Multivariate Calculus Quizzes: Master Calculus for Multiple Variables

Question 1

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

Accepted Answer

1/√2

Answer

√2

Answer

2

Answer

4

Question 2

What is the formula for the gradient of a function f(x, y)?

Accepted Answer

∇f(x, y) = (∂f/∂x, ∂f/∂y)

Answer

∇f(x, y) = (∂f/∂y, ∂f/∂x)

Answer

∇f(x, y) = (∂²f/∂x², ∂²f/∂y²)

Answer

∇f(x, y) = (f(x) + f(y))

Question 3

Determine the gradient of the function f(x, y) = x^2 + y^2 at the point (1, 2).

Accepted Answer

(2, 4)

Answer

(4, 2)

Answer

(1, 2)

Answer

(0, 0)

Question 4

For a function of two variables, which condition ensures the existence of a local minimum at a specific point?

Accepted Answer

The Hessian matrix is positive definite at that point.

Answer

The gradient vector points in the direction of a positive slope.

Answer

The first-order partial derivatives are continuous at that point.

Answer

The second-order partial derivatives are all negative at that point.

Question 5

Find the gradient of the function F(x, y, z) = x^2 + y^2 + z^2 at the point (1, 2, 3).

Accepted Answer

<1, 2, 3>

Answer

0

Answer

<1, 1, 1>

Answer

<-1, -2, -3>

Question 6

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

Accepted Answer

2√2

Answer

1

Answer

2

Answer

4

Question 7

Which theorem in multivariate calculus does NOT relate to vector fields?

Accepted Answer

The Inverse Function Theorem

Answer

The Gradient Theorem

Answer

Stokes' Theorem

Answer

Green's Theorem

Question 8

Calculate the directional derivative of the function f(x,y) at the point (1,2) in the direction of the unit vector u = (1/sqrt(2), 1/sqrt(2)).

Accepted Answer

sqrt(2) * (fx(1,2) + fy(1,2))

Answer

fx(1,2) + fy(1,2)

Answer

fx(1,2) - fy(1,2)

Answer

0

Question 9

Calculate the gradient of a function f(x, y) using the appropriate formula.

Accepted Answer

∇f = (f_x, f_y)

Answer

∇f = (f_y, f_x)

Answer

∇f = (f_x, f_x)

Question 10

Find the directional derivative of the function f(x,y) = x² + y² at the point (1,2) in the direction of the unit vector u = (6/10)î + (8/10)ĵ.

Accepted Answer

14

Answer

10

Answer

2

Answer

-2

Question 11

What are the necessary and sufficient conditions for a multivariable function to attain an absolute maximum or absolute minimum value?

Accepted Answer

The function must be continuous on a closed and bounded domain and have continuous first-order partial derivatives at each point in the domain.

Answer

The function must have continuous first-order partial derivatives at each point in the domain.

Answer

The function must have zero gradients at all points in the domain.

Answer

The function must be differentiable at each point in the domain.

Question 12

Given a differentiable function f(x, y), determine the directional derivative of f at the point (1, 2) along the direction specified by the unit vector u = (cos(θ), sin(θ)).

Accepted Answer

cos(θ) * f_x(1, 2) + sin(θ) * f_y(1, 2)

Answer

||u|| * f(1, 2)

Answer

f_x(1, 2) + f_y(1, 2)

Answer

0

Question 13

Which of the following is a sufficient condition for a critical point of a function f(x, y) to be a local minimum?

Accepted Answer

The Hessian matrix of f is positive definite at the critical point.

Answer

The first partial derivatives of f are both zero at the critical point.

Answer

The function f is concave up at the critical point.

Answer

The second partial derivatives of f are both negative at the critical point.

Question 14

Which of the following correctly defines the gradient of a function f(x, y)?

Accepted Answer

The vector composed of the partial derivatives of f

Answer

The set of points where f(x, y) = 0

Answer

The linear approximation of f(x, y) near a given point

Answer

The direction of the steepest ascent

Question 15

What expression represents the surface integral of a scalar function f(x, y, z) over a surface S?

Accepted Answer

∫∫f(x, y, z) dA

Answer

∫∫f(x, y) dA

Answer

∫∫f(x, y, z) dV

Answer

∫∫f(x, y) dV

Question 16

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

Accepted Answer

(y - z, z - x, x - y)

Answer

(0, 0, 0)

Answer

(2x, 2y, 2z)

Question 17

Which of the following is NOT a necessary condition for the existence of a line integral for a vector field?

