Gradient and Tangent Planes: Interactive Learning Quizzes

Question 1

Determine the equation of the tangent plane to the surface z = x^2 + y^2 at the point (1, 2, 5).

Accepted Answer

z = 5 + 2x + 4y

Answer

z = 5 - 2x - 4y

Answer

z = 5 + x + 2y

Answer

z = 5 - x - 2y

Question 2

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

Accepted Answer

8

Answer

4

Answer

16

Answer

32

Question 3

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

Accepted Answer

(2, -2, 4)

Answer

(-2, 2, -4)

Answer

(1, -1, 2)

Answer

(-1, 1, -2)

Question 4

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

Accepted Answer

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Answer

∇f = (f'(x), f'(y), f'(z))

Answer

∇f = ∂f/∂z

Answer

∇f = (x, y, z)

Question 5

What is the gradient of the function f(x,y,z) = sin(x + y + z)?

Accepted Answer

∇f = (cos(x + y + z), cos(x + y + z), cos(x + y + z))

Answer

∇f = (sin(x + y + z), sin(x + y + z), sin(x + y + z))

Answer

∇f = (x, y, z)

Answer

∇f is undefined.

Question 6

If the gradient vector of a function f(x,y) is zero at a point (a,b), what can be concluded about the function at that point?

Accepted Answer

The function is constant in the neighborhood of (a,b).

Answer

The function has a local maximum at (a,b).

Answer

The function has a local minimum at (a,b).

Answer

The function has a saddle point at (a,b).

Question 7

Indicate which property of the gradient vector is correct.

Accepted Answer

It indicates the direction of maximum rate of change.

Answer

It is perpendicular to the level curve of the function.

Answer

Its magnitude equals the directional derivative in any direction.

Answer

It is zero at critical points.

Question 8

Determine the rate of change of f(x, y) = e^(x-y) at the point (1, 0) in the direction of the vector ⟨2, 1⟩.

Accepted Answer

e^(2/3)

Answer

e

Answer

e^(1/3)

Answer

e^(1/2)

Question 9

For a constant function, which of the following statements about its gradient vector is true?

Accepted Answer

It is a zero vector.

Answer

It is undefined.

Answer

It is parallel to the x-axis.

Answer

It is parallel to the y-axis.

Question 10

For f(x, y) = x^3 + y^3, calculate the directional derivative at the point (1, -1) in the direction of the vector ⟨2, 1⟩.

Accepted Answer

4

Answer

6

Answer

2

Answer

0

Question 11

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

Accepted Answer

A vector that points in the direction of the greatest positive rate of change of f

Answer

A scalar that represents the average rate of change of f in all directions

Answer

A constant that determines the shape of the graph of f

Answer

A matrix that contains the second derivatives of f

Question 12

What is the local maximum of the function f(x, y) = x^3 - 3xy + y^3?

Accepted Answer

(1, 1)

Answer

(0, 0)

Answer

(-1, -1)

Answer

The function has no local maximum

Question 13

What is the absolute minimum of the function f(x, y) = x^2 + y^2 on the closed square [0, 1] x [0, 1]?

Accepted Answer

0

Answer

1

Answer

2

Answer

4

Question 14

Which of the following is a level surface of the function f(x, y, z) = x^2 + y^2 + z^2?

Accepted Answer

The sphere x^2 + y^2 + z^2 = 1

Answer

The plane z = 1

Answer

The cylinder x^2 + y^2 = 1

Answer

The paraboloid z = x^2 + y^2

Question 15

Which of the following accurately defines the gradient of a function f(x,y) at a point (a,b)?

Accepted Answer

A vector pointing in the direction of the greatest rate of change of f at (a,b)

Answer

A vector perpendicular to the level surface of f at (a,b)

Answer

A vector matching the magnitude of the directional derivative of f in any direction at (a,b)

Answer

A vector parallel to the tangent plane to the level surface of f at (a,b)

Question 16

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

Accepted Answer

sqrt(2)

Answer

0

Answer

1

Answer

2

Question 17

Consider the function f(x, y) = x^2 + y^2. Which of the following is the correct definition of the gradient of f at a point (a, b)?

