Master Partial Derivatives: Enhance Your Calculus Skills

Question 1

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

Accepted Answer

2x

Answer

2y

Answer

x^2

Answer

y^2

Question 2

If f(x, y) is a function with continuous partial derivatives, which statement holds true?

Accepted Answer

f is differentiable at all points

Answer

f is differentiable at some points but not all

Answer

f is not differentiable at any point

Answer

The differentiability of f cannot be determined from the provided information

Question 3

For the function f(x, y) = x^2 + y^2, determine the gradient at the point (2, 3).

Accepted Answer

(4, 6)

Answer

(-2, -3)

Answer

(2, 3)

Answer

(0, 0)

Question 4

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

Accepted Answer

2x

Answer

3y^2

Answer

1

Question 5

Identify which of the following is not a property of partial derivatives.

Accepted Answer

Symmetry

Answer

Linearity

Answer

Product Rule

Answer

Chain Rule

Question 6

Which test is appropriate for determining the local maximum or minimum of a multivariable function?

Accepted Answer

Second Derivative Test

Answer

First Derivative Test

Answer

Lagrange Multiplier Test

Answer

Cauchy-Schwarz Inequality

Question 7

Calculate the Hessian matrix of the function f(x, y) = x^4 + y^4 - 2x^2y^2.

Accepted Answer

[4x^3 - 4xy^2, -4xy^3 + 4y^3]

Answer

[4x^3 - 4xy^2, -4xy^3 - 4y^3]

Answer

[4x^3 + 4xy^2, -4xy^3 + 4y^3]

Answer

[4x^3 + 4xy^2, -4xy^3 - 4y^3]

Question 8

Which of the following is an invalid notation for the partial derivative of ( f(x,y) ) with respect to ( y )?

Accepted Answer

( rac{dy}{dx} )

Answer

( f_y )

Answer

( rac{partial f}{partial y} )

Answer

( D_y f )

Question 9

For a differentiable function ( f(x,y) ), which statement is correct?

Accepted Answer

Partial derivatives may not exist at some points.

Answer

Partial derivatives exist at every point.

Answer

Partial derivatives are always constant.

Answer

Partial derivatives are always 0.

Question 10

Find the total differential of ( f(x,y) = e^{xy} ).

Accepted Answer

e^{xy} dx + e^{xy} dy

Answer

dx + dy

Answer

xdy + ydx

Answer

xy dx + xy dy

Question 11

What is the partial derivative of f(x, y) = x^2 + y^2 with respect to x?

Accepted Answer

2x

Answer

x

Answer

y

Answer

2y

Question 12

Which of the following is the linear approximation of the function f(x, y) = sin(x + y) at the point (π/4, π/4)?

Accepted Answer

L(x, y) = (x - π/4) + (y - π/4)

Answer

L(x, y) = (x - π/4) - (y - π/4)

Answer

L(x, y) = -((x - π/4) + (y - π/4))

Answer

L(x, y) = ((x - π/4) - (y - π/4)) / √2

Question 13

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

Accepted Answer

2x

Answer

3y^2

Answer

y^3

Answer

x^3

Question 14

Use partial derivatives to find the local maximum and minimum values of the function h(x, y) = x^2 + y^2 - x.

Accepted Answer

Local maximum: (0, 0); Local minimum: (1/2, 0)

Answer

Local maximum: (0, 0); Local minimum: (0, 0)

Answer

Local maximum: (1/2, 0); Local minimum: (0, 0)

Answer

No local maximum or minimum

Question 15

Find the total differential of f(x, y) = ln(x^2 + y^2) and evaluate it at (1, 1).

Accepted Answer

2x dx + 2y dy

Answer

dx + 2y dy

Answer

x dx + y dy

Answer

2x dx - 2y dy

Question 16

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

Accepted Answer

-2y

Answer

0

Answer

3x^2

Answer

1

Question 17

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

Accepted Answer

Both f_x(a, b) = 0 and f_y(a, b) = 0, and the Hessian matrix of f at (a, b) is positive definite.

Answer

Either f_x(a, b) = 0 or f_y(a, b) = 0, and the Hessian matrix of f at (a, b) is indefinite.

Answer

Both f_x(a, b) = 0 and f_y(a, b) = 0, and the Hessian matrix of f at (a, b) is negative definite.