Accepted Answer

The vector field must be conservative.

Answer

The vector field must be continuous.

Answer

The path must be piecewise smooth.

Answer

The path must be closed.

Question 18

What is the surface integral of the function f(x, y, z) = z over the surface of the unit sphere?

Accepted Answer

4π

Answer

0

Answer

2π

Answer

π

Question 19

What is the dot product of the vectors A = (2, -1, 3) and B = (4, 0, -2)?

Accepted Answer

5

Answer

2

Answer

10

Answer

15

Question 20

Which of the following is the equation of the tangent plane to the surface f(x, y) = x^3 + y^3 at the point (1, 1)?

Accepted Answer

z = 3x + 3y - 5

Answer

z = 3x - 3y + 5

Answer

z = x + y - 1

Answer

z = -x - y + 1

Question 21

Consider the vector field F(x, y, z) = (y, z, x). Calculate the curl of F and interpret its physical meaning in terms of the circulation of F around a small loop.

Accepted Answer

The curl of F is (z, x, y), indicating that the circulation of F around a small loop is given by the line integral of F around the loop.

Answer

The curl of F is (0, 0, 0), indicating that F is a conservative vector field.

Answer

The curl of F is (-z, x, -y), indicating the counterclockwise circulation of F around a small loop.

Answer

The curl of F is (1, 1, 1), indicating the net flux of F through a small loop.

Question 22

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

Accepted Answer

The work done is path-dependent.

Answer

The curl is zero.

Answer

The line integral around any closed path is zero.

Answer

It is the gradient of a scalar potential function.

Question 23

True or False: The order of differentiation does not affect the result for mixed partial derivatives.

Accepted Answer

True

Answer

False

Question 24

Apply Green's Theorem to evaluate the line integral of the vector field F(x, y) = (x^2, y^2) around the boundary of the region R enclosed by the circle x^2 + y^2 = 1.

Accepted Answer

2π

Answer

0

Answer

π

Answer

4π

Question 25

Which of the following is NOT a partial derivative of the function f(x, y, z) = x^2 + y^2 + z^2?

Accepted Answer

∂f/∂u = 0

Answer

∂f/∂x = 2x

Answer

∂f/∂y = 2y

Answer

∂f/∂z = 2z

Question 26

Find the gradient of the function g(x, y) = e^(xy).

Accepted Answer

∇g = (ye^(xy), xe^(xy))

Answer

∇g = (xye^(xy), e^(xy))

Answer

∇g = (e^(xy), e^(xy))

Answer

∇g = (0, 0)

Question 27

Use the Divergence Theorem to calculate the outward flux of the vector field F(x, y, z) = (x^2, y^2, z^2) through the boundary of the cube defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 1.

Accepted Answer

24

Answer

6

Answer

12

Answer

0

Question 28

Find the curl of the vector field F(x, y, z) = (xz, xy, yz).

Accepted Answer

∇×F = (y, -x, x - y)

Answer

∇×F = (x, y, z)

Answer

∇×F = (0, 0, 0)

Answer

∇×F cannot be defined

Question 29

Which of the following is a Jacobian matrix?

Accepted Answer

The matrix of partial derivatives of a vector-valued function with respect to its input variables

Answer

The matrix of partial derivatives of a scalar-valued function with respect to its input variables

Answer

The inverse of the Hessian matrix

Answer

The determinant of a square matrix

Question 30

Which of the following is a necessary condition for a function f(x, y) to have a maximum at a point (a, b)?

Accepted Answer

∇f(a, b) = 0 and the Hessian matrix of f at (a, b) is positive definite

Answer

∇f(a, b) = 0 and the Hessian matrix of f at (a, b) is negative definite

Answer

∇f(a, b) ≠ 0 and the Hessian matrix of f at (a, b) is positive definite

Answer

∇f(a, b) ≠ 0 and the Hessian matrix of f at (a, b) is negative definite

Question 31

Which of the following is a line integral?

Accepted Answer

An integral of a vector function along a curve

Answer

An integral of a scalar function along a curve

Answer

An integral of a scalar function over a surface

Answer

An integral of a vector function over a surface

Question 32

The gradient of a scalar function f(x, y) is defined as:

Accepted Answer

(∂f/∂x, ∂f/∂y)

Answer

(-∂f/∂x, -∂f/∂y)

Answer

(∂f/∂y, ∂f/∂x)

Answer

(-∂f/∂y, ∂f/∂x)

Question 33

Which of these integrals calculates the volume under the surface z = f(x, y) and above the rectangular region R in the xy-plane?

