Master Multivariable Functions: Functions of Several Variables Quiz

Question 1

A farmer wants to build a rectangular enclosure along a river, using 100 meters of fencing for the three sides. What is the maximum area the farmer can enclose?

Accepted Answer

625 square meters

Answer

500 square meters

Answer

600 square meters

Answer

700 square meters

Question 2

Define a function of two variables and illustrate with an example.

Accepted Answer

A function of two variables assigns a unique real number z to each ordered pair of real numbers (x, y).

Answer

A graph depicting the relationship between two variables is not a function of two variables.

Answer

An equation with two variables is not a function of two variables.

Answer

A set of ordered pairs does not constitute a function of two variables.

Question 3

Which of the following is a critical point of the function f(x, y) = x^2 + y^2 - 4x + 2y?

Accepted Answer

(0, 0)

Answer

(1, 1)

Answer

(2, -1)

Answer

(0, 1)

Question 4

Find the directional derivative of the function f(x, y) = e^(x-y) at the point (1, 0) in the direction of the vector v = <3, 4>

Accepted Answer

9

Answer

5

Answer

7

Answer

11

Question 5

Which of the following is a sufficient condition for a function of several variables to be differentiable at a point?

Accepted Answer

The partial derivatives exist and are continuous at that point.

Answer

The function is continuous at that point.

Answer

The limit of the function exists at that point.

Answer

The directional derivatives exist at that point.

Question 6

Calculate the Jacobian of the transformation T(x, y) = (u = x^2 + y^2, v = x - y).

Accepted Answer

2x + 2y

Answer

2x - 2y

Answer

-2x + 2y

Answer

-2x - 2y

Question 7

Which characteristic does NOT apply to functions with multiple variables?

Accepted Answer

They have a single independent variable.

Answer

They can be represented in higher dimensions.

Answer

Their graphs are typically surfaces or curves.

Answer

Their values depend on multiple independent variables.

Question 8

Find the directional derivative of f(x, y) = sin(xy) at the point (1, π/2) in the direction of the vector (1, 1).

Accepted Answer

√2

Answer

0

Answer

1

Answer

-1

Question 9

Find the partial derivatives of f(x, y) = x^2 sin(y) and interpret their geometric significance, specifically the gradient.

Accepted Answer

(∂f/∂x, ∂f/∂y) = (2x sin(y), x^2 cos(y))

Answer

(∂f/∂x, ∂f/∂y) = (x^2, sin(y))

Question 10

Let f(x, y) = x^2 - y^2. What is the partial derivative of f with respect to x?

Accepted Answer

2x

Answer

2y

Answer

0

Answer

x^2

Question 11

Find the direction of steepest ascent for the function f(x, y) = xy at the point (1, 2).

Accepted Answer

Vector (1, 2)

Answer

Vector (-1, 2)

Answer

Vector (2, 1)

Answer

Vector (-2, 1)

Question 12

Evaluate the integral ∬R (x + y) dA over the rectangular region R = {(x, y) | 0 ≤ x ≤ 2, 0 ≤ y ≤ 3}

Accepted Answer

15

Answer

9

Answer

18

Answer

21

Question 13

Which equation corresponds to a level curve of the paraboloid defined by (z = x^2 + y^2)?

Accepted Answer

(z = c)

Answer

(x + y = c)

Answer

(x^2 - y^2 = c)

Question 14

Find the partial derivative of the function f(x, y, z) = x^2 + y^3 + z^4 with respect to y.

Accepted Answer

3y^2

Answer

2x

Answer

4z^3

Answer

0

Question 15

Determine if the function f(x, y) = sin(x) + cos(y) is continuous at the point (0, 0).

Accepted Answer

Yes

Answer

Continuous only at y = 0

Answer

No

Answer

Continuous only at x = 0

Question 16

Find the directional derivative of f(x, y) = x^2 - y^2 at the point (1, 1) in the direction of the unit vector in the direction of <1, 1>.

Accepted Answer

2

Answer

0

Answer

-2

Answer

1

Question 17

Which of the following is the chain rule for the composite function f(g(x, y))?

Accepted Answer

df/dx = (df/dg) * (dg/dx) + (df/dy) * (dg/dy)

Answer

df/dx = df/dg * dg/dx

Answer

df/dy = df/dg * dg/dy

Answer

df/dx = df/dy * dg/dx

Question 18

Which of the following is the Implicit Function Theorem?

Accepted Answer

If f(x, y, z) is differentiable at (x0, y0, z0) and f(x0, y0, z0) = 0, then there exists a unique differentiable function g(x, y) such that f(x, y, g(x, y)) = 0 in a neighborhood of (x0, y0).

Answer

If f(x, y, z) is continuous at (x0, y0, z0) and f(x0, y0, z0) = 0, then there exists a unique differentiable function g(x, y) such that f(x, y, g(x, y)) = 0 in a neighborhood of (x0, y0).

