Question 1

Evaluate the limit .

Accepted Answer

A)  1 
B)  2 
C)  4 
D)  The limit does not exist. 
E)  None of the above 
A)  1 
B)  2 
C)  4 
D)  The limit does not exist. 
E)  None of the above

Question 2

Near what points can the equation 2x2 + xy + y2 = 7 not be solved for y as a function of x?

Accepted Answer

A)  (2, 1) and (-2, -1) 
B)  (2, -1) and (-2, 1) 
C)  (2, 1), (-2, -1), (2, -1), and (-2, 1) 
D)  (2, 1) and (2, -1) 
E)  (2, -1), (-2, 1), and (0, 0) 
A)  (2, 1) and (-2, -1) 
B)  (2, -1) and (-2, 1) 
C)  (2, 1), (-2, -1), (2, -1), and (-2, 1) 
D)  (2, 1) and (2, -1) 
E)  (2, -1), (-2, 1), and (0, 0)

Question 3

Find   (x, y) and   (x, y) if f(x, y) = ln   .

Accepted Answer

A)    (x, y) = -   ,   (x, y) = -   
B)    (x, y) = -   ,   (x, y) = -   
C)    (x, y) = -   ,   (x, y) = -   
D)    (x, y) = -   ,   (x, y) = -   
E)    (x, y) = -   ,   (x, y) = -   
A)    (x, y) = -   ,   (x, y) = -   
B)    (x, y) = -   ,   (x, y) = -   
C)    (x, y) = -   ,   (x, y) = -   
D)    (x, y) = -   ,   (x, y) = -   
E)    (x, y) = -   ,   (x, y) = -

Question 4

Find all horizontal planes that are tangent to the surface z = x3 - 3xy2 + 6y2 + 1.

Accepted Answer

A)  z = 1 and z = 3 
B)  z = 1 
C)  z = 3 
D)  z = -15 
E)  z = 9 
A)  z = 1 and z = 3 
B)  z = 1 
C)  z = 3 
D)  z = -15 
E)  z = 9

Question 5

If W =   , where x = t + sin(t) and y = 2t -1, then the value of   at t = 0 is equal to:

Accepted Answer

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

Question 6

Find the domain of the function f(x) =

Accepted Answer

A)  {(x, y) : x + y > 0} 
B)  {(x, y) : y $\neq$ -x and |x| $\neq$ |y|} 
C)  {(x, y) : y $\neq$ -x and y $\neq$ x} 
D)  {(x, y) : x + y > 0 and x - y > 0} 
E)  {(x, y) : x > 0 and y > 0} 
A)  {(x, y) : x + y > 0} 
B)  {(x, y) : y $\neq$ -x and |x| $\neq$ |y|} 
C)  {(x, y) : y $\neq$ -x and y $\neq$ x} 
D)  {(x, y) : x + y > 0 and x - y > 0} 
E)  {(x, y) : x > 0 and y > 0}

Question 7

Evaluate the limit.

Accepted Answer

A)  -2 
B)  0 
C)  -1 
D)  $\infty$ 
E)  The limit does not exist. 
A)  -2 
B)  0 
C)  -1 
D)  $\infty$ 
E)  The limit does not exist.

Question 8

Evaluate the limit     .

Accepted Answer

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

Question 9

Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( $\pi$, 1/4) in the direction of the vector   i + j.

Accepted Answer

A)  -   -   
B)  -   -   
C)  -   +   
D)  -   -   
E)  -   -   
A)  -   -   
B)  -   -   
C)  -   +   
D)  -   -   
E)  -   -

Question 10

Evaluate the limit .

Accepted Answer

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

Question 11

The electric potential V at time t at any point (x, y) in the plane is given by V =   sin   . An observer is moving in the plane so that at time t her position is given by x = t3, y = 2 -t4. How fast does she experience V changing at time t = 1?

Accepted Answer

A)  -   +   
B)    +   
C)  -   -   
D)  -   -   
E)    +   
A)  -   +   
B)    +   
C)  -   -   
D)  -   -   
E)    +

Question 12

Find the Taylor series of the function exy about the point (2, 3), by using Taylor series for functions of one variable.

Accepted Answer

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

Question 13

When the ellipse b2x2 + a2y2 = a2b2 is rotated about the x-axis, the volume V of the spheroid is 4  $\pi$ab2/3. If a and b are each increased by 1%, use differentials to find the approximate percentage change in V.

