Master Multivariable Calculus: Enhance Your Mathematical Skills

Question 1

Find the direction of the maximum rate of change of the function f(x, y) = sin(x) + cos(y) at the point (π/4, π/3). Select the correct option for the direction of maximum rate of change.

Accepted Answer

(√2/2, √2/2)

Answer

(-√2/2, √2/2)

Answer

(√2/2, -√2/2)

Answer

(-√2/2, -√2/2)

Question 2

The surface area of a sphere of radius 'r' is given by 4πr^2. Find the rate of change of the surface area with respect to the radius. Select the correct option for the rate of change.

Accepted Answer

8πr

Answer

4πr

Answer

2πr

Answer

πr

Question 3

Determine whether the vector field F(x, y) = (y, -x) is conservative or not. Select the correct option for the nature of the vector field.

Accepted Answer

Conservative

Answer

Not Conservative

Answer

Cannot be determined

Question 4

Which of the following is the equation of the tangent plane to the surface z = x^2 - y^2 at the point (1, 1, 0)? Select the correct option for the equation of the tangent plane.

Accepted Answer

z = 2x - 2y - 1

Answer

z = 2x + 2y - 1

Answer

z = x - y - 1

Answer

z = x + y - 1

Question 5

Which of the following is the definition of the partial derivative of a function f(x, y) with respect to x?

Accepted Answer

The rate of change of f(x, y) with respect to x when y is held constant

Answer

The rate of change of f(x, y) with respect to y when x is held constant

Answer

The total change of f(x, y) when both x and y change

Answer

The average rate of change of f(x, y) over a given interval

Question 6

Which of the following is the formula for the double integral of a function f(x, y) over a region R?

Accepted Answer

∫∫R f(x, y) dA

Answer

∫∫R f(x) dx dy

Answer

∫∫R f(y) dx dy

Answer

∫∫R f(x, y) dx + dy

Question 7

Which of the following is a necessary condition for a function of two variables to have a maximum or minimum?

Accepted Answer

The partial derivatives of the function must be zero at that point

Answer

The partial derivatives of the function must be nonzero at that point

Answer

The second partial derivatives of the function must be equal at that point

Answer

The function must be continuous at that point

Question 8

Find the critical points of the function f(x, y) = x^3 + y^2 - 3x.

Accepted Answer

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

Answer

(0, 0)

Answer

(-1, 1)

Answer

No critical points

Question 9

Which of the following is the definition of the gradient of a function f(x, y)?

Accepted Answer

A vector whose components are the partial derivatives of the function

Answer

A vector whose magnitude is the directional derivative of the function

Answer

A vector whose direction is perpendicular to the level curves of the function

Answer

A vector whose length is the maximum value of the function

Question 10

Which of the following is the Chain Rule for partial derivatives?

Accepted Answer

∂f/∂x = ∂g/∂x * ∂f/∂g

Answer

∂f/∂x = ∂f/∂y * ∂y/∂x

Answer

∂f/∂x = ∂x/∂f * ∂f/∂g

Answer

∂f/∂x = ∂f/∂g * ∂g/∂y

Question 11

What is the curl of the vector field F(x, y) = (x + y)i + (x - y)j?

Accepted Answer

(0, 2)

Answer

(2, 0)

Answer

(-2, 0)

Answer

(0, -2)

Question 12

Which of the following is a necessary condition for a function to have a local maximum?

Accepted Answer

Gradient vector is zero

Answer

Divergence vector is zero

Answer

Curl vector is zero

Answer

Laplacian operator is positive

Question 13

Which of the following is a line integral?

Accepted Answer

∫f(x, y) dr

Answer

∫∫f(x, y) dA

Answer

∫∫∫f(x, y, z) dV

Answer

∫f(x) dx

Question 14

Calculate the directional derivative of the function f(x, y) = x³y² + xy at the point (1, 1) in the direction of the unit vector v = (3/5)i - (4/5)j.

Accepted Answer

7

Answer

5

Answer

3

Answer

1

Question 15

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

Accepted Answer

2xy + 3y^2

Answer

xy + y^3

Answer

x^2 + 3y^2

Answer

x^2y

Question 16

Evaluate the double integral ∬R (x^2 + y^2) dA, where R is the region bounded by the circle with radius 2 centered at the origin.

Accepted Answer

π

Answer

π/4

Answer

2π

Answer

4π

Question 17

Which of the following is the gradient of the function f(x, y) = sin(xy)?

