Exam 2 Theorems Flashcards
What is Clairaut’s Theorem?
Clairaut’s Theorem states that if the mixed partial derivatives of a function are continuous, then the order of differentiation does not matter.
True or False: Clairaut’s Theorem applies to all functions regardless of their continuity.
False
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.
Clairaut
Which condition must a function satisfy for Clairaut’s Theorem to hold?
The mixed partial derivatives must be continuous.
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.
B) The mixed partial derivatives can be interchanged.
What is Fubini’s Theorem primarily used for in calculus?
Fubini’s Theorem is used to evaluate double integrals by allowing the integration to be performed iteratively.
True or False: Fubini’s Theorem can only be applied to continuous functions.
False: Fubini’s Theorem can be applied to integrable functions, not just continuous ones.
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 _______.
the iterated integral of f(x,y) with respect to x and then y, or vice versa.
What are the conditions under which Fubini’s Theorem holds?
Fubini’s Theorem holds when the function is integrable over the region of integration.
If f(x,y) = xy, what is the iterated integral of f over the rectangle [0,1] x [0,1] using Fubini’s Theorem?
The iterated integral is ∫ from 0 to 1 (∫ from 0 to 1 xy dy) dx = 1/4.
What does the Extreme Value Theorem state?
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.
True or False: The Extreme Value Theorem applies to functions that are not continuous.
False
Fill in the blank: The Extreme Value Theorem guarantees the existence of extreme values on a closed interval if the function is ______.
continuous
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.
B) The function must be continuous
In the context of calculus, how can the Extreme Value Theorem be practically used?
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.
What does the Extreme Value Theorem state for functions of two variables?
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.
True or False: The Extreme Value Theorem applies to functions that are continuous only on open regions.
False
Fill in the blank: For the Extreme Value Theorem, the region must be ______ and ______.
closed; bounded
What is the first step to finding extreme values of a function of two variables?
Calculate the partial derivatives and set them equal to zero to find critical points.
Multiple Choice: Which of the following methods can be used to test for extrema in a function of two variables?
The Second Derivative Test
What is a local maximum in a function of two variables?
A function has a local maximum at (a, b) if f(z, y) < f(a, b) when (z, y) is near (a, b)
What is a local minimum in a function of two variables?
A function has a local minimum at (a, b) if f(z, y) > f(a, b) when (z, y) is near (a, b)
What is the local maximum value at (a, b)?
The number f(a, b) is called a local maximum value
What is the local minimum value at (a, b)?
The number f(a, b) is called a local minimum value
What indicates an absolute maximum or minimum in a function?
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)
What does Theorem 2 state about local maxima or minima?
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
What is the gradient vector notation related to Theorem 2?
The conclusion of Theorem 2 can be stated as Vf(a, b) = 0
Fill in the blank: A function has a _______ at (a, b) if f(z, y) < f(a, b) for points near (a, b).
local maximum
Fill in the blank: A function has a _______ at (a, b) if f(z, y) > f(a, b) for points near (a, b).
local minimum
What is a partial derivative?
A partial derivative is the derivative of a function with respect to one variable while keeping other variables constant.
True or False: The notation for the partial derivative of a function f(x, y) with respect to x is ∂f/∂x.
True
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.
f with respect to y
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
B) Differentiate f with respect to x
What is the significance of partial derivatives in multivariable calculus?
Partial derivatives provide insights into how a function changes as one variable changes, allowing for the analysis of functions with multiple inputs.