Accepted Answer

∫∫R f(x, y) dA

Answer

∫∫R f(y, x) dA

Answer

∫∫R f(x, y) dy dx

Answer

∫∫R f(y, x) dx dy

Question 34

Which of these theorems links line integrals to double integrals?

Accepted Answer

Green's theorem

Answer

Divergence theorem

Answer

Stokes' theorem

Answer

Fundamental theorem of calculus

Question 35

For a function to have a local minimum, which of these is a necessary condition?

Accepted Answer

The gradient is zero at that point

Answer

The Hessian matrix is positive definite at that point

Answer

The Laplacian is positive at that point

Answer

The determinant of the Hessian matrix is zero at that point

Question 36

Which of these transformations is a linear transformation?

Accepted Answer

f(x, y) = (x + y, x - y)

Answer

f(x, y) = (x^2, y^2)

Answer

f(x, y) = (sin(x), cos(y))

Answer

f(x, y) = (x, y^3)

Question 37

What is the Jacobian matrix of the transformation f(x, y) = (u = x + y, v = x - y)?

Accepted Answer

[[1, 1], [1, -1]]

Answer

[[1, -1], [1, 1]]

Answer

[[1, 0], [0, 1]]

Answer

[[0, 1], [1, 0]]

Question 38

Identify the conservative vector field among the following:

Accepted Answer

F(x, y) = (e^x + y)i + (e^y + x)j

Answer

F(x, y) = (2x + y)i + (x - 2y)j

Answer

F(x, y) = (x^2 + y)i + (x - y^2)j

Answer

F(x, y) = (x - y)i + (x + y)j

Question 39

The gradient of a function f(x, y) is defined as:

Accepted Answer

∇f = (∂f/∂x, ∂f/∂y)

Answer

∇f = (∂²f/∂x², ∂²f/∂y²)

Answer

∇f = (∂f/∂y, ∂f/∂x)

Answer

∇f = (f/∂x, f/∂y)

Question 40

The surface integral ∫∫[S] f(x, y, z) dS, where S is a surface represented by the equation z = x² + y², can be evaluated using the parametrization:

Accepted Answer

r(u, v) = (u, v, u² + v²)

Answer

r(u, v) = (v, u, u + v)

Answer

r(u, v) = (u, v, u + v)

Question 41

Which of the following is a necessary condition for a vector field to be irrotational?

Accepted Answer

The curl of the vector field must be zero.

Answer

The vector field must be conservative.

Answer

The divergence of the vector field must be zero.

Question 42

Evaluate the double integral ∫∫_D (x + y) dx dy over the region D bounded by y = x^2 and y = 4

Accepted Answer

32/3

Answer

16

Answer

8

Answer

4

Question 43

Which of the following is the correct definition of the Laplace operator?

Accepted Answer

∇^2 f = ∂^2 f / ∂x^2 + ∂^2 f / ∂y^2

Answer

∇^2 f = (∂f / ∂x)^2 + (∂f / ∂y)^2

Answer

∇^2 f = ∂^2 f / ∂x^2 - ∂^2 f / ∂y^2

Answer

∇^2 f = ∂^2 f / ∂x + ∂^2 f / ∂y

Question 44

Find the divergence of the vector field F(x, y, z) = x^2 î + y^3 ĵ + z^4 k̂

Accepted Answer

2x + 3y^2 + 4z^3

Answer

x + y + z

Answer

0

Answer

x^2 + y^3 + z^4

Question 45

Which of the following is the correct formula for the curl of a vector field?

Accepted Answer

∇ x F = (∂F_z / ∂y - ∂F_y / ∂z) î + (∂F_x / ∂z - ∂F_z / ∂x) ĵ + (∂F_y / ∂x - ∂F_x / ∂y) k̂

Answer

∇ x F = (∂F_x / ∂y - ∂F_y / ∂x) î

Answer

∇ x F = (∂F_x / ∂x + ∂F_y / ∂y + ∂F_z / ∂z) î

Question 46

Find the surface integral of the vector field F(x, y, z) = x î + y ĵ + z k̂ over the part of the sphere x^2 + y^2 + z^2 = 1 that lies above the xy-plane

Accepted Answer

2π

Answer

π

Answer

0

Answer

4π

Question 47

Which of the following is the correct formula for the line integral of a vector field?