Accepted Answer

The vector (f_x(a, b), f_y(a, b))

Answer

The vector (f_y(a, b), -f_x(a, b))

Answer

The scalar f_x(a, b) + f_y(a, b)

Answer

The scalar f_x(a, b) - f_y(a, b)

Question 18

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

Accepted Answer

The partial derivatives of f exist at (a, b).

Answer

The gradient of f exists at (a, b).

Answer

The Hessian matrix of f exists at (a, b).

Answer

The Laplacian of f exists at (a, b).

Question 19

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

Accepted Answer

The gradient of f is zero at (a, b) and the Hessian matrix of f has both positive and negative eigenvalues at (a, b).

Answer

The gradient of f is nonzero at (a, b) and the Hessian matrix of f has both positive and negative eigenvalues at (a, b).

Answer

The gradient of f is zero at (a, b) and the Hessian matrix of f has only positive eigenvalues at (a, b).

Answer

The gradient of f is zero at (a, b) and the Hessian matrix of f has only negative eigenvalues at (a, b).

Question 20

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

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 21

Consider the function f(x, y) = sin(x - y). Find the equation of the tangent plane to the surface z = f(x, y) at the point (π/4, π/4, 0).

Accepted Answer

z = x - y

Answer

z = -x - y

Answer

z = x + y

Answer

z = -x + y

Question 22

What is the directional derivative of the function f(x, y) = xy^2 at the point (1, 2) in the direction of the vector (3, 4)?

Accepted Answer

28

Answer

14

Answer

10

Answer

8

Question 23

Which of the following is a necessary condition for a function to be differentiable at a point?

Accepted Answer

The partial derivatives exist and are continuous at that point.

Answer

The gradient vector exists at that point.

Answer

The function is continuous at that point.

Answer

The level curves of the function are smooth at that point.

Question 24

Given the function f(x, y) = x² + y² - 4, determine the gradient of f at the point (1, 1).

Accepted Answer

(2, 2)

Answer

(1, -1)

Answer

(1, 1)

Answer

(2, -2)

Question 25

Determine 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 + (x - 1) - 2(y - 1)

Answer

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

Answer

z = 2 + 2x + 2y

Question 26

If the directional derivative of f(x, y) at the point (1, 2) in the direction of the vector (3, 4) is 5, what is the relationship between the gradient vector of f at (1, 2) and the vector (3, 4)?

Accepted Answer

The gradient vector is a scalar multiple of (3, 4)

Answer

The gradient vector is equal to (3, 4)

Answer

The gradient vector is equal to (5, 0)

Answer

The gradient vector is equal to (0, 5)

Question 27

If the gradient of a function f(x, y) at a point is the zero vector, what is the orientation of the tangent plane at that point?

Accepted Answer

The tangent plane is horizontal.

Answer

The tangent plane does not exist.

Answer

The tangent plane is vertical.

Answer

The tangent plane is undefined.

Question 28

Which of the following is the gradient of the function f(x, y) = x^2 + y^3 at the point (1, 2)?

Accepted Answer

<2, 6>

Answer

<1, 4>

Answer

<2, 8>

Answer

<1, 3>

Question 29

What 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)

Answer

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

Answer

z = 2 + 2(y - 1)

Question 30

Find the directional derivative of the function f(x, y) = xy at the point (2, 3) in the direction of the vector v = <1, 2>.

Accepted Answer

13

Answer

9

Answer

11

Answer

15

Question 31

Which of the following statements is true about the gradient of a function?

Accepted Answer

It points in the direction of the greatest rate of change.

Answer

It is a constant vector.

Answer

It is always parallel to the x-axis.

Question 32

What is the gradient of the function f(x, y, z) = x^2 + y^3 + z^4 at the point (1, 2, 3)?

