Multivariable Calculus: Explore Functions of Multiple Variables

Question 1

Express the chain rule for calculating partial derivatives.

Accepted Answer

$$\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} + \frac{\partial z}{\partial v} \cdot \frac{\partial v}{\partial x}$$

Answer

$$\frac{\partial z}{\partial x} = \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial x}$$

Answer

$$\frac{\partial z}{\partial x} = \frac{\partial y}{\partial x} \cdot \frac{\partial z}{\partial y}$$

Answer

$$\frac{\partial z}{\partial x} = \frac{\partial x}{\partial z} \cdot \frac{\partial z}{\partial y}$$

Question 2

Which of the following is the gradient of the function f(x, y) = x^2 + y^3?

Accepted Answer

(2x, 3y)

Answer

(x^2, y^3)

Answer

(1, 1)

Answer

(0, 0)

Question 3

What is the directional derivative of the function f(x, y) = sin(x + y) at the point (π/2, π/4) in the direction of the vector (1, 1)?

Accepted Answer

√2/2

Answer

-√2/2

Answer

1

Answer

-1

Question 4

What is the Lagrange multiplier for the constraint function g(x, y) = x^2 + y^2 - 1 when optimizing the function f(x, y) = x + y subject to the constraint?

Accepted Answer

λ

Answer

1

Answer

-1

Answer

-λ

Question 5

What is the surface integral of the vector field F(x, y, z) = (x, y, z) over the surface S given by the equation z = x^2 + y^2, 0 ≤ z ≤ 1?

Accepted Answer

π

Answer

π/2

Answer

2π

Answer

4π

Question 6

Which of the following is the curl of the vector field F(x, y, z) = (x^2, y^2, z^2)?

Accepted Answer

(-2y, 2x, 0)

Answer

(0, 0, 0)

Answer

(2x, 2y, 2z)

Answer

(2y, -2x, 0)

Question 7

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

Accepted Answer

No, not differentiable

Answer

Yes, differentiable

Answer

Cannot be determined

Answer

Only differentiable if x and y are both non-zero

Question 8

In multivariable calculus, what's the main reason we use implicit differentiation?

Accepted Answer

To find the derivative of a function defined by an equation.

Answer

To locate a function's critical points.

Answer

It's not used in multivariable calculus.

Answer

To calculate a function's integral.

Question 9

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

Accepted Answer

(2x, 3y)

Answer

(x^2, y^3)

Answer

(2x, 2y)

Answer

(x, 3y^2)

Question 10

Which of the following is the integral of the function f(x, y) = x + y over the region R = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 11

Which of the following is the Jacobian of the transformation x = u + v, y = u - v?

Accepted Answer

2

Answer

1

Answer

-1

Answer

-2

Question 12

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

Accepted Answer

z = x + 2y + 1

Answer

z = x + 2y - 3

Answer

z = 2x + y - 1

Answer

z = 2x + y + 3

Question 13

Which integral calculates the volume of the solid region bounded by the surface f(x, y) and the region R in the xy-plane?

Accepted Answer

∫∫R f(x, y) dA

Answer

∫∫∫R f(x, y, z) dV

Answer

∫C f(x, y) ds

Answer

∫[a, b] f(x) dx

Question 14

Which of the following is NOT a partial derivative of the given function f(x, y) = x^2 + y^3?

Accepted Answer

f(x, y)

Answer

∂f/∂x

Answer

∂f/∂y

Answer

∂^2f/∂x∂y

Question 15

Find the gradient of the function g(x, y) = sin(x) * cos(y) at the point (π/4, π/3).

Accepted Answer

(cos(π/4) * cos(π/3), -sin(π/4) * sin(π/3))

Answer

(sin(π/4) * cos(π/3), cos(π/4) * sin(π/3))

Answer

(cos(π/4) * sin(π/3), -sin(π/4) * cos(π/3))

Answer

(sin(π/4) * sin(π/3), cos(π/4) * cos(π/3))

Question 16

Which technique is appropriate for evaluating the integral ∬R (x + y) dA, where R is the region bounded by the curves y = x and y = x^2?

