Question 1

Find the domain and range of the function   .

Accepted Answer

The answer of Find the domain and range of the...

Question 2

Given that   , find   and   .

Accepted Answer

The answer of Given that   , find ...

Question 3

Let   be the function   . 
A) Find   . 
B) Is   continuous?

Accepted Answer

A) @#IMG-DLM& B) @#IMG-DLM& is

Question 4

Find the domain of the function and sketch it in the   plane.

Accepted Answer

The answer of Find the domain of the function and...

Question 5

Let   . 
A) Find   and   , if they exist. 
B) Use the definition of differentiability at a point to determine whether   is differentiable at the origin.

Accepted Answer

A) @#IMG-DLM& B) @#IMG-DLM& is

Question 6

The function   is defined near the point   by the equation   The maximum value of the directional derivative of   (among all possible directions) is which of the following?

Accepted Answer

A)    
B)  5 
C)  3 
D)    
E)  There is no maximum value. 
A)    
B)  5 
C)  3 
D)    
E)  There is no maximum value.

Question 7

Find the linearization of   at the point   and use it to approximate

Accepted Answer

The answer of Find the linearization of   at...

Question 8

If   and   change by   and   , respectively, from   and   , then the approximate change   in the angle   is which of the following?

Accepted Answer

A)    
B)  0.464 
C)  0.003 
D)  1.373 
E)  0.002 
A)    
B)  0.464 
C)  0.003 
D)  1.373 
E)  0.002

Question 9

A radioactive radiation with strength   is suddenly discharged. A man standing at the point   must run away, in the direction of maximum decrease of radiation. 
A) What direction should he choose?
B) The man decided to run along the path   . Find the directional derivative of   in the direction of the path at   .

Accepted Answer

A) He should choose

Question 10

Find the linearization of   at the point   and use it to approximate

Accepted Answer

The answer of Find the linearization of   at...

Question 11

Let   be a differentiable function, and let   be the function   Compute   at   .

Accepted Answer

The answer of Let   be a differentiable function,...

Question 12

Let   .
Which of the following statements is correct?

Accepted Answer

A)    and   are continuous. 
B)    is continuous, but   is not. 
C)    is continuous, but   is not. 
D)    and   are not continuous. 
E)    is continuous at the origin, hence so are   and   . 
A)    and   are continuous. 
B)    is continuous, but   is not. 
C)    is continuous, but   is not. 
D)    and   are not continuous. 
E)    is continuous at the origin, hence so are   and   .

Question 13

Find the derivative of   at the point   with respect to the vector   , where   ,   , and   .

Accepted Answer

The answer of Find the derivative of   at...

Question 14

Determine whether the following function is continuous.

Accepted Answer

The answer of Determine whether the following function is continuous....

Question 15

Evaluate the limit or state that it does not exist.
A)   
B)

Accepted Answer

A) 0
B) Th

Find the domain and range of the function .

Given that , find and .

Let be the function . A) Find . B) Is continuous?

Find the domain of the function and sketch it in the plane.

Let . A) Find and , if they exist. B) Use the definition of differentiability at a point to determine whether is differentiable at the origin.

The function is defined near the point by the equation The maximum value of the directional derivative of (among all possible directions) is which of the following?

Find the linearization of at the point and use it to approximate

If and change by and , respectively, from and , then the approximate change in the angle is which of the following?

Find the linearization of at the point and use it to approximate

Let be a differentiable function, and let be the function Compute at .

Let . Which of the following statements is correct?

Find the derivative of at the point with respect to the vector , where , , and .

Determine whether the following function is continuous.

Evaluate the limit or state that it does not exist. A) B)

Precalculus Review

Limits

Differentiation

Applications of the Derivative

The Integral

Applications of the Integral

Exponential Functions

Techniques of Integration

Further Applications of the Integral and Taylor Polynomials

Introduction to Differential Equations

Infinite Series

Parametric Equations, Polar Coordinates, and Conic Sections

Vector Geometry

Calculus of Vector-Valued Functions

Multiple Integration

Line and Surface Integrals

Fundamental Theorems of Vector Analysis

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Exam 15: Differentiation in Several Variables

Find the domain and range of the function .

Given that , find and .

Let be the function . A) Find . B) Is continuous?

Find the domain of the function and sketch it in the plane.

Let . A) Find and , if they exist. B) Use the definition of differentiability at a point to determine whether is differentiable at the origin.

The function is defined near the point by the equation The maximum value of the directional derivative of (among all possible directions) is which of the following?

Find the linearization of at the point and use it to approximate

If and change by and , respectively, from and , then the approximate change in the angle is which of the following?

Find the linearization of at the point and use it to approximate

Let be a differentiable function, and let be the function Compute at .

Let . Which of the following statements is correct?

Find the derivative of at the point with respect to the vector , where , , and .

Determine whether the following function is continuous.

Evaluate the limit or state that it does not exist. A) B)

Precalculus Review

Limits

Differentiation

Applications of the Derivative

The Integral

Applications of the Integral

Exponential Functions

Techniques of Integration

Further Applications of the Integral and Taylor Polynomials

Introduction to Differential Equations

Infinite Series

Parametric Equations, Polar Coordinates, and Conic Sections

Vector Geometry

Calculus of Vector-Valued Functions

Multiple Integration

Line and Surface Integrals

Fundamental Theorems of Vector Analysis

Filters