Accepted Answer

A)  increase of 1% 
B)  increase of 2% 
C)  increase of 3% 
D)  increase of 4% 
E)  increase of 5% 
A)  increase of 1% 
B)  increase of 2% 
C)  increase of 3% 
D)  increase of 4% 
E)  increase of 5%

Question 14

A particle is moving so that at time t its position is given by   . Find the derivative of f(x, y, z) = xyz at the location of the particle at time t =   in the direction in which the particle is moving at that time.

Accepted Answer

A)  -   
B)    
C)  - $\pi$ 
D)    
E)  3 $\pi$ 
A)  -   
B)    
C)  - $\pi$ 
D)    
E)  3 $\pi$

Question 15

For what value(s) of the constant k is the function f(x, y, z) =   sin(kz) a harmonic function in 3-space? Note that a harmonic function f in 3-space satisfies   +   +   = 0.

Accepted Answer

A)  k = 7 
B)  k = ± 1 
C)  k = ± 5 
D)  k = 0 
E)  k = ±4 
A)  k = 7 
B)  k = ± 1 
C)  k = ± 5 
D)  k = 0 
E)  k = ±4

Question 16

Find all points where the surface z = xy   has a horizontal tangent plane.

Accepted Answer

A)    ,   ,   ,   , (0, 0) 
B)    ,   ,   ,   
C)    ,   , (0, 0) 
D)    ,   
E)  The tangent plane is never horizontal. 
A)    ,   ,   ,   , (0, 0) 
B)    ,   ,   ,   
C)    ,   , (0, 0) 
D)    ,   
E)  The tangent plane is never horizontal.

Question 17

Let f(x, y, z) =   -2   - 3y + 4 and let P be the point (1, -1, 3). Which of the following statements is false?

Accepted Answer

A)    
B)  The maximum rate of change of f at P is 9. 
C)  There is no direction in which the rate of change of f at P is equal to 10. 
D)  There is a direction in which the rate of change of f at P is equal to -12. 
E)  The function f increases most rapidly at P in the direction of the vector -6 i - 3 j + 6 k. 
A)    
B)  The maximum rate of change of f at P is 9. 
C)  There is no direction in which the rate of change of f at P is equal to 10. 
D)  There is a direction in which the rate of change of f at P is equal to -12. 
E)  The function f increases most rapidly at P in the direction of the vector -6 i - 3 j + 6 k.

Question 18

The area of a triangle is given by the formula A =   ab sin $\theta$, where $\theta$ is the angle between the sides having lengths a and b. If measurements indicate that a = 4 m with error ± 1 cm,b = 3 m with error ± 1 cm, and $\theta$ = 60° with error ± 2°, use differentials to determine the approximate maximum error in the calculated area of the triangle.

Accepted Answer

A)  about 0.135 m2 
B)  about 6.015 m2 
C)  about 0.603 m2 
D)  about 0.014 m2 
E)  about 0.862 m2 
A)  about 0.135 m2 
B)  about 6.015 m2 
C)  about 0.603 m2 
D)  about 0.014 m2 
E)  about 0.862 m2

Question 19

Find   (x, y) and   (x, y) if f(x, y) = ln   .

Accepted Answer

A)    (x, y) =   ;   (x, y) =   
B)    (x, y) =   ;   (x, y) =   
C)    (x, y) = 1 -   ;   (x, y) = 1 +   
D)    (x, y) =   ;   (x, y) =   
E)    (x, y) =   (xy) -   ;   (x, y) =   (xy) +   
A)    (x, y) =   ;   (x, y) =   
B)    (x, y) =   ;   (x, y) =   
C)    (x, y) = 1 -   ;   (x, y) = 1 +   
D)    (x, y) =   ;   (x, y) =   
E)    (x, y) =   (xy) -   ;   (x, y) =   (xy) +

Question 20

Do the equations x2 + y2s + yt2 =13 and y2 + x2s + xt2 = 9 define s and t as functions of x and y near the point (x, y, s, t) = (1, 2, 1, -2)? If so, find   at that point.

Accepted Answer

A)  2 
B)  1 
C)  0 
D)  -2 
E)  The equations do not define s and t as functions of x and y near the given point. 
A)  2 
B)  1 
C)  0 
D)  -2 
E)  The equations do not define s and t as functions of x and y near the given point.

Evaluate the limit .

Near what points can the equation 2x² + xy + y² = 7 not be solved for y as a function of x?

Find (x, y) and (x, y) if f(x, y) = ln .

Find all horizontal planes that are tangent to the surface z = x³ - 3xy² + 6y² + 1.

If W = , where x = t + sin(t) and y = 2t -1, then the value of at t = 0 is equal to:

Find the domain of the function f(x) =

Evaluate the limit.

