 board-feet 
D)  8,820 board-feet 
E)  35,280 board-feet 
A)  9,680 board-feet 
B)  2,205 board-feet 
C)  440 board-feet 
D)  8,820 board-feet 
E)  35,280 board-feet

Question 18

Find the absolute extrema of   on the region bounded by the square with vertices   and   . ​

Accepted Answer

A)  absolute minimum:  
Absolute maximum:  
​ 
B)  absolute minimum:  
Absolute maximum:  
​ 
C)  absolute minimum:  
Absolute maximum:  
​ 
D)  absolute minimum:  
Absolute maximum:  
​ 
E)  absolute minimum:  
Absolute maximum:  
​ 
A)  absolute minimum:  
Absolute maximum:  
​ 
B)  absolute minimum:  
Absolute maximum:  
​ 
C)  absolute minimum:  
Absolute maximum:  
​ 
D)  absolute minimum:  
Absolute maximum:  
​ 
E)  absolute minimum:  
Absolute maximum:  
​

Question 19

Use Lagrange multipliers to find the maximum value of   where   and   subject to the constraint   . ​

Accepted Answer

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

Question 20

Use the gradient to find a normal vector to the graph of the equation at the given point.

Accepted Answer

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

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

Suppose the centripetal acceleration of a particle moving in a circle is , where v is the velocity and r is the radius of the circle. Approximate the maximum percent error in measuring the acceleration due to errors of 7% in v and 5% in r.

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

Find and simplify for the given function .

A rectangular box with an open top has a length of x feet, a width of y feet, and a height of z feet. It costs $1.80 per square foot to build the base and $0.45 per square foot to build the sides. Write the cost C of constructing the box as a function of x, y, and z.

Find the angle of inclination of the tangent plane to the surface at the point . Round your answer to two decimal places.

Find the least squares regression line for the points shown in the graph.

Find the total differential for the function .

Find the path of a heat-seeking particle placed at point on a metal plate with a temperature field .

Find the partial derivative for the function .

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

Find the limit for the function .

Determine the continuity of the composite function where and .

Find the highest point on the curve of intersection of the following surfaces. Cone: , Plane:

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

Assume a rule that is one of several methods used to determine the lumber yield of a log (in board-feet) in terms of its diameter d (in inches) and its length L (in feet). The number of board-feet is . Find the number of board-feet of lumber in a log 46 inches in diameter and 20 feet in length.

Find the absolute extrema of on the region bounded by the square with vertices and .

Use Lagrange multipliers to find the maximum value of where and subject to the constraint .

Use the gradient to find a normal vector to the graph of the equation at the given point.

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

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

Suppose the centripetal acceleration of a particle moving in a circle is , where v is the velocity and r is the radius of the circle. Approximate the maximum percent error in measuring the acceleration due to errors of 7% in v and 5% in r. ​

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

Find and simplify for the given function . ​

A rectangular box with an open top has a length of x feet, a width of y feet, and a height of z feet. It costs $1.80 per square foot to build the base and $0.45 per square foot to build the sides. Write the cost C of constructing the box as a function of x, y, and z.

Find the angle of inclination of the tangent plane to the surface at the point . Round your answer to two decimal places.

Find the least squares regression line for the points shown in the graph.

Find the total differential for the function .

Find the path of a heat-seeking particle placed at point on a metal plate with a temperature field .

Find the partial derivative for the function .

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

Find the limit for the function . ​

Determine the continuity of the composite function where and . ​

Find the highest point on the curve of intersection of the following surfaces. Cone: , Plane:

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

Assume a rule that is one of several methods used to determine the lumber yield of a log (in board-feet) in terms of its diameter d (in inches) and its length L (in feet). The number of board-feet is . Find the number of board-feet of lumber in a log 46 inches in diameter and 20 feet in length.

Find the absolute extrema of on the region bounded by the square with vertices and . ​

Use Lagrange multipliers to find the maximum value of where and subject to the constraint . ​

Use the gradient to find a normal vector to the graph of the equation at the given point.

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

Suppose the centripetal acceleration of a particle moving in a circle is , where v is the velocity and r is the radius of the circle. Approximate the maximum percent error in measuring the acceleration due to errors of 7% in v and 5% in r.

Find and simplify for the given function .

Find the limit for the function .

Determine the continuity of the composite function where and .

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

Find the absolute extrema of on the region bounded by the square with vertices and .

Use Lagrange multipliers to find the maximum value of where and subject to the constraint .