Question 1

Find the partial derivative   for the function   . ​

Accepted Answer

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

Question 2

The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $400.00, find the dimensions of the box of largest volume that can be made.

Accepted Answer

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

Question 3

Find   by using the limits   and   . ​

Accepted Answer

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

Question 4

Examine the function   for relative extrema and saddle points. ​   ​

Accepted Answer

A)  saddle point:   
B)  relative minimum:   
C)  relative maximum:   
D)  relative minimum:   
E)  no critical points 
A)  saddle point:   
B)  relative minimum:   
C)  relative maximum:   
D)  relative minimum:   
E)  no critical points

Question 5

Use Lagrange multipliers to minimize the function   subject to the following constraint. ​   ​
Assume that x, y, and z are positive.
​

Accepted Answer

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

Question 6

Find the directional derivative of the function   at   in the direction of   . Round your answer to two decimal places. ​

Accepted Answer

A)  -8.02 
B)  -1.71 
C)  -5.67 
D)  -1.03 
E)  -5.14 
A)  -8.02 
B)  -1.71 
C)  -5.67 
D)  -1.03 
E)  -5.14

Question 7

Sketch the surface given by the function   .

Accepted Answer

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

Question 8

Discuss the continuity of the function.

Accepted Answer

A)  f is continuous everywhere except at those points   such that   . 
B)  f is continuous everywhere except at those points   such that   . 
C)  f is continuous at those points   such that   . 
D)  f is continuous only when   . 
E)  f is continuous everywhere in the xy-plane. 
A)  f is continuous everywhere except at those points   such that   . 
B)  f is continuous everywhere except at those points   such that   . 
C)  f is continuous at those points   such that   . 
D)  f is continuous only when   . 
E)  f is continuous everywhere in the xy-plane.

Question 9

The temperature at the point   on a metal plate is modeled by   ,   . Find the directions of no change in heat on the plate from the point   .

Accepted Answer

A)  There will be no change in directions perpendicular to the gradient   . 
B)  There will be no change in directions parallel to the gradient   . 
C)  There will be no change in directions parallel to the gradient   . 
D)  There will be no change in directions perpendicular to the gradient   . 
E)  There will be no change in directions perpendicular to the gradient   . 
A)  There will be no change in directions perpendicular to the gradient   . 
B)  There will be no change in directions parallel to the gradient   . 
C)  There will be no change in directions parallel to the gradient   . 
D)  There will be no change in directions perpendicular to the gradient   . 
E)  There will be no change in directions perpendicular to the gradient   .

Question 10

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

Accepted Answer

A)    ​ 
B)  0
​ 
C)    ​ 
D)    ​ 
E)  no absolute minimum
​ 
A)    ​ 
B)  0
​ 
C)    ​ 
D)    ​ 
E)  no absolute minimum
​

Question 11

Find   using the appropriate Chain Rule for   where   and   . ​

Accepted Answer

A)  104t 
B)  52t 
C)  80t 
D)  44t 
E)  156t 
A)  104t 
B)  52t 
C)  80t 
D)  44t 
E)  156t

Question 12

The surface of a mountain is modeled by the equation   . A mountain climber is at the point   . In what direction should the climber move in order to ascend at the greatest rate? Round all numerical values in your answer to one decimal place.

Accepted Answer

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

Question 13

Suppose the period T of a pendulum of length L is   where g is the acceleration due to gravity. A pendulum is moved from the Canal Zone, where   feet per second per second, to Greenland, where   feet per second per second. Because of the change in temperature, the length of the pendulum changes from 2.5 feet to 2.43 feet. Approximate the change in the period of the pendulum. Round your answer to four decimal places. ​

Accepted Answer

A)  4.9300 seconds 
B)  -0.0350 second 
C)  0.9650 second 
D)  7.9650 seconds 
E)  -1.0700 seconds 
A)  4.9300 seconds 
B)  -0.0350 second 
C)  0.9650 second 
D)  7.9650 seconds 
E)  -1.0700 seconds

Question 14

Find the absolute extrema of the function   over the triangular region in the xy-plane with vertices   and   .

