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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #983 : Partial Derivatives

Find $f_{yyx}$ of the following function:

f(x,y,z)=x^3z5^y

Possible Answers:

$f_{yyx}=(\ln 5)^2x^2z5^y$

$f_{yyx}=3(\ln 5)^2x^25^y$

$f_{yyx}=3(\ln 5)^2x^2z5^y$

$f_{yyx}=3(\ln 5)x^2z5^y$

Correct answer:

$f_{yyx}=3(\ln 5)^2x^2z5^y$

Explanation:

The first thing to do is to take the partial derivative with respect to one variable while everything else behaves as a constant. In this case, you can take the partial derivative with respect to which is:

$f_y=x^3z5^y(\ln5)$

Then, you can take the partial derivative with respect to again because the problem expresses it twice. This would be:

$f_{yy}=x^3z5^y(\ln5)(\ln5)$

Finally, you must take the partial derivative with respect to which is:

$f_{yyx}=3(\ln5)^2x^2z5^y$

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Example Question #984 : Partial Derivatives

Find $f_{xyz}$ of the following function:

$f(x,y,z)=(\cos x)(\sin y)(\cos z)$

Possible Answers:

$f_{xyz}=(\sin x)(\cos y)(\cos z)$

$f_{xyz}=(\sin x)(\cos y)(\sin z)$

$f_{xyz}=-(\sin x)(\cos y)(\sin z)$

$f_{xyz}=-(\cos x)(\cos y)(\sin z)$

Correct answer:

$f_{xyz}=(\sin x)(\cos y)(\sin z)$

Explanation:

The first thing that you need to do is to take the partial derivative of the function with respect to one variable while everything else remains constant. In this case, you can take the partial derivative of which is:

$f_x=(-\sin x)(\sin y)(\cos z)$

Then, you can take the partial derivative with respect to which is:

$f_{xy}=(-\sin x)(\cos y)(\cos z)$

Finally, you must take the partial derivative with respect to :

$f_{xy}=(\sin x)(\cos y)(\sin z)$

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Example Question #984 : Partial Derivatives

Find $f_{xyz}$ of the following function:

$f(x,y,z)=5e^{2x}y^3z$

Possible Answers:

$f_{xyz}=30e^{2x}y^3z$

$f_{xyz}=30e^{2x}y^2z$

$f_{xyz}=10e^{2x}y^2$

$f_{xyz}=30e^{2x}y^2$

Correct answer:

$f_{xyz}=30e^{2x}y^2$

Explanation:

$f_x=10e^{2x}y^3z$

Then, you can take the partial derivative with respect to which is:

$f_{xy}=30e^{2x}y^2z$

Finally, you must take the partial derivative with respect to which is:

$f_{xyz}=30e^{2x}y^2$

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Example Question #986 : Partial Derivatives

Find $f_{xxy}$ of the following function:

$f(x,y,z)=16x^2e^{y^2}z$

Possible Answers:

$f_{xxy}=64ye^{y^2}z$

$f_{xxy}=64xye^{y^2}z$

$f_{xxy}=32ye^{y^2}z$

$f_{xxy}=64e^{y^2}z$

Correct answer:

$f_{xxy}=64ye^{y^2}z$

Explanation:

$f_x=32xe^{y^2}z$

Then, you can take the partial derivative of again because the problem expresses twice. This would be:

$f_{xx}=32e^{y^2}z$

Finally, you must take the partial derivative of which is:

$f_{xxy}=64ye^{y^2}z$

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Example Question #985 : Partial Derivatives

Find $f_{xxy}$ of the function x^3+3x^2yz

Possible Answers:

6x+6yz

6xz

Correct answer:

Explanation:

To find $f_{xxy}$ of the function, we take three consecutive partial derivatives: $\frac{\partial }{\partial x},\frac{\partial }{\partial x},\frac{\partial }{\partial y}$ .

$\frac{\partial }{\partial x}(x^3+3x^2yz)=3x^2+6xyz$

$\frac{\partial }{\partial x}(3x^2+6xyz)=6x+6yz$

$\frac{\partial }{\partial y}(6x+6yz)=6z$

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Example Question #986 : Partial Derivatives

Find $f_{xyz}$ of the function xy^3+3xy^2z^3+xyz

Possible Answers:

xyz+9xy^2

9xy^2z

18yz^2+1

Correct answer:

18yz^2+1

Explanation:

To find $f_{xyz}$ of the function, we take three consecutive partial derivatives: $\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}$ .