Accepted Answer

Change of variables

Answer

Integration by parts

Answer

Trigonometric substitution

Answer

Partial fractions

Question 17

Determine the area of the region in the xy-plane bounded by the curves y = e^x and y = x^2.

Accepted Answer

∫[0, e] (e^x - x^2) dx

Answer

∫[0, 1] (e^x - x^2) dx

Answer

∫[1, e] (x^2 - e^x) dx

Answer

∫[0, e] (x^2 - e^x) dx

Question 18

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

Accepted Answer

(2, 4, 6)

Answer

(1, 2, 3)

Answer

(x, y, z)

Answer

(0, 0, 0)

Question 19

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

Accepted Answer

2x + 2y

Answer

x + 2y

Answer

x^2 + 2y

Answer

3x + 2y

Question 20

Find the gradient of the function f(x, y) = sin(x) + cos(y).

Accepted Answer

(sin(x), cos(y))

Answer

(cos(x), sin(y))

Answer

(-cos(x), sin(y))

Answer

(-sin(x), -cos(y))

Question 21

Which of the following tests can be used to determine if a function is differentiable at a point?

Accepted Answer

Partial derivative test

Answer

Cauchy-Riemann equations

Answer

Chain rule

Answer

Implicit function theorem

Question 22

Identify the technique that is NOT used for evaluating multiple integrals.

Accepted Answer

Integration by Substitution

Answer

Fubini's Theorem

Answer

Change of Variables

Answer

Polar Coordinates

Question 23

Calculate the directional derivative of f(x, y) = e^(x+y) at the point (0, 1) in the direction of the unit vector ⟨1/√2, 1/√2⟩.

Accepted Answer

e√2

Answer

√2

Answer

2

Answer

e

Question 24

Identify the curl of the vector field F = ⟨z, x, y⟩.

Accepted Answer

⟨y-z, z-x, x-y⟩

Answer

⟨z-x, x-y, y-z⟩

Answer

⟨x-y, y-z, z-x⟩

Answer

⟨y-z, x-z, z-y⟩

Question 25

Determine a potential function for the vector field F = ⟨y, -x⟩.

Accepted Answer

x^2 - y^2

Answer

xy

Answer

x+y

Answer

sin(x)cos(y)

Question 26

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

Accepted Answer

2x

Answer

2y

Answer

x^2

Answer

y^2

Question 27

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

Accepted Answer

The first partial derivatives are zero.

Answer

The second partial derivatives are positive.

Answer

The second partial derivatives are negative.

Answer

The function is differentiable everywhere.

Question 28

Calculate the line integral of the vector field F(x, y) = (x, y) along the curve C given by the upper half of the unit circle centered at the origin.

Accepted Answer

2π

Answer

0

Answer

π

Answer

4π

Question 29

Which of the following is the correct formula for calculating the divergence of a vector field F(x, y, z)?

Accepted Answer

∇·F = (∂F/∂x) + (∂F/∂y) + (∂F/∂z)

Answer

∇·F = (∂F/∂x)² + (∂F/∂y)² + (∂F/∂z)²

Answer

∇·F = (∂F/∂x)·(∂F/∂y)·(∂F/∂z)

Answer

∇·F = 0

Question 30

Determine whether the vector field F(x, y) = (y, -x) is conservative.

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Answer

Only in the first quadrant.

Question 31

Calculate the partial derivative of f(x, y) = x² + y³ with respect to y.

Accepted Answer

df/dy = 3y²

Answer

df/dx = x

Answer

df/d²x = 2x

Answer

df/d²y = 6y

Question 32

Determine the gradient of the function f(x, y) = e^(x+y).

Accepted Answer

∇f = (e^x, e^y)

Answer

∇f = (e^(x+y), e^(x+y))

Answer

∇f = (x+y, x+y)

Answer

∇f = (1, 1)

Question 33

Which property is not associated with conservative vector fields?

Accepted Answer

Curl is zero

Answer

Line integrals are path-independent

Answer

Work done in a closed path is always zero

Answer

Gradient is irrotational

Question 34

Identify the correct form of the divergence theorem.

