Question 1

Sketch f(x, y) = 1 - y2 in xyz space and state its natural domain.

Accepted Answer

The natural domain includes all real values of x and y.

Question 2

What is/are the domain restriction(s) for   ?

Accepted Answer

A) x > 0, y $\neq$0 B) x > 0 and y > 0, or x < 0 and y < 0 C) x $\neq$ 0, y $\neq$ 0 D) No restrictions exist E) x $\neq$ 0, y < 0 A) x > 0, y $\neq$0 B) x > 0 and y > 0, or x < 0 and y < 0 C) x $\neq$ 0, y $\neq$ 0 D) No restrictions exist E) x $\neq$ 0, y < 0 B

Question 3

Let   . There is a critical point at

Accepted Answer

A)  (0, 0) 
B)    
C)    
D)  none exist 
E)  (1, 1) 
A)  (0, 0) 
B)    
C)    
D)  none exist 
E)  (1, 1) A

Question 4

Let   . Describe the level surface passing through   .

Accepted Answer

A)  Hyperboloid of two sheets 
B)  Hyperboloid of one sheet 
C)  Elliptical cone 
D)  Elliptic paraboloid 
E)  Hyperbolic paraboloid 
A)  Hyperboloid of two sheets 
B)  Hyperboloid of one sheet 
C)  Elliptical cone 
D)  Elliptic paraboloid 
E)  Hyperbolic paraboloid

Question 5

Evaluate   .

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  Does not exist 
E)  4 
A)  0 
B)  1 
C)  2 
D)  Does not exist 
E)  4

Question 6

The legs of a right triangle are measured to be   and   inches with a maximum error of   inches in each measurement. Use differentials to estimate the maximum possible error in the calculated value of the hypotenuse.

Accepted Answer

A)    is the maximum error in the hypotenuse. 
B)    is the maximum error in the hypotenuse. 
C)    is the maximum error in the hypotenuse. 
D)    is the maximum error in the hypotenuse. 
E)    is the maximum error in the hypotenuse. 
A)    is the maximum error in the hypotenuse. 
B)    is the maximum error in the hypotenuse. 
C)    is the maximum error in the hypotenuse. 
D)    is the maximum error in the hypotenuse. 
E)    is the maximum error in the hypotenuse.

Question 7

Compute the total differential of f(x, y, z) = 3 + x4yz2.

Accepted Answer

A)  4x3yz2dx dy dz 
B)  4x3yz2dx + x4yz2dy + 2x4yz2dz 
C)  4x3yz dx + x4yz2dy + x4yz2dz 
D)  4x3yz2dx + x4z2dy + 2x4yz dz 
E)  4xyzdx + xzdy + 2xyz dz 
A)  4x3yz2dx dy dz 
B)  4x3yz2dx + x4yz2dy + 2x4yz2dz 
C)  4x3yz dx + x4yz2dy + x4yz2dz 
D)  4x3yz2dx + x4z2dy + 2x4yz dz 
E)  4xyzdx + xzdy + 2xyz dz

Question 8

Let   . Find   if   .

Accepted Answer

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

Question 9

Use the Lagrange multiplier method to find the point on the plane x + 2y + 8z = 1 that is closest to the point (1, 1, 0).

Accepted Answer

The closes

Question 10

Use the Lagrange multiplier method to find the maximum sum of 2(x2 + y2 + z2) if
2(x + 2y + 2z) = 24.

Accepted Answer

Question 11

Use Lagrange multipliers to find all the locations of the extreme values of   subject to   .

Accepted Answer

A)  (0, 0, 0) 
B)    
C)    
D)    
E)  There are none 
A)  (0, 0, 0) 
B)    
C)    
D)    
E)  There are none

Question 12

Let   . There is a critical point at

Accepted Answer

A)  (0, 0) 
B)    
C)    
D)  none exist 
E)  (1, 1) 
A)  (0, 0) 
B)    
C)    
D)  none exist 
E)  (1, 1)

Question 13

Let z = 3 + e5xy . Find   .

