Exam 2 Theorems

Question 1

What is Clairaut’s Theorem?

Answer 1

Clairaut’s Theorem states that if the mixed partial derivatives of a function are continuous, then the order of differentiation does not matter.

Question 2

True or False: Clairaut’s Theorem applies to all functions regardless of their continuity.

Answer 2

False

Question 3

Fill in the blank: If ( f(x, y) ) has continuous second partial derivatives, then ( f_{xy} = f_{yx} ). This is an example of ________’s Theorem.

Answer 3

Clairaut

Question 4

Which condition must a function satisfy for Clairaut’s Theorem to hold?

Answer 4

The mixed partial derivatives must be continuous.

Question 5

Multiple choice: Which of the following is a consequence of Clairaut’s Theorem? A) The function must be linear, B) The mixed partial derivatives can be interchanged, C) The function must be differentiable.

Answer 5

B) The mixed partial derivatives can be interchanged.

Question 6

What is Fubini’s Theorem primarily used for in calculus?

Answer 6

Fubini’s Theorem is used to evaluate double integrals by allowing the integration to be performed iteratively.

Question 7

True or False: Fubini’s Theorem can only be applied to continuous functions.

Answer 7

False: Fubini’s Theorem can be applied to integrable functions, not just continuous ones.

Question 8

Fill in the blank: Fubini’s Theorem states that the double integral of a function f(x,y) over a rectangular region can be expressed as _______.

Answer 8

the iterated integral of f(x,y) with respect to x and then y, or vice versa.

Question 9

What are the conditions under which Fubini’s Theorem holds?

Answer 9

Fubini’s Theorem holds when the function is integrable over the region of integration.

Question 10

If f(x,y) = xy, what is the iterated integral of f over the rectangle [0,1] x [0,1] using Fubini’s Theorem?

Answer 10

The iterated integral is ∫ from 0 to 1 (∫ from 0 to 1 xy dy) dx = 1/4.

Question 11

What does the Extreme Value Theorem state?

Answer 11

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must attain a maximum and a minimum value on that interval.

Question 12

True or False: The Extreme Value Theorem applies to functions that are not continuous.

Answer 12

False

Question 13

Fill in the blank: The Extreme Value Theorem guarantees the existence of extreme values on a closed interval if the function is ______.

Answer 13

continuous

Question 14

Which of the following is a necessary condition for the Extreme Value Theorem to hold? A) The function must be differentiable, B) The function must be continuous, C) The function must be periodic.

Answer 14

B) The function must be continuous

Question 15

In the context of calculus, how can the Extreme Value Theorem be practically used?

Answer 15

The Extreme Value Theorem can be used to find the maximum and minimum values of a function within a specific interval, which is essential in optimization problems.

Question 16

What does the Extreme Value Theorem state for functions of two variables?

Answer 16

The Extreme Value Theorem states that if a function is continuous on a closed and bounded region, then it attains both a maximum and minimum value in that region.

Question 17

True or False: The Extreme Value Theorem applies to functions that are continuous only on open regions.

Answer 17

False

Question 18

Fill in the blank: For the Extreme Value Theorem, the region must be ______ and ______.

Answer 18

closed; bounded

Question 19

What is the first step to finding extreme values of a function of two variables?

Answer 19

Calculate the partial derivatives and set them equal to zero to find critical points.

Question 20

Multiple Choice: Which of the following methods can be used to test for extrema in a function of two variables?

Answer 20

The Second Derivative Test

Question 21

What is a local maximum in a function of two variables?

Answer 21

A function has a local maximum at (a, b) if f(z, y) < f(a, b) when (z, y) is near (a, b)

Question 22

What is a local minimum in a function of two variables?

Answer 22

A function has a local minimum at (a, b) if f(z, y) > f(a, b) when (z, y) is near (a, b)

Question 23

What is the local maximum value at (a, b)?

Answer 23

The number f(a, b) is called a local maximum value

Question 24

What is the local minimum value at (a, b)?

Answer 24

The number f(a, b) is called a local minimum value

Question 25

What indicates an absolute maximum or minimum in a function?

Answer 25

If the inequalities hold for all points (z, y) in the domain of f, then f has an absolute maximum or absolute minimum at (a, b)

Question 26

What does Theorem 2 state about local maxima or minima?

Answer 26

If f has a local maximum or minimum at (a, b) and the first-order partial derivatives exist, then fa(a, b) = 0 and fy(a, b) = 0

Question 27

What is the gradient vector notation related to Theorem 2?

Answer 27

The conclusion of Theorem 2 can be stated as Vf(a, b) = 0

Question 28

Fill in the blank: A function has a _______ at (a, b) if f(z, y) < f(a, b) for points near (a, b).

Answer 28

local maximum

Question 29

Fill in the blank: A function has a _______ at (a, b) if f(z, y) > f(a, b) for points near (a, b).

Answer 29

local minimum

Question 30

What is a partial derivative?

Answer 30

A partial derivative is the derivative of a function with respect to one variable while keeping other variables constant.

Question 31

True or False: The notation for the partial derivative of a function f(x, y) with respect to x is ∂f/∂x.

Answer 31

True

Question 32

Fill in the blank: To calculate the partial derivative of f(x, y) with respect to y, you differentiate ______ while treating x as a constant.

Answer 32

f with respect to y

Question 33

Which of the following is the correct algorithm step to find the partial derivative of a function f(x, y) with respect to x? A) Differentiate f with respect to y B) Differentiate f with respect to x C) Evaluate f at a point

Answer 33

B) Differentiate f with respect to x

Question 34

What is the significance of partial derivatives in multivariable calculus?

Answer 34

Partial derivatives provide insights into how a function changes as one variable changes, allowing for the analysis of functions with multiple inputs.