Accepted Answer

∮S F · dS = ∭V ∇ · F dV

Answer

∮S F · dS = ∬R F · dr

Answer

∮S F · dS = ∭V F · dS

Answer

∮S F · dS = ∬R ∇ · F dV

Question 35

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

Accepted Answer

2x

Answer

2x + 3y^2

Answer

3y^2

Answer

0

Question 36

Determine whether f(x, y) = x^3 - y^2 is continuous at the point (0, 0).

Accepted Answer

Yes

Answer

No

Question 37

Determine if the integral ∫∫[0,1][0,x] e^(x+y) dy dx is convergent.

Accepted Answer

Yes

Answer

No

Question 38

Choose the appropriate substitution for evaluating the integral ∫∫[0,1][0,√(1-x^2)] x^2 + y^2 dy dx.

Accepted Answer

x = r cos(θ), y = r sin(θ)

Answer

x = r sin(θ), y = r cos(θ)

Answer

x = θ cos(r), y = θ sin(r)

Answer

x = θ sin(r), y = θ cos(r)

Question 39

What is the gradient of the function g(x, y) = e^(x-2y)?

Accepted Answer

(e^(x-2y), -2e^(x-2y))

Answer

(2e^(x-2y), -e^(x-2y))

Answer

(-e^(x-2y), 2e^(x-2y))

Answer

(e^(x-2y), 2e^(x-2y))

Question 40

Determine the gradient of the function g(x,y) = sin(x) + cos(y).

Accepted Answer

(sin(x), cos(y))

Answer

(cos(x), -sin(y))

Answer

(-sin(x), -cos(y))

Answer

(cos(x), sin(y))

Question 41

If the double integral of h(x,y) over a region R is 10, what is the value of the double integral of 2h(x,y) over the same region?

Accepted Answer

20

Answer

5

Answer

10

Answer

40

Question 42

Find the partial derivative of the function f(x, y) = x^2 + y^3 with respect to x (Calculate the rate of change of f with respect to x, assuming y is constant).

Accepted Answer

2x

Answer

3y^2

Answer

x^2

Answer

y^3

Question 43

Calculate the gradient of the function f(x, y) = e^(xy) (Find a vector that points in the direction of the greatest rate of change of f at a given point).

Accepted Answer

(ye^xy, xe^xy)

Answer

(e^xy, e^xy)

Answer

(xy, xy)

Answer

(x, y)

Question 44

Find the local maximum value of the function f(x, y) = x^2 - y^2 on the closed disk x^2 + y^2 ≤ 1. (Determine the largest value of f within the circle of radius 1 centered at the origin).

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 45

Which of the following is NOT a potential application of multivariable calculus? (Which area does NOT use multivariable calculus?)

Accepted Answer

Linear algebra

Answer

Fluid dynamics

Answer

Optimization

Answer

Chemistry

Question 46

For the function  *f(x, y) = x² + y²*, what is the partial derivative of *f* with respect to *x* at the point *(2, 1)*?

Accepted Answer

4

Answer

6

Answer

8

Answer

2

Question 47

What is the gradient of the function *f(x, y) = e^(x-y)* at the point *(0, 0)*?

Accepted Answer

(1, -1)

Answer

(1, 1)

Answer

(-1, 1)

Answer

(-1, -1)

Question 48

Which expression correctly represents the change in the value of *f(x, y)* when *x* increases by *Δx* and *y* increases by *Δy*?

Accepted Answer

f(x + Δx, y + Δy) - f(x, y)

Answer

Δx * f(x, y) + Δy * f(x, y)

Answer

ΔxΔy * f(x, y)

Answer

f(x, y) - f(x + Δx, y + Δy)

Question 49

What is the divergence of the vector field *F(x, y) = (x, y)*?

Accepted Answer

2

Answer

-1

Answer

0

Answer

1

Question 50

What is the domain of the function *f(x, y) = ln(x² + y²)*?

Accepted Answer

x² + y² > 0

Answer

x ≥ 0 and y ≥ 0

Answer

x² + y² ≥ 0

Answer

All real numbers