Accepted Answer

∫_C F · dr = ∫_a^b F(x(t), y(t)) · (dx/dt î + dy/dt ĵ) dt

Answer

∫_C F · dr = ∫_C F(x î + y ĵ) ds

Answer

∫_C F · dr = ∫_a^b F(x, y) dx dy

Question 48

Which condition is necessary for a function f(x, y) to have a critical point at (a, b)?

Accepted Answer

∇f(a, b) = 0

Answer

f(a, b) = 0

Answer

f(x, y) is continuous at (a, b)

Answer

∇f(a, b) ≠ 0

Question 49

Provide the correct parametrization for the surface given by the equation z = x^2 + y^2.

Accepted Answer

r(u, v) = (u, v, u^2 + v^2)

Answer

r(u, v) = (u, v, u + v)

Answer

r(u, v) = (v, u, u^2 + v^2)

Answer

r(u, v) = (u^2 + v^2, u, v)

Question 50

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

Accepted Answer

√2

Answer

0

Answer

2

Answer

1

Question 51

Which theorem is a sufficient condition for a vector field to be conservative?

Accepted Answer

The Gradient Theorem

Answer

Green's Theorem

Answer

Stokes' Theorem

Question 52

Which vector operator calculates the magnitude of the rate of change of a scalar field in a specific direction?

Accepted Answer

Gradient

Answer

Laplacian

Answer

Divergence

Answer

Curl

Question 53

Which theorem provides a sufficient condition for the existence of a multiple integral over a closed and bounded region?

Accepted Answer

Fubini's Theorem

Answer

Gauss's Theorem

Answer

Cauchy's Integral Theorem

Answer

Stokes' Theorem

Question 54

What is the curl of the vector field F(x, y, z) = (xy, yz, zx)?

Accepted Answer

(y - z, z - x, x - y)

Answer

(x, y, z)

Answer

(z - y, x - z, y - x)

Answer

(0, 0, 0)

Question 55

What is the surface integral of the scalar field f(x, y, z) = z over the surface of the unit sphere?

Accepted Answer

4π

Answer

2π

Answer

0

Answer

π

Question 56

What is the volume of the solid enclosed by the surfaces z = x^2 + y^2 and z = 4?

Accepted Answer

16π/3

Answer

4π

Answer

2π

Answer

8π

Question 57

State Green's Theorem for the line integral of the vector field F(x, y) = (P, Q) around a closed curve C.

Accepted Answer

∫C P dx + Q dy = ∫∫R (∂Q/∂x - ∂P/∂y) dA

Answer

∫C P dx - Q dy = ∫∫R (∂P/∂y - ∂Q/∂x) dA

Answer

∫C P dy - Q dx = ∫∫R (∂P/∂y + ∂Q/∂x) dA

Answer

∫C P dy + Q dx = ∫∫R (∂P/∂x + ∂Q/∂y) dA

Question 58

Which of the following options is a recognized technique for finding the extreme values of a function of two variables?

Accepted Answer

Method of Lagrange Multipliers

Answer

Method of Undetermined Coefficients

Answer

Method of Partial Fractions

Question 59

Find the curl of the vector field F(x, y, z) = (x^2, y^3, z^4).

Accepted Answer

(2z - 3y, -2x + 4z, 3y - 4x)

Answer

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

Answer

(2x, 3y, 4z)

Question 60

Which of the following mathematical expressions represents a surface integral?

Accepted Answer

∬_S f(x, y, z) dS

Answer

∫∫_R f(x, y) dA

Answer

∫_a^b f(x) dx

Answer

∫_C f(x, y) ds

Question 61

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

Accepted Answer

**Corrected:** The vector of partial derivatives (∇f = (∂f/∂x, ∂f/∂y))

Answer

The vector of second derivatives

Answer

The scalar derivative

Answer

The normal vector to the surface

Question 62

**Enhanced:** Determine the value of the double integral ∬R f(x, y) dA, where R is the region bounded by the curves y = x and y = x^2.

Accepted Answer

**Refined:** 1/3

Answer

1/4

Answer

1/2

Answer

1

Question 63

**Revised:** Which of the following theorems provides sufficient conditions for a function to have a local maximum or minimum?

Accepted Answer

**Corrected:** Second Derivative Test

Answer

Extreme Value Theorem

Answer

Rolle's Theorem

Question 64

**Simplified:** Calculate the divergence of the vector field F(x, y, z) = (x, y, z).

Accepted Answer

**Concise:** 3

Answer

x + y + z

Answer

0

Answer

1