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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 #3391 : Calculus 3

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

f(x, y, z)=x^2+2xy^2+3yz^

Possible Answers:

2+4y

2x+4xy

Correct answer:

2+4y

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must find the partial derivative of the function with respect to x:

f_x=2x+4xy

Finally, we take the partial derivative of the above function with respect to x:

$f_{xx}=2+4y$

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Example Question #3392 : Calculus 3

Find $f_{xy}$ for the following function:

$f(x,y,z)=z^3\sin(xyz)+x^2y^2z$

Possible Answers:

$z^4\cos(xyz)+xyz^5\sin(xyz)+4xyz$

$z^4\cos(xyz)-xyz^5\sin(xyz)+4xyz$

$z^4\cos(xyz)-yz^4\sin(xyz)+4xyz$

Correct answer:

$z^4\cos(xyz)-xyz^5\sin(xyz)+4xyz$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

We must find the partial derivative of the function with respect to x:

$f_x=yz^4\cos(xyz)+2xy^2z$

Finally, we find the partial derivative of the function above with respect to y:

$f_{xy}=z^4\cos(xyz)-xyz^5\sin(xyz)+4xyz$

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Example Question #3393 : Calculus 3

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

Possible Answers:

2y+3z

y^2+3z

y+3

2y^2+3y

Correct answer:

2y+3z

Explanation:

To find $f_{xy}$ of the function, you must take two consecutive partial derivatives:

$\frac{\partial }{\partial x},\frac{\partial }{\partial y}$

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

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

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Example Question #3394 : Calculus 3

Find $f_{xyy}$ of the following function: $x\cos(y)+4x^2y^3$

Possible Answers:

$-\cos(y)+48xy$

$-\cos(y)+16xyz$

$-\sin(y)+48xy$

$-\cos(y)+4xy$

Correct answer:

$-\cos(y)+48xy$

Explanation:

To find $f_{xyy}$ of the function, we take three consecutive partial derivatives:

$\frac{\partial }{\partial x}(x\cos(y)+4x^2y^3)=\cos(y)+8xy^3$

$\frac{\partial }{\partial y}(\cos(y)+8xy^3)=-\sin(y)+24xy^2$

$\frac{\partial }{\partial y}(-\sin(y)+24xy^2)=-\cos(y)+48xy$

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Example Question #3395 : Calculus 3

Find $f_{zyz}$ for the following function:

f(x, y, z)=8z^2e^zy

Possible Answers:

e^zy(8z^2+32z+16)

e^z(z+8z^2)

e^z(8z^2+32z+16)

8z^2+32z+16

Correct answer:

e^z(8z^2+32z+16)

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

First, we must find the partial derivative of the function with respect to z:

f_z=y(16ze^z+8z^2e^z)

Next, we find the partial derivative of that function with respect to y:

$f_{zy}=16ze^z+8z^2e^z$

Finally, we find the derivative of the function above with respect to z:

$f_{zyz}=16e^z+16ze^z+16ze^z+8z^2e^z=e^z(16+32z+8z^2)$

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Example Question #3396 : Calculus 3

Find $f_{yy}$ for the following function:

f(x, y)=5x+4xy^2

Possible Answers:

5+4y^2

8xy

Correct answer:

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must find the partial derivative of the function with respect to y:

f_y=8xy

Then, we find the partial derivative of the function above with respect to y:

$f_{yy}=8x$

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Example Question #3397 : Calculus 3

Find $f_{xy}$ for the following function:

f(x, y)=x^2+5xy^2+20xy+4y

Possible Answers:

10y+24

10y+20

2x+10y+20

Correct answer:

10y+20

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

First, we find the partial derivative of the function with respect to x:

f_x=2x+5y^2+20y

Next, we find the partial derivative of this function with respect to y:

$f_{xy}=10y+20$

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Example Question #1023 : Partial Derivatives

Find f_x for the following function:

$f(x, y, z)=x^2y+\sqrt{xz}$

Possible Answers:

$2xy+\frac{z}{2}(xz)^{\frac{1}{2}}$

$2xy+\frac{z}{2}(xz)^{-\frac{1}{2}}$

$2xy+\sqrt{z}$

$2xy+\frac{(xz)^{-\frac{1}{2}}}{2}$

Correct answer:

$2xy+\frac{z}{2}(xz)^{-\frac{1}{2}}$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivative of the function with respect to x is

$f_x=2xy+\frac{z}{2}(xz)^{-\frac{1}{2}}$

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Example Question #3399 : Calculus 3

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

$f(x, y, z)=x+x^2y^2+xy^3+x\cos(z)$

Possible Answers:

2x^2+3x^2y

4x+6y

2x^2+6xy

Correct answer:

4x+6y

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

First, we find the partial derivative of the function with respect to y:

f_y=2x^2+3xy^2

Next, we find the partial derivative of this function with respect to y:

$f_{yy}=2x^2+6xy$

Finally, we find the partial derivative of the function above with respect to x:

$f_{yyx}=4x+6y$

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Example Question #3400 : Calculus 3

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

Possible Answers:

6y^2+3z

6y^2+6z^2

6y^2+4z

y^2+4z

Correct answer:

6y^2+4z

Explanation:

To find $f_{xxy}$ we take three consecutive partial derivatives

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

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

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

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #3391 : Calculus 3

Example Question #3392 : Calculus 3

Example Question #3393 : Calculus 3

Example Question #3394 : Calculus 3

Example Question #3395 : Calculus 3

Example Question #3396 : Calculus 3

Example Question #3397 : Calculus 3

Example Question #1023 : Partial Derivatives

Example Question #3399 : Calculus 3

Example Question #3400 : Calculus 3

All Calculus 3 Resources

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