Question 18

Given the function  *z = sin(xy) + x²y*, find the partial derivative of *z* with respect to *y*, denoted as ∂z/∂y.

Accepted Answer

x cos(xy) + x²

Answer

x cos(xy) + 2xy

Answer

cos(xy) + 2x

Answer

y cos(xy) + x²

Question 19

The temperature *T* (in °C) at a point *(x, y)* on a metal plate is given by *T(x, y) = 100/(1 + x² + y²)*. Calculate the rate of change of temperature at the point *(1, 1)* in the direction of the vector <1, -1>.

Accepted Answer

0

Answer

-25

Answer

25

Answer

50

Question 20

Determine all the critical points of the function *f(x, y) = x³ - 3xy + y³*.

Accepted Answer

(0, 0), (1, 1)

Answer

(0, 0)

Answer

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

Answer

(1, 1)

Question 21

The equation *z² = x² + y²* represents which type of quadric surface?

Accepted Answer

Cone

Answer

Ellipsoid

Answer

Hyperboloid of two sheets

Answer

Hyperboloid of one sheet

Question 22

What is the maximum rate of change of the function *f(x, y) = e^(xy)* at the point *(1, 0)*?

Accepted Answer

√2

Answer

e

Answer

1

Answer

0

Question 23

For the function *f(x, y) = ln(x² + y²)*, find the value of ∂²f/∂x² + ∂²f/∂y² at the point *(1, 1)*.

Accepted Answer

0

Answer

2

Answer

-1

Answer

1

Question 24

A rectangular box has dimensions *x*, *y*, and *z*. If *x* and *y* are increasing at rates of 2 cm/s and 3 cm/s respectively, while *z* is decreasing at a rate of 1 cm/s, what is the rate of change of the volume of the box when *x = 1 cm*, *y = 2 cm*, and *z = 3 cm*?

Accepted Answer

8 cm³/s

Answer

5 cm³/s

Answer

-1 cm³/s

Answer

11 cm³/s

Question 25

The function *f(x, y) = x² + y²* subject to the constraint *g(x, y) = x + y - 1 = 0* has a:

Accepted Answer

Minimum at (1/2, 1/2)

Answer

Minimum at (0, 0)

Answer

Maximum at (1/2, 1/2)

Answer

Saddle point at (1/2, 1/2)

Question 26

Calculate the partial derivative of *f(x, y) = x²y³ + 2xy* with respect to *x*.

Accepted Answer

2xy³ + 2y

Answer

x²y³ + 2y

Answer

3xy² + 2

Answer

3x²y² + 2x

Question 27

Given the function *f(x, y) = sin(xy)*, find ∂f/∂y.

Accepted Answer

xcos(xy)

Answer

cos(xy)

Answer

-xcos(xy)

Answer

ysin(xy)

Question 28

Which of the following statements is TRUE about a function f(x, y) if ∂f/∂x = 0 and ∂f/∂y = 0 at a point (a, b)?

Accepted Answer

The function f(x, y) may have a critical point at (a, b).

Answer

The function f(x, y) has a minimum at (a, b).

Answer

The function f(x, y) has a maximum at (a, b).

Question 29

The equation of the tangent plane to the surface z = f(x, y) = x² + y² at the point (1, 1, 2) is:

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 30

The directional derivative of the function f(x, y) = x²y at the point (1, 2) in the direction of the vector ⟨1, 1⟩ is:

Accepted Answer

4

Answer

6

Answer

8

Answer

2

Question 31

Find the gradient of the function f(x, y) = ln(x² + y²).

Accepted Answer

⟨2x/(x² + y²), 2y/(x² + y²)⟩

Answer

⟨1/(x² + y²), 1/(x² + y²)⟩

Answer

⟨2x, 2y⟩

Answer

⟨x/(x² + y²), y/(x² + y²)⟩

Question 32

The mixed partial derivatives ∂²f/∂x∂y and ∂²f/∂y∂x are equal for a function f(x, y) if:

Accepted Answer

The second-order partial derivatives are continuous in a neighborhood of the point.

Answer

The function f(x, y) is continuous.

Answer

The function f(x, y) is differentiable.

Question 33

Which of the following functions has a saddle point at the origin (0, 0)?

Accepted Answer

f(x, y) = x² - y²

Answer

f(x, y) = x⁴ + y⁴

Answer

f(x, y) = x² + 2y²

Answer

f(x, y) = x² + y²