Question 19

Which of the following equations represents a function of two variables?

Accepted Answer

f(x, y) = x² + y²

Answer

x² + y² = 1

Answer

x² - y² = 0

Answer

y = x

Question 20

Classify the critical point(s) of the function f(x, y) = x⁴ + y⁴ - 4xy + 1.

Accepted Answer

(0, 0): Saddle point

Answer

(0, 0): Local minimum

Answer

(0, 0): Local maximum

Answer

No critical points

Question 21

Is the function  *f(x, y) = x²y* continuous at the point (0, 0)?

Accepted Answer

Yes

Answer

No

Answer

Continuous only for values of *y*

Answer

Continuous only for values of *x*

Question 22

What is the partial derivative of *f(x, y) = sin(xy)* with respect to *x*?

Accepted Answer

y cos(xy)

Answer

sin(xy)

Answer

cos(xy)

Answer

x cos(xy)

Question 23

Identify the critical points of the function *f(x, y) = x³ + y³ - 3xy*.

Accepted Answer

(0, 0), (1, 1), (-1, -1)

Answer

(0, 0), (2, 2)

Answer

(1, 1), (-1, -1)

Answer

(0, 0), (1, 2)

Question 24

Using the Chain Rule, find the derivative of *f(x, y) = e^(x² + y²)* with respect to *y*.

Accepted Answer

2ye^(x² + y²)

Answer

e^(x² + y²)

Answer

x²e^(x² + y²)

Answer

y²e^(x² + y²)

Question 25

Calculate the directional derivative of *f(x, y) = xy²* at the point (2, 1) in the direction of the vector <1, 2>.

Accepted Answer

7

Answer

5

Answer

11

Answer

9

Question 26

Given the vector field *F(x, y) = <x, y>*, find the line integral of *F* along the curve *C* defined by *y = x²* from (0, 0) to (1, 1).

Accepted Answer

1/3

Answer

1

Answer

1/6

Answer

1/2

Question 27

Calculate the curl of the vector field *F(x, y) = <x², y²>*.

Accepted Answer

<0, 0, 2>

Answer

<0, 2, 0>

Answer

<2, 0, 0>

Answer

<2, 2, 0>

Question 28

Use Green's Theorem to evaluate the line integral of *F(x, y) = <xy, x²>* around the triangle with vertices (0, 0), (1, 0), and (0, 1).

Accepted Answer

1/2

Answer

1/4

Answer

1

Answer

2

Question 29

What is the domain of the function *f(x, y) = √(9 - x² - y²)*?

Accepted Answer

{(x, y) | x² + y² ≤ 9}

Answer

{(x, y) | x² + y² ≥ 9}

Answer

All real numbers

Answer

{(x, y) | x² + y² < 9}

Question 30

Find the limit as (x, y) approaches (0, 0) of the function *f(x, y) = (x²y)/(x⁴ + y²)*, if it exists.

Accepted Answer

The limit does not exist.

Answer

0

Answer

∞

Answer

1

Question 31

Is the function  f(x, y) = (x²y) / (x⁴ + y²) continuous at the point (0, 0)?

Accepted Answer

No, the function is not continuous at (0, 0).

Answer

The function is continuous only along certain paths.

Answer

The function is discontinuous everywhere.

Answer

Yes, the function is continuous at (0, 0).

Question 32

What is the partial derivative of the function f(x, y) = x³y² + 2xy with respect to x?

Accepted Answer

3x²y² + 2y

Answer

3x²y² + 2xy

Answer

6xy

Answer

x³y² + 2y

Question 33

Find the gradient of the function f(x, y) = sin(xy) at the point (π/2, 1).

Accepted Answer

(0, (π/2))

Answer

(1, 0)

Answer

(0, -π/2)

Answer

(1, π/2)

Question 34

What is the equation of the tangent plane to the surface z = x² + y² at the point (1, 1, 2)?

Accepted Answer

z = 2 + 2(x - 1) + 2(y - 1)

Answer

z = 2 - 2(x - 1) - 2(y - 1)

Answer

z = 2 + 2(x - 1) - 2(y - 1)

Answer

z = 2 + (x - 1) + (y - 1)

Question 35

Using the second partial derivative test, classify the critical point (0, 0) of the function f(x, y) = x² - 2xy + 2y².

Accepted Answer

Local Minimum

Answer

Saddle Point

Answer

Local Maximum

Answer

Test inconclusive

Question 36

What are the absolute maximum and minimum values of the function f(x, y) = x² + y² on the closed unit disk x² + y² ≤ 1?

Accepted Answer

Maximum: 1, Minimum: 0

Answer

Maximum: 2, Minimum: -1

Answer

Maximum: 2, Minimum: 0

Answer

Maximum: 1, Minimum: -1

Question 37

Which of the following is the correct notation for a function of three variables, x, y, and z?