Evaluate the limit .

Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( $\pi$ , 1/4) in the direction of the vector $Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( \pi , 1/4) in the direction of the vector i + j.$ i + j.

Evaluate the limit .

The electric potential V at time t at any point (x, y) in the plane is given by V = sin . An observer is moving in the plane so that at time t her position is given by x = t³, y = 2 -t⁴. How fast does she experience V changing at time t = 1?

Find the Taylor series of the function e^xy about the point (2, 3), by using Taylor series for functions of one variable.

When the ellipse b²x² + a²y² = a²b² is rotated about the x-axis, the volume V of the spheroid is 4 $\pi$ ab²/3. If a and b are each increased by 1%, use differentials to find the approximate percentage change in V.

A particle is moving so that at time t its position is given by . Find the derivative of f(x, y, z) = xyz at the location of the particle at time t = in the direction in which the particle is moving at that time.

For what value(s) of the constant k is the function f(x, y, z) = sin(kz) a harmonic function in 3-space? Note that a harmonic function f in 3-space satisfies + + = 0.

Find all points where the surface z = xy has a horizontal tangent plane.

Let f(x, y, z) = -2 - 3y + 4 and let P be the point (1, -1, 3). Which of the following statements is false?

Find (x, y) and (x, y) if f(x, y) = ln .

Do the equations x² + y²s + yt² =13 and y² + x²s + xt² = 9 define s and t as functions of x and y near the point (x, y, s, t) = (1, 2, 1, -2)? If so, find at that point.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 13: Partial Differentiation

Evaluate the limit .

Near what points can the equation 2x2 + xy + y2 = 7 not be solved for y as a function of x?

Find (x, y) and (x, y) if f(x, y) = ln .

Find all horizontal planes that are tangent to the surface z = x3 - 3xy2 + 6y2 + 1.

If W = , where x = t + sin(t) and y = 2t -1, then the value of at t = 0 is equal to:

Find the domain of the function f(x) =

Evaluate the limit.

Evaluate the limit .

Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( π\piπ , 1/4) in the direction of the vector i + j.

Evaluate the limit .

The electric potential V at time t at any point (x, y) in the plane is given by V = sin . An observer is moving in the plane so that at time t her position is given by x = t3, y = 2 -t4. How fast does she experience V changing at time t = 1?

Find the Taylor series of the function exy about the point (2, 3), by using Taylor series for functions of one variable.

When the ellipse b2x2 + a2y2 = a2b2 is rotated about the x-axis, the volume V of the spheroid is 4 π\piπ ab2/3. If a and b are each increased by 1%, use differentials to find the approximate percentage change in V.

A particle is moving so that at time t its position is given by . Find the derivative of f(x, y, z) = xyz at the location of the particle at time t = in the direction in which the particle is moving at that time.

For what value(s) of the constant k is the function f(x, y, z) = sin(kz) a harmonic function in 3-space? Note that a harmonic function f in 3-space satisfies + + = 0.

Find all points where the surface z = xy has a horizontal tangent plane.

Let f(x, y, z) = -2 - 3y + 4 and let P be the point (1, -1, 3). Which of the following statements is false?

Find (x, y) and (x, y) if f(x, y) = ln .

Do the equations x2 + y2s + yt2 =13 and y2 + x2s + xt2 = 9 define s and t as functions of x and y near the point (x, y, s, t) = (1, 2, 1, -2)? If so, find at that point.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Near what points can the equation 2x² + xy + y² = 7 not be solved for y as a function of x?

Find all horizontal planes that are tangent to the surface z = x³ - 3xy² + 6y² + 1.

Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( $\pi$ , 1/4) in the direction of the vector $Find the second directional derivative of f(x, y) = cos(xy) + sin(xy) at ( \pi , 1/4) in the direction of the vector i + j.$ i + j.

The electric potential V at time t at any point (x, y) in the plane is given by V = sin . An observer is moving in the plane so that at time t her position is given by x = t³, y = 2 -t⁴. How fast does she experience V changing at time t = 1?

Find the Taylor series of the function e^xy about the point (2, 3), by using Taylor series for functions of one variable.

When the ellipse b²x² + a²y² = a²b² is rotated about the x-axis, the volume V of the spheroid is 4 $\pi$ ab²/3. If a and b are each increased by 1%, use differentials to find the approximate percentage change in V.

Do the equations x² + y²s + yt² =13 and y² + x²s + xt² = 9 define s and t as functions of x and y near the point (x, y, s, t) = (1, 2, 1, -2)? If so, find at that point.