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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 #3401 : Calculus 3

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

Possible Answers:

11x+4y

20x+4y^2

18x+4y

18x+5y

Correct answer:

18x+4y

Explanation:

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

$\frac{\partial }{\partial x}(3x^3y+x^2y^2)=9x^2y+2xy^2$

$\frac{\partial }{\partial x}(9x^2y+2xy^2)=18xy+2y^2$

$\frac{\partial }{\partial y}(18xy+2y^2)=18x+4y$

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Example Question #3402 : Calculus 3

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

$f(x,y,z)=10z\sec(xy)$

Possible Answers:

$10y\sec(xy)\tan(xy)$

$10\sec(xy)$

$-10x\sec(xy)\tan(xy)$

$10x\sec(xy)\tan(xy)$

Correct answer:

$10x\sec(xy)\tan(xy)$

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 find the partial derivative of the function with respect to z:

$f_z=10\sec(xy)$

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

$f_{zy}=10x\sec(xy)\tan(xy)$

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Example Question #3403 : Calculus 3

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

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

Possible Answers:

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

$-\frac{2}{x^2}$

$\frac{2}{x}+yz$

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

$f_x=\frac{2}{x}+yz$

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

$f_{xx}=-\frac{2}{x^2}$

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

$f_{xxy}=0$

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Example Question #3404 : Calculus 3

Find $(f_{xy})^2$ of the following function:

Possible Answers:

$4x^2z^2(\cos^2(y)-y^2(\sin^2(y)))$

$4x^2z^2(\cos^2(y)+y^2(\sin^2(y)))$

$2xz(\cos(y)-y\sin(y))$

$4x^2z^2(\cos(y)-y\sin(y))^2$

Correct answer:

$4x^2z^2(\cos(y)-y\sin(y))^2$

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:

$f_x=2xy\cos(y)z$

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

$f_{xy}=2xz(\cos(y)-y\sin(y))$

Finally, we square this function:

$(f_{xy})^2=4x^2z^2(\cos(y)-y\sin(y))^2$

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Example Question #3405 : Calculus 3

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

f(x,y)=x^5ye^y

Possible Answers:

x^5e^y(1+y)

x^5e^y(2+y)

x^5(2e^y+y)

2x^5e^y

Correct answer:

x^5e^y(2+y)

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=x^5(e^y+ye^y)

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

$f_{yy}=x^5(e^y+e^y+ye^y)=x^5e^y(2+y)$

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Example Question #3406 : Calculus 3

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

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

Possible Answers:

$\cos(x^2)-2x\sin(x^2)$

$y\cos(x^2)-2x^2y\sin(x^2)$

$\cos(x^2)+2x^2\sin(x^2)$

$\cos(x^2)-2x^2\sin(x^2)$

Correct answer:

$\cos(x^2)-2x^2\sin(x^2)$

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:

$f_x=y\cos(x^2)-2x^2y\sin(x^2)$

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

$f_{xy}=\cos(x^2)-2x^2\sin(x^2)$

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Example Question #3407 : Calculus 3

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

$f(x,y,z)=xz\cos(y^2)$

Possible Answers:

$-2y\sin(y^2)$

$z\cos(y^2)$

$\cos(y^2)$

Correct answer:

$-2y\sin(y^2)$

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:

$f_x=z\cos(y^2)$

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

$f_{xz}=\cos(y^2)$

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

$f_{xzy}=-2y\sin(y^2)$

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Example Question #3408 : Calculus 3

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

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

Possible Answers:

$x^2z^2\cos(xyz)$

$-x^2z^2\cos(xyz)$

$-xz\sin(xyz)$

$-x^2z^2\sin(xyz)$

Correct answer:

$-x^2z^2\cos(xyz)$

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:

$f_y=-xz\sin(xyz)$

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

$f_{yy}=-x^2z^2\cos(xyz)$

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Example Question #3409 : Calculus 3

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

f(x,y,z)=6y^2e^z+xz^2

Possible Answers:

6y^2e^z+2x

e^z+2x

6y^2e^z+2xz

Correct answer:

6y^2e^z+2x

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 z:

f_z=6y^2e^z+2xz

Then, we find the partial derivative of this function with respect to z:

$f_{zz}=6y^2e^z+2x$

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Example Question #3410 : Calculus 3

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

$f(x,y,z)=\cos(xz)+x^2y^2e^z$

Possible Answers:

$-z^2\cos(xz)+2y^2e^z$

4ye^z

4e^z

$-z\sin(xz)+2xye^z$

Correct answer:

4ye^z

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:

$f_x=-z\sin(xz)+2xy^2e^z$

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

$f_{xx}=-z^2\cos(xz)+2y^2e^z$

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

$f_{xxy}=4ye^z$

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

$f_{xxyz}=4ye^z$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #3401 : Calculus 3

Example Question #3402 : Calculus 3

Example Question #3403 : Calculus 3

Example Question #3404 : Calculus 3

Example Question #3405 : Calculus 3

Example Question #3406 : Calculus 3

Example Question #3407 : Calculus 3

Example Question #3408 : Calculus 3

Example Question #3409 : Calculus 3

Example Question #3410 : Calculus 3

All Calculus 3 Resources

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