Accepted Answer

A)  5xe5xy 
B)  5e5xy 
C)  5ye5xy 
D)  5xye5xy 
E)  e5xyz 
A)  5xe5xy 
B)  5e5xy 
C)  5ye5xy 
D)  5xye5xy 
E)  e5xyz

Question 14

Compute the total differential of the function   .

Accepted Answer

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

Question 15

Let   ;   Find   .

Accepted Answer

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

Question 16

Let   . Find   if   .

Accepted Answer

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

Question 17

Use implicit differentiation to evaluate   at (1, -2, 1) if x3z - 3xy2 - (yz)3 = -11.

Accepted Answer

Question 18

Let   ;   Find   .

Accepted Answer

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

Question 19

Let   . Find   .

Accepted Answer

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

Question 20

Let   . Find f(7, 4, 5).

Accepted Answer

A)  3 
B)  27 
C)  9 
D)    
E)  81 
A)  3 
B)  27 
C)  9 
D)    
E)  81

Sketch f(x, y) = 1 - y² in xyz space and state its natural domain.

What is/are the domain restriction(s) for ?

Let . There is a critical point at

Let . Describe the level surface passing through .

Evaluate .

The legs of a right triangle are measured to be and inches with a maximum error of inches in each measurement. Use differentials to estimate the maximum possible error in the calculated value of the hypotenuse.

Compute the total differential of f(x, y, z) = 3 + x⁴yz².

Let . Find if .

Use the Lagrange multiplier method to find the point on the plane x + 2y + 8z = 1 that is closest to the point (1, 1, 0).

Use the Lagrange multiplier method to find the maximum sum of 2(x² + y² + z²) if 2(x + 2y + 2z) = 24.

Use Lagrange multipliers to find all the locations of the extreme values of subject to .

Let . There is a critical point at

Let z = 3 + e^5xy . Find .

Compute the total differential of the function .

Let ; Find .

Let . Find if .

Use implicit differentiation to evaluate at (1, -2, 1) if x³z - 3xy² - (yz)³ = -11.

Let ; Find .

Let . Find .

Let . Find f(7, 4, 5).

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Vector-Valued Functions

Multiple Integrals

Topics in Vector Calculus

Filters

Exam 13: Partial Derivatives

Sketch f(x, y) = 1 - y2 in xyz space and state its natural domain.

What is/are the domain restriction(s) for ?

Let . There is a critical point at

Let . Describe the level surface passing through .

Evaluate .

The legs of a right triangle are measured to be and inches with a maximum error of inches in each measurement. Use differentials to estimate the maximum possible error in the calculated value of the hypotenuse.

Compute the total differential of f(x, y, z) = 3 + x4yz2.

Let . Find if .

Use the Lagrange multiplier method to find the point on the plane x + 2y + 8z = 1 that is closest to the point (1, 1, 0).

Use the Lagrange multiplier method to find the maximum sum of 2(x2 + y2 + z2) if 2(x + 2y + 2z) = 24.

Use Lagrange multipliers to find all the locations of the extreme values of subject to .

Let . There is a critical point at

Let z = 3 + e5xy . Find .

Compute the total differential of the function .

Let ; Find .

Let . Find if .

Use implicit differentiation to evaluate at (1, -2, 1) if x3z - 3xy2 - (yz)3 = -11.

Let ; Find .

Let . Find .

Let . Find f(7, 4, 5).

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Vector-Valued Functions

Multiple Integrals

Topics in Vector Calculus

Filters

Sketch f(x, y) = 1 - y² in xyz space and state its natural domain.

Compute the total differential of f(x, y, z) = 3 + x⁴yz².

Use the Lagrange multiplier method to find the maximum sum of 2(x² + y² + z²) if 2(x + 2y + 2z) = 24.

Let z = 3 + e^5xy . Find .

Use implicit differentiation to evaluate at (1, -2, 1) if x³z - 3xy² - (yz)³ = -11.