$\frac{\partial }{\partial x}(xy^3+3xy^2z^3+xyz)=y^3+3y^2z^3+yz$

$\frac{\partial }{\partial y}(y^3+3y^2z^3+yz)=3y^2+6yz^3+z$

$\frac{\partial }{\partial z}(3y^2+6yz^3+z)=18yz^2+1$

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Example Question #987 : Partial Derivatives

Find $f_{xyz}$ of the function 2x^2y^5z+x^5yz^2

Possible Answers:

2x^3y+z^3

10x^4yz+1

xyz+2xz^2

20xy^4+10x^4z

Correct answer:

20xy^4+10x^4z

Explanation:

To find $f_{xyz}$ of the function, we take three consecutive partial derivatives: $\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}$ .

$\frac{\partial }{\partial x}(2x^2y^5z+x^5yz^2)=4xy^5z+5x^4yz^2$

$\frac{\partial }{\partial y}(4xy^5z+5x^4yz^2)=20xy^4z+5x^4z^2$

$\frac{\partial }{\partial z}(20xy^4z+5x^4z^2)=20xy^4+10x^4z$

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Example Question #988 : Partial Derivatives

Find $f_{xxy}$ of the function x^2y+x^3y^4z^3

Possible Answers:

2+24x^2y^2z^3

4+16xyz

1+24xy^3z^2

2+24xy^3z^3

Correct answer:

2+24xy^3z^3

Explanation:

To find $f_{xxy}$ of the function, we take three consecutive partial derivatives: $\frac{\partial }{\partial x},\frac{\partial }{\partial x},\frac{\partial }{\partial y}$ .

$\frac{\partial }{\partial x}(x^2y+x^3y^4z^3)=2xy+3x^2y^4z^3$

$\frac{\partial }{\partial x}(2xy+3x^2y^4z^3)=2y+6xy^4z^3$

$\frac{\partial }{\partial y}(2y+6xy^4z^3)=2+24xy^3z^3$

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Example Question #991 : Partial Derivatives

If $f(x,y) = \frac{\ln(xy)}{z^2}$ , evaluate f_x .

Possible Answers:

$\frac{1}{x^2y}$

$\frac{\ln(xy)}{xyz^2}$

$\frac{1}{xy}$

$\frac{1}{yz}$

$\frac{1}{xz^2}$

Correct answer:

$\frac{1}{xz^2}$

Explanation:

To evaluate the partial derivative with respect to , we take the ordinary derivative of $\frac{\ln(xy)}{z^2}$ while treating y,z as constants

$f_x = \frac{\partial}{\partial x}(\frac{\ln(xy)}{z^2}).$ Start.

$= \frac{1}{z^2}\frac{\partial}{\partial x}(\ln(xy))$ . Factor out $\frac{1}{z^2}$ .

$= \frac{1}{z^2}\frac{y}{xy}$ . Use the Chain Rule.

$=\frac{1}{xz^2}$ . Simplify

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Example Question #992 : Partial Derivatives

Find $f_{xxy}$ of the function $\sin(x)+x^2yz^5$

Possible Answers:

2z^5

$-\sin(x)+2yz^4$

$\cos(x)+2z^5$

$-\cos(x)+z^5$

Correct answer:

2z^5

Explanation:

To find $f_{xxy}$ of the function, we take three consecutive partial derivatives: $\frac{\partial }{\partial x},\frac{\partial }{\partial x},\frac{\partial }{\partial y}$ .

$\frac{\partial }{\partial x}(\sin(x)+x^2yz^5)=\cos(x)+2xyz^5$

$\frac{\partial }{\partial x}(\cos(x)+2xyz^5)=-\sin(x)+2yz^5$

$\frac{\partial }{\partial y}(-\sin(x)+2yz^5)=2z^5$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #983 : Partial Derivatives

Example Question #984 : Partial Derivatives

Example Question #984 : Partial Derivatives

Example Question #986 : Partial Derivatives

Example Question #985 : Partial Derivatives

Example Question #986 : Partial Derivatives

Example Question #987 : Partial Derivatives

Example Question #988 : Partial Derivatives

Example Question #991 : Partial Derivatives

Example Question #992 : Partial Derivatives

All Calculus 3 Resources

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