Accepted Answer

<2, 6, 12>

Answer

<1, 4, 8>

Answer

<1, 3, 4>

Answer

<2, 8, 16>

Question 33

Determine the equation of the tangent plane to the surface z = sin(x)cos(y) at the point (π/4, π/3, 1/2).

Accepted Answer

z = 1/2 - (1/2)√3(x - π/4) + (1/2)√3(y - π/3)

Answer

z = 1/2 + (1/2)√3(x - π/4) + (1/2)√3(y - π/3)

Answer

z = 1/2 + 2(x - π/4) + 2(y - π/3)

Question 34

What is the directional derivative of the function f(x, y) = x^2y at the point (1, 2) in the direction of the vector v = <3, 4>?

Accepted Answer

22

Answer

14

Answer

26

Answer

18

Question 35

Which of the following is a necessary condition for a point to lie on the tangent plane to a surface at a given point?

Accepted Answer

The gradient of the function at that point is parallel to the vector from the given point to the candidate point.

Answer

The value of the function at that point is equal to the value of the function at the given point.

Question 36

What is the equation of the tangent plane to the surface z = xy^2 at the point (2, 3, 18)?

Accepted Answer

z = 18 + 6(x - 2) + 12(y - 3)

Answer

z = 18 + 3(x - 2) + 6(y - 3)

Answer

z = 18 + 12(x - 2) + 6(y - 3)

Answer

z = 18 + 6(x - 2) - 12(y - 3)

Question 37

What does the gradient of a function of two variables represent geometrically?

Accepted Answer

A vector pointing in the direction of the greatest rate of increase of the function.

Answer

The slope of the tangent line to the function.

Answer

A vector perpendicular to the tangent plane of the function.

Question 38

Let f(x, y) = x² + xy + y². What is the gradient of f at the point (1, 2)?

Accepted Answer

<4, 5>

Answer

<5, 6>

Answer

<2, 2>

Answer

<3, 4>

Question 39

If the gradient of a function f at a point P is zero, what can be concluded about P?

Accepted Answer

P could be a local maximum, local minimum, or a saddle point.

Answer

P must be a local minimum.

Answer

P must be a local maximum.

Question 40

What is the equation of the tangent plane to the surface z = x² + 2y² at the point (1, 1, 3)?

Accepted Answer

z = 2x + 4y - 3

Answer

z = x + 2y

Answer

z = 2x + 4y + 3

Answer

z = 2x + 4y

Question 41

The directional derivative of f(x, y) at a point P in the direction of a unit vector u represents:

Accepted Answer

The rate of change of f at P in the direction of u.

Answer

The slope of the tangent line to the level curve of f at P.

Answer

The gradient of f at P.

Question 42

Find the directional derivative of f(x, y) = 3x²y at the point (1, 2) in the direction of the vector <3, 4>.

Accepted Answer

30

Answer

36

Answer

12

Answer

24

Question 43

In which direction does the function f(x, y) = x² - y² increase most rapidly at the point (1, 1)?

Accepted Answer

<2, -2>

Answer

<-1, -1>

Answer

<-2, 2>

Answer

<1, 1>

Question 44

The tangent plane to the level surface of a function f(x, y, z) = k at a point P is:

Accepted Answer

Perpendicular to the gradient of f at P.

Answer

Always horizontal.

Answer

Tangent to the level curve of f at P.

Answer

Parallel to the gradient of f at P.

Question 45

If the gradient of f(x, y) = x² + y² - 2x - 4y is <0, 0> at a point P, what is the nature of P?

Accepted Answer

P is a local minimum.

Answer

P is a saddle point.

Answer

P is a local maximum.

Answer

P is an inflection point.

Question 46

The directional derivative of a function f(x, y) at a point P is zero in every direction. What can be concluded about P?

Accepted Answer

P is a critical point of f.

Answer

The gradient of f at P is non-zero.

Answer

P is a local minimum of f.

Answer

P is a local maximum of f.