Accepted Answer

(cos(xy), xsin(xy))

Answer

(ysin(xy), xcos(xy))

Answer

(xcos(xy), ysin(xy))

Answer

(-ysin(xy), xcos(xy))

Question 18

Determine if the following integral exists: ∬D (1/sqrt(x^2 + y^2)) dA, where D is the region inside the unit circle.

Accepted Answer

Diverges

Answer

Converges

Answer

Cannot be determined

Answer

Exists only at the origin

Question 19

Determine if the vector field F(x, y) = (y^2, x^2) is conservative.

Accepted Answer

Conservative

Answer

Not conservative

Answer

Cannot be determined

Answer

Conservative only in the first quadrant

Question 20

Which of the following is the partial derivative of f(x, y, z) = x^2 + y^3 + z^4 with respect to x?

Accepted Answer

2x

Answer

3y2

Answer

4z3

Answer

None of these

Question 21

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

Accepted Answer

4

Answer

0

Answer

2

Answer

6

Question 22

Which of the following is the correct formula for the change of variables in a double integral?

Accepted Answer

∫∫R' f(x, y) dx dy = ∫∫R f(u, v) |J| du dv

Answer

∫∫R f(x, y) dx dy = ∫∫R' f(u, v) |J| du dv

Answer

∫∫R' f(u, v) dx dy = ∫∫R f(x, y) |J| du dv

Question 23

Determine whether the following function f(x, y) = x^2y + y^3 has a local minimum at the origin.

Accepted Answer

No

Answer

Yes

Answer

Cannot be determined

Question 24

Green's Theorem connects line integrals around a closed curve to double integrals over the enclosed region. Which statement accurately describes this theorem?

Accepted Answer

Line integrals around a closed curve are related to double integrals over the enclosed region by Green's Theorem.

Answer

Green's Theorem is not applicable to positively oriented closed curves.

Answer

Green's Theorem is only valid for negatively oriented closed curves.

Question 25

Identify the correct formula for calculating the gradient of a function f(x, y, z):

Accepted Answer

(df/dx, df/dy, df/dz)

Answer

(dx/df, dy/df, dz/df)

Answer

(df/dx, df/dy)

Answer

(dx/df, dy/df)

Question 26

Which theorem guarantees the existence of the limit of a multivariable function?

Accepted Answer

Cauchy's Limit Theorem

Answer

Chain Rule

Answer

Mean Value Theorem

Answer

Cauchy-Schwarz Inequality

Question 27

Identify the correct formula for the directional derivative of f(x, y) in the direction of the unit vector u = (cos θ, sin θ):

Accepted Answer

∇f ⋅ u

Answer

f(x + cos θ, y + sin θ)

Answer

f'(x + cos θ, y + sin θ)

Answer

f(x, y) + ∇f ⋅ u

Question 28

Which method is used to find the absolute maximum and minimum values of a multivariable function on a closed and bounded region?

Accepted Answer

Closed Interval Method

Answer

Extreme Value Theorem

Answer

Rolle's Theorem

Answer

Newton's Method

Question 29

Define the partial derivative of a multivariable function f(x, y) with respect to x.

Accepted Answer

The rate of change of f in the x direction, holding y constant

Answer

The rate of change of f in the y direction, holding x constant

Answer

The directional derivative of f in the direction of the vector (1, 0)

Answer

The gradient of f

Question 30

Calculate the partial derivative of f(x, y) = x^2y + y^3 with respect to x.

Accepted Answer

2xy + 3y^2

Answer

2xy

Answer

x^2

Answer

y^3

Question 31

Find the gradient of the function f(x, y, z) = x^2 + y^2 + z^2.

Accepted Answer

(2x, 2y, 2z)

Answer

(x^2, y^2, z^2)

Answer

(x + y + z, x + y + z, x + y + z)

Answer

(1, 1, 1)

Question 32

State the Fundamental Theorem of Line Integrals.

Accepted Answer

If a vector field is conservative, the line integral around a closed curve is zero.

Answer

The line integral of a vector field along a path is equal to the work done by the vector field along the path.

Answer

The line integral of a scalar field around a closed curve is equal to the circulation of the scalar field around the curve.

Answer

The line integral of a vector field along a path is always positive.

Question 33

Define the triple integral of a function f(x, y, z) over a solid region E.

Accepted Answer

The limit of a sum of Riemann sums over a partition of E

Answer

The volume under the surface defined by f(x, y, z) over E

Answer

The surface area of the solid bounded by f(x, y, z) over E

Answer

The work done by the vector field F(x, y, z) = (f(x, y, z), 0, 0) along the boundary of E

Question 34

Which of the following is NOT an application of multivariable calculus?

