Functions of Several Variables: Master the Concepts with Interactive Quizzes

Question 1

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

Accepted Answer

(1, 1)

Answer

(0, 0)

Answer

(e, e)

Answer

(-e, -e)

Question 2

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

Accepted Answer

The gradient is zero.

Answer

The Hessian matrix is negative definite.

Answer

The function is continuous.

Answer

The function is differentiable.

Question 3

Which of the following is a sufficient condition for a function of several variables to have a maximum?

Accepted Answer

The Hessian matrix is positive definite.

Answer

The function is bounded above.

Answer

The function is continuous.

Answer

The function has a unique critical point.

Question 4

State the chain rule for finding the derivative of a function of a function.

Accepted Answer

d/dx[f(g(x))] = (df/dg)(dg/dx)

Answer

(df/dg)(dg/dx) = d/dx[f(g(x))]

Answer

d/dx[f(g(x))] = (df/dx)(dg/dx)

Answer

(df/dx)(dg/dx) = d/dx[f(g(x))]

Question 5

State the inverse function theorem.

Accepted Answer

If f(x, y) has a continuous Jacobian at (a, b) and f(a, b) is invertible, then f is invertible in a neighborhood of (a, b).

Answer

If f(x, y) has a continuous Jacobian at (a, b), then f is invertible in a neighborhood of (a, b).

Answer

If f(x, y) is differentiable at (a, b), then f is invertible in a neighborhood of (a, b).

Answer

If f(x, y) is continuous at (a, b), then f is invertible in a neighborhood of (a, b).

Question 6

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

Accepted Answer

(2, 2, 2)

Answer

(1, 1, 1)

Answer

(x, y, z)

Question 7

How can the Jacobian matrix be used to determine if a differentiable function of several variables is invertible at a given point?

Accepted Answer

If the Jacobian matrix has a non-zero determinant at that point.

Answer

If the Jacobian matrix is a square matrix.

Answer

If the trace of the Jacobian matrix is non-zero.

Answer

If the Jacobian matrix is diagonalizable.

Question 8

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

Accepted Answer

3y^2

Answer

2x

Answer

y

Answer

x

Question 9

Calculate the directional derivative of f(x, y) = sin(x + y) at the point (π/4, π/4) in the direction of the unit vector v = (1, 1).

Accepted Answer

√2/2

Answer

1

Answer

0

Answer

-√2/2

Question 10

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

Accepted Answer

The gradient of f at (a, b) is zero.

Answer

The Hessian matrix of f at (a, b) is positive definite.

Answer

The function f is continuous at (a, b).

Answer

The Laplacian of f at (a, b) is negative.

Question 11

Find the area of the region bounded by the curves y = x^2 and y = 4x.

Accepted Answer

64/3

Answer

64

Answer

16/3

Answer

128/3

Question 12

Consider the function f(x, y) = x^2 - y^2. Find the equation of the level curve of f corresponding to the level z = 1.

Accepted Answer

x^2 - y^2 = 1

Answer

x^2 + y^2 = 1

Answer

x - y = 1

Question 13

Which of the following functions is differentiable at every point in its domain? A function is differentiable if its partial derivatives exist and are continuous.

Accepted Answer

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

Answer

f(x, y) = xy/(x^2 + y^2), which has a discontinuity at (0,0)

Answer

f(x, y) = |x| + |y|, which is not differentiable at points where x or y is zero

Answer

f(x, y) = max(x, y), which is not differentiable at points where x = y

Question 14

Given the function f(x, y) = e^(-x^2 - y^2), find the critical point(s) of f.

Accepted Answer

(0, 0)

Answer

(0, 1)

Answer

(1, 0)

Answer

No critical points

Question 15

Which of the following is NOT a property of a differentiable function?

Accepted Answer

Continuous at every point in its domain

Answer

Has a gradient vector at every point in its domain

Answer

Satisfies the Mean Value Theorem

Answer

Has partial derivatives at every point in its domain

Question 16

Which of the following is NOT a concept related to functions of several variables?

Accepted Answer

Integral

Answer

Directional derivative

Answer

Partial derivative

Answer

Gradient

Question 17

Given the function f(x, y) = x^2 + y^2, what is the partial derivative of f with respect to x at the point (1, 2)?

Accepted Answer

2

Answer

4

Answer

1

Question 18

What is the shape of the level curves of the function f(x, y) = x² - y²?

Accepted Answer

Hyperbolas

Answer

Lines

Answer

Circles

Answer

Parabolas

Question 19

The partial derivative ∂f/∂x represents the rate of change of f with respect to:

Accepted Answer

x, holding y constant

Answer

both x and y

Answer

y, holding x constant

Answer

neither x nor y

Question 20

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

Accepted Answer

z = 2x + 2y - 2

Answer

z = x + y

Answer

z = x² + y²

Answer

z = 2x + 2y + 2

Question 21

The Hessian matrix of a function f(x, y) is used to determine:

Accepted Answer

The nature of critical points (maximum, minimum, saddle point)

Answer

The directional derivative

Answer

The gradient of the function

Question 22

A function f(x, y) is said to be differentiable at a point (a, b) if:

Accepted Answer

The partial derivatives exist and are continuous at (a, b)

Answer

The gradient exists at (a, b)

Answer

The function is continuous at (a, b)

Question 23

The chain rule for functions of several variables states that:

Accepted Answer

The derivative of a composite function is the product of the derivatives of the individual functions, with each derivative evaluated at the appropriate point.

Answer

The derivative of a composite function is the sum of the derivatives of the individual functions.