Accepted Answer

A)  absolute maximum:   at   absolute minimum:   at   and along the line   
B)  absolute maximum:   at   absolute minimum:   at   and along the line   
C)  absolute maximum:   at   absolute minimum:   at   and along the line   
D)  absolute maximum:   at   absolute minimum:   at   
E)  absolute maximum:   at   absolute minimum:   at   
A)  absolute maximum:   at   absolute minimum:   at   and along the line   
B)  absolute maximum:   at   absolute minimum:   at   and along the line   
C)  absolute maximum:   at   absolute minimum:   at   and along the line   
D)  absolute maximum:   at   absolute minimum:   at   
E)  absolute maximum:   at   absolute minimum:   at

Question 15

Find and simplify the function   at the given value   .

Accepted Answer

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

Question 16

Examine the function   for relative extrema.

Accepted Answer

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

Question 17

Examine the function   for relative extrema and saddle points.

Accepted Answer

A)  saddle point:   ; relative minimum:   
B)  saddle point:   ; relative minimum:   
C)  relative minimum:   ; relative maximum:   
D)  relative minimum:   ; relative maximum:   
E)  saddle points:   ,   
A)  saddle point:   ; relative minimum:   
B)  saddle point:   ; relative minimum:   
C)  relative minimum:   ; relative maximum:   
D)  relative minimum:   ; relative maximum:   
E)  saddle points:   ,

Question 18

Find   and use the total differential to approximate the quantity   . Round your answer to two decimal places. ​

Accepted Answer

A)  5.80 
B)  3.80 
C)  4.40 
D)  2.40 
E)  0.40 
A)  5.80 
B)  3.80 
C)  4.40 
D)  2.40 
E)  0.40

Question 19

Find the limit.

Accepted Answer

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

Question 20

Differentiate implicitly to find   , given   . ​

Accepted Answer

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

Find the partial derivative for the function .

The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $400.00, find the dimensions of the box of largest volume that can be made.

Find by using the limits and .

Examine the function for relative extrema and saddle points.

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

Find the directional derivative of the function at in the direction of . Round your answer to two decimal places.

Sketch the surface given by the function .

Discuss the continuity of the function.

The temperature at the point on a metal plate is modeled by , . Find the directions of no change in heat on the plate from the point .

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

Find using the appropriate Chain Rule for where and .

The surface of a mountain is modeled by the equation . A mountain climber is at the point . In what direction should the climber move in order to ascend at the greatest rate? Round all numerical values in your answer to one decimal place.

Find the absolute extrema of the function over the triangular region in the xy-plane with vertices and .

Find and simplify the function at the given value .

Examine the function for relative extrema.

Examine the function for relative extrema and saddle points.

Find and use the total differential to approximate the quantity . Round your answer to two decimal places.

Find the limit.

Differentiate implicitly to find , given .

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

Find the partial derivative for the function . ​

The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $400.00, find the dimensions of the box of largest volume that can be made.

Find by using the limits and . ​

Examine the function for relative extrema and saddle points. ​ ​

Use Lagrange multipliers to minimize the function subject to the following constraint. ​ ​ Assume that x, y, and z are positive. ​

Find the directional derivative of the function at in the direction of . Round your answer to two decimal places. ​

Sketch the surface given by the function .

Discuss the continuity of the function.

The temperature at the point on a metal plate is modeled by , . Find the directions of no change in heat on the plate from the point .

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

Find using the appropriate Chain Rule for where and . ​

The surface of a mountain is modeled by the equation . A mountain climber is at the point . In what direction should the climber move in order to ascend at the greatest rate? Round all numerical values in your answer to one decimal place.

Find the absolute extrema of the function over the triangular region in the xy-plane with vertices and .

Find and simplify the function at the given value .

Examine the function for relative extrema.

Examine the function for relative extrema and saddle points.

Find and use the total differential to approximate the quantity . Round your answer to two decimal places. ​

Find the limit.

Differentiate implicitly to find , given . ​

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

Find the partial derivative for the function .

Find by using the limits and .

Examine the function for relative extrema and saddle points.

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

Find the directional derivative of the function at in the direction of . Round your answer to two decimal places.

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

Find using the appropriate Chain Rule for where and .

Find and use the total differential to approximate the quantity . Round your answer to two decimal places.

Differentiate implicitly to find , given .