Accepted Answer

f(x, y, z)

Answer

h(x + y + z)

Answer

g(x y z)

Answer

j(x * y * z)

Question 38

What is the domain of the function f(x, y) = √(x² - y²)?

Accepted Answer

{(x, y) | x² ≥ y²}

Answer

{(x, y) | x² ≤ y²}

Answer

{(x, y) | x ≤ y}

Answer

{(x, y) | x ≥ y}

Question 39

Calculate the limit of the function f(x, y) = (x² - y²) / (x - y) as (x, y) approaches (1, 1).

Accepted Answer

2

Answer

-1

Answer

1

Answer

Undefined

Question 40

What is the partial derivative of the function f(x, y) = x³ * y² + 2xy with respect to x?

Accepted Answer

3x² * y² + 2y

Answer

x^3 * 2y + 2x

Answer

x^2 * y^2 + 2

Answer

y^2 * 3x^2 + 2

Question 41

Find the gradient of the function f(x, y, z) = xyz at the point (1, -1, 2).

Accepted Answer

(-1, -2, 1)

Answer

(1, 2, -1)

Answer

(-1, 2, -1)

Question 42

Which of the following is the equation of the tangent plane to the surface z = x^2 + y^2 at the point (1, 1, 2)?

Accepted Answer

z = 2 + 2(x - 1) + 2(y - 1)

Answer

z = 2 + 2(x + 1) + 2(y + 1)

Answer

z = 2 + (x - 1) + (y - 1)

Answer

z = 2 + (x + 1) + (y + 1)

Question 43

Determine whether the function f(x, y) = x^2 * y + y^3 is differentiable at the point (0, 0).

Accepted Answer

Not differentiable

Answer

Conditionally differentiable

Answer

Partially differentiable

Answer

Differentiable

Question 44

Find the directional derivative of the function f(x, y) = e^(x+y) in the direction of the vector u = (1, 1) at the point (0, 0).

Accepted Answer

sqrt(2)

Answer

0

Answer

1

Answer

2

Question 45

Which of the following is the correct formula for the curl of a vector field F = (P, Q, R)?

Accepted Answer

(del x F) = (Rz - Qy, Px - Rz, Qy - Px)

Answer

(del x F) = (Qx - Rz, Px - Qy, Rz - Px)

Answer

(del x F) = (Qy + Rz, Px - Qy + Rz, Px + Qy - Rz)

Answer

(del x F) = (Px + Qy + Rz, Px - Qy + Rz, Px + Qy - Rz)

Question 46

Calculate the divergence of the vector field F = (x^2, y^2, z^2).

Accepted Answer

2x + 2y + 2z

Answer

x^2 + y^2 + z^2

Answer

2

Answer

1

Question 47

Find the limit of the function f(x, y) = (x^2 - y^2)/(x - y) as (x, y) approaches (1, 1).

Accepted Answer

0

Answer

2

Answer

1

Question 48

Which of the following points is a critical point of the function f(x, y) = x^3 + y^3 - 3xy?

Accepted Answer

(0, 0)

Answer

(2, 2)

Answer

(3, 3)

Answer

(1, 1)

Question 49

Calculate the partial derivative of f(x, y) = sin(xy) with respect to x.

Accepted Answer

y*cos(xy)

Answer

cos(xy)

Answer

sin(xy)

Answer

x*cos(xy)

Question 50

Find the directional derivative of f(x, y) = x^2 + y^2 at the point (1, 2) in the direction of the vector (1, 1).

Accepted Answer

5

Answer

10

Answer

3

Answer

8

Question 51

Determine if the function f(x, y) = x^2 - 2xy + y^2 is continuous at the point (0, 0).

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

Only if x = y

Question 52

Which of the following is a level curve of the function f(x, y) = x^2 + y^2 - 4?

Accepted Answer

A circle with center at (0, 0) and radius 2

Answer

An ellipse

Answer

A straight line

Answer

A parabola opening up

Question 53

Find the absolute maximum value of the function f(x, y) = x*y on the rectangle defined by 0 ≤ x ≤ 2 and 0 ≤ y ≤ 3.

Accepted Answer

6

Answer

8

Answer

12

Answer

4

Question 54

Which of the following is the integral of the function f(x, y) = xy over the region R bounded by the lines x = 0, x = 2, y = 0, and y = 1?

Accepted Answer

1

Answer

4

Answer

3

Answer

2

Question 55

Find the volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 2 - x about the x-axis.

Accepted Answer

π/3

Answer

π/6

Answer

Cannot be determined from the given information

Answer

2π/3

Question 56

Let f(x, y) = x^2 + y^2 and g(x, y) = xy. Find the gradient of the function h(x, y) = f(x, y) + g(x, y).

Accepted Answer

(2x + y, 2y + x)

Answer

(x + y, y + x)

Answer

(2x, 2y)

Answer

(x^2 + y^2, xy)