# Derivatives Of Multivariable Functions

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Are you familiar with the mathematical rule, which wants that when all other variables are held constant, then the partial derivative rules for dealing with coefficients, simple powers of variables, constants, and sums/difference of function remain the same, and are used to determine the function of the slope for each independent variable? This more or less, sums up derivatives and multivariable functions. Take our quiz to see how much you really known about them.

• 1.

### What's the particularity of the function f(x,y) = x^2(y)/ ((x^4) +( y^2))?

• A.

It is that it approaches "100" along any line through the origin.

• B.

It is that it approaches "1" along any line through the origin.

• C.

It is that it approaches "0" along any line through the origin.

• D.

It is that it approaches "-1" along any line through the origin.

C. It is that it approaches "0" along any line through the origin.
Explanation
The particularity of the function f(x,y) = x^2(y)/ ((x^4) +( y^2)) is that it approaches "0" along any line through the origin. This means that as the values of x and y approach infinity or negative infinity, the value of the function approaches zero. Additionally, as x and y approach zero, the function also approaches zero. This behavior is consistent for any line passing through the origin, making "0" the correct answer.

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• 2.

### What happens when the origin is approached along a parabola y=x^2?

• A.

It has a limit of 0.5

• B.

It has a limit of 1

• C.

It is negative

• D.

It is positive

A. It has a limit of 0.5
Explanation
As the parabola y=x^2 approaches the origin (0,0), the y-values get closer and closer to 0. However, they never actually reach 0. The limit of the parabola as it approaches the origin is 0.5. This means that as x gets closer and closer to 0, y gets closer and closer to 0.5. Therefore, the correct answer is that the parabola has a limit of 0.5 as it approaches the origin.

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• 3.

### If function "f" is symmetric with regard to x and y, what happens to f(x)?

• A.

F(x) is constant.

• B.

F(x) is continuous.

• C.

F(x) is curved.

• D.

F(x) discontinous

B. F(x) is continuous.
Explanation
If a function is symmetric with regard to x and y, it means that if you swap the x and y values, the function remains the same. This implies that the graph of the function is symmetric about the line y = x. Since a continuous function does not have any breaks or jumps in its graph, it is the most likely choice for a function that is symmetric. Therefore, the correct answer is that f(x) is continuous.

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• 4.

### What is a partial derivative of a multivariable function?

• A.

It is a derivative with respect to 2 variables, with all other variable held constant.

• B.

It is a derivative with respect to 1 variable, with all other variable equal to 0.

• C.

It is a derivative with respect to 1 variable, with all other variable held constant.

• D.

It is a derivative with respect to 1 variable, with 1 variable held constant.

C. It is a derivative with respect to 1 variable, with all other variable held constant.
Explanation
A partial derivative of a multivariable function is a derivative with respect to 1 variable, while holding all other variables constant. This means that we are only considering the rate of change of the function with respect to one variable, while treating all other variables as fixed or constant.

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• 5.

### What does the multiple integral expand to?

• A.

To the concept of the integral to function of a distinctive number of variables.

• B.

To the concept of the integral to function of certain variables.

• C.

To the concept of the integral to function of any number of variables.

• D.

To the concept of the integral to function of any number equal to 0.

C. To the concept of the integral to function of any number of variables.
Explanation
The correct answer is "To the concept of the integral to function of any number of variables." This answer suggests that the multiple integral expands to the concept of integrating a function that depends on any number of variables. This means that the integral can be taken over a region in multi-dimensional space, integrating with respect to each variable.

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• 6.

### What is a gradient theorem?

• A.

It says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

• B.

It says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the function.

• C.

It says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the equation.

• D.

It says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the shape.

A. It says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
Explanation
The gradient theorem states that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. This means that instead of integrating over the entire curve, we can simply evaluate the scalar field at the starting and ending points of the curve to find the line integral.

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• 7.

### What definition better fit the Stokes theorem?

• A.

The surface integration of the curl of a vector field over a surface Σ in Euclidena three-space to the line integral of the vector field over its curve.

• B.

The surface integration of the curl of a vector field over a surface Σ in Euclidena three-space to the line integral of the vector field over its origin.

• C.

The surface integration of the curl of a vector field over a surface Σ in Euclidena three-space to the line integral of the vector field over its axis.

• D.

The surface integration of the curl of a vector field over a surface Σ in Euclidena three-space to the line integral of the vector field over its boundary.

D. The surface integration of the curl of a vector field over a surface Σ in Euclidena three-space to the line integral of the vector field over its boundary.
Explanation
The Stokes theorem states that the surface integration of the curl of a vector field over a surface Σ in Euclidean three-space is equal to the line integral of the vector field over its boundary. This means that the circulation of the vector field around the boundary of the surface is related to the curl of the vector field over the entire surface.

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• 8.

### What is the best definition of the Green's theorem?

• A.

It is known as the 3rd integrand of both sides in the integral in terms of P and Q.

• B.

It is known as the 3rd formula of both sides in the integral in terms of P, Q, and R.

• C.

It is known as the 3rd integral of both sides in the integral in terms of P, Q, and R.

• D.

It is known as the 3rd integrand of both sides in the integral in terms of P, Q, and R.

D. It is known as the 3rd integrand of both sides in the integral in terms of P, Q, and R.
Explanation
The Green's theorem is a fundamental result in vector calculus that relates a line integral around a simple closed curve to a double integral over the plane region bounded by the curve. It states that the line integral of a vector field F around a closed curve C is equal to the double integral of the curl of F over the region D bounded by C. The given answer correctly states that the Green's theorem is known as the 3rd integrand of both sides in the integral in terms of P, Q, and R.

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• 9.

### What's the other name for the divergence theorem?

• A.

Gauss Theorem

• B.

• C.

Gaul theorem

• D.

Ostro-Gauss theorem

Explanation
The other name for the divergence theorem is the Ostrogradsky-Gauss theorem.

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• 10.

### How would you define Green's theorem?

• A.

The one that gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C.

• B.

The one that gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by B.

• C.

The one that gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by X.

• D.

The one that gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by A.

A. The one that gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C.
Explanation
Green's theorem is a fundamental result in vector calculus that relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It states that the line integral of a vector field around a closed curve is equal to the double integral of the curl of the vector field over the region enclosed by the curve. This theorem is used to evaluate various physical quantities such as circulation and flux in fluid dynamics and electromagnetism.

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• Current Version
• Mar 18, 2023
Quiz Edited by
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• Jan 14, 2018
Quiz Created by
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