Accepted Answer

Simplifying functions with only one variable

Answer

Analyzing the behavior of dynamic systems with multiple inputs

Answer

Solving complex problems involving multiple variables

Answer

Understanding the geometry of surfaces

Question 35

Determine the surface integral of F = (x, y, z) over the surface of the unit sphere

Accepted Answer

4π/3

Answer

π

Answer

2π

Answer

0

Question 36

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

Accepted Answer

lim(h->0) [f(x+h, y) - f(x, y)]/h

Answer

lim(h->0) [f(x, y+h) - f(x, y)]/h

Answer

lim(k->0) [f(x+k, y) - f(x, y)]/k

Answer

lim(k->0) [f(x, y+k) - f(x, y)]/k

Question 37

Which of the following is NOT a theorem in multivariable calculus?

Accepted Answer

Pythagorean Theorem

Answer

Green's Theorem

Answer

Chain Rule

Answer

Gauss's Theorem

Question 38

What is the gradient of the vector field F(x, y, z) = (x, y, z)?

Accepted Answer

∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z)

Answer

∇F = (∂F/∂y, ∂F/∂z, ∂F/∂x)

Answer

∇F = (∂F/∂z, ∂F/∂x, ∂F/∂y)

Answer

∇F = (x, y, z)

Question 39

What is the Laplacian of the function f(x, y, z)?

Accepted Answer

∇^2f = ∂^2f/∂x^2 + ∂^2f/∂y^2 + ∂^2f/∂z^2

Answer

∇^2f = ∂f/∂x + ∂f/∂y + ∂f/∂z

Answer

∇^2f = x^2 + y^2 + z^2

Answer

∇^2f = 0

Question 40

Determine if the vector field F(x, y) = (2x + y)i + (x - 2y)j is conservative?

Accepted Answer

Yes, because ∂(P)/∂y = ∂(Q)/∂x

Answer

No, because ∂(P)/∂y ≠ ∂(Q)/∂x

Answer

Yes, because ∂(P)/∂x = ∂(Q)/∂y

Answer

No, because the curl of F is not zero

Question 41

Find the maximum value of f(x, y, z) = x^2 + y^2 - z^2 on the unit sphere: x^2 + y^2 + z^2 = 1?

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 42

Determine if f(x, y) = e^x + y^2 is a potential function for the vector field F(x, y) = (e^x)i + (2y)j?

Accepted Answer

Yes, because ∂f/∂x = e^x = P and ∂f/∂y = 2y = Q

Answer

No, because ∂f/∂x = e^x ≠ P

Answer

Yes, because ∂f/∂y = 2y = Q

Answer

No, because ∂f/∂x = e^x ≠ P and ∂f/∂y = 2y ≠ Q

Question 43

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

Accepted Answer

2xy + y^2

Answer

x^2 + y^2

Answer

x^2y

Answer

xy^2

Question 44

Find the critical points of the function f(x, y) = x^3 + y^3 - 3xy.

Accepted Answer

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

Answer

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

Answer

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

Answer

No critical points

Question 45

Determine if the function f(x, y) = 2x^2 + 3y^2 is homogeneous.

Accepted Answer

Yes, of degree 2

Answer

Yes, of degree 1

Answer

No, it is not homogeneous

Answer

Cannot be determined from the given options

Question 46

Which of the following accurately defines the partial derivative of a function f(x, y) with respect to x?

Accepted Answer

The rate of change of f relative to x while y is held constant

Answer

The rate of change of f with respect to both x and y simultaneously

Answer

The rate of change of f with respect to y while x is held constant

Question 47

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

Accepted Answer

2xy

Answer

2xy + 9y^2

Answer

x^2 + 9y^2

Answer

2x + 9y^2

Question 48

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

Accepted Answer

(0, 0)

Answer

(2, 2)

Answer

(1, 1)

Answer

(3, 3)

Question 49

Which of the following is NOT a property of a conservative vector field?

Accepted Answer

The line integral of the vector field is dependent on the path taken.

Answer

The curl of the vector field is zero.

Answer

The vector field can be expressed as the gradient of a scalar function.

Question 50

Find the surface area of the part of the paraboloid z = 1 - x^2 - y^2 that lies above the xy-plane.

Accepted Answer

π(√2 + 1)

Answer

π/2 (√2 + 1)

Answer

π/2 (√2 - 1)

Answer

π(√2 - 1)

Question 51

The divergence of the vector field F(x, y, z) = <x^2, y^2, z^2> is:

Accepted Answer

2x + 2y + 2z

Answer

2x^2 + 2y^2 + 2z^2

Answer

x^2 + y^2 + z^2

Answer

x + y + z