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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 #105 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (6,1,2) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(6.08,9.46^{\circ},2)$

$(6.08,170.54^{\circ},2)$

$(7,170.54^{\circ},2)$

$(7,-170.54^{\circ},2)$

$(7,9.46^{\circ},2)$

Correct answer:

$(6.08,9.46^{\circ},2)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative values by convention create a negative $\theta$ , while negative values lead to $|\theta|>90^{\circ};\frac{\pi}{2}$

For our coordinates (6,1,2)

$r=\sqrt{x^2+y^2}\rightarrow6.08$

$\theta=arctan(\frac{y}{x})\rightarrow 9.46^{\circ}$ (Bearing in mind sign convention)

z=2

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Example Question #106 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-8,5,6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(-3,-32.01^{\circ},6)$

$(-3,32.01^{\circ},6)$

$(9.43,-32.01^{\circ},6)$

$(9.43,147.99^{\circ},6)$

$(9.43,-147.99^{\circ},6)$

Correct answer:

$(9.43,147.99^{\circ},6)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-8,5,6)

$r=\sqrt{x^2+y^2}\rightarrow9.43$

$\theta=arctan(\frac{y}{x})\rightarrow 147.99^{\circ}$ (Bearing in mind sign convention)

z=6

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Example Question #21 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-3,-9,6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(9.49,71.57^{\circ},6)$

$(9.49,-108.43^{\circ},6)$

$(9.49,108.43^{\circ},6)$

$(9.49,-71.57^{\circ},6)$

Correct answer:

$(9.49,-108.43^{\circ},6)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-3,-9,6)

$r=\sqrt{x^2+y^2}\rightarrow9.49$

$\theta=arctan(\frac{y}{x})\rightarrow -108.43^{\circ}$ (Bearing in mind sign convention)

z=6

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Example Question #107 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-7,7,-2) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(7.7,135^{\circ},-2)$

$(9.9,-45^{\circ},-2)$

$(7.7,-45^{\circ},-2)$

$(9.9,135^{\circ},-2)$

$(7.7,45^{\circ},-2)$

Correct answer:

$(9.9,135^{\circ},-2)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-7,7,-2)

$r=\sqrt{x^2+y^2}\rightarrow9.9$

$\theta=arctan(\frac{y}{x})\rightarrow 135^{\circ}$ (Bearing in mind sign convention)

z=-2

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Example Question #1771 : Calculus 3

A point in space is located, in Cartesian coordinates, at (-4,5,-3) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(5,-51.34^{\circ},-5)$

$(6.4,-51.34^{\circ},-3)$

$(5,128.66^{\circ},-5)$

$(5,51.34^{\circ},-5)$

$(6.4,128.66^{\circ},-3)$

Correct answer:

$(6.4,128.66^{\circ},-3)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-4,5,-3)

$r=\sqrt{x^2+y^2}\rightarrow6.4$

$\theta=arctan(\frac{y}{x})\rightarrow 128.66^{\circ}$ (Bearing in mind sign convention)

z=-3

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Example Question #22 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at (-2,-9,3) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(9.22,77.47^{\circ},3)$

$(9.22,-102.53^{\circ},3)$

$(3,-77.47^{\circ},9.22)$

$(9.22,102.53^{\circ},3)$

$(3,77.47^{\circ},9.22)$

Correct answer:

$(9.22,-102.53^{\circ},3)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-2,-9,3)

$r=\sqrt{x^2+y^2}\rightarrow9.22$

$\theta=arctan(\frac{y}{x})\rightarrow -102.53^{\circ}$ (Bearing in mind sign convention)

z=3

Report an Error

Example Question #111 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (-9,3,9) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(90,-18.44^{\circ},9)$

$(9.49,161.57^{\circ},9)$

$(9.49,-18.44^{\circ},9)$

$(90,18.44^{\circ},9)$

$(90,161.57^{\circ},9)$

Correct answer:

$(9.49,161.57^{\circ},9)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (-9,3,9)

$r=\sqrt{x^2+y^2}\rightarrow9.49$

$\theta=arctan(\frac{y}{x})\rightarrow 161.57^{\circ}$ (Bearing in mind sign convention)

z=9

Report an Error

Example Question #112 : 3 Dimensional Space

A point in space is located, in Cartesian coordinates, at (1,-5,-6) . What is the position of this point in cylindrical coordinates?

Possible Answers:

$(-4,101.31^{\circ},-6)$

$(5.1,101.31^{\circ},-6)$

$(5.1,-78.69^{\circ},-6)$

$(-4,-101.31^{\circ},-6)$

$(5.1,78.69^{\circ},-6)$

Correct answer:

$(5.1,-78.69^{\circ},-6)$

Explanation:

When given Cartesian coordinates of the form (x,y,z) to cylindrical coordinates of the form $(r,\theta,z)$ , the first and third terms are the most straightforward.

$r=\sqrt{x^2+y^2}$

z=z

Care should be taken, however, when calculating $\theta$ . The formula for it is as follows:

$\theta=arctan(\frac{y}{x})$

However, it is important to be mindful of the signs of both and , bearing in mind which quadrant the point lies; this will determine the value of $\theta$ :

Quadrants

For our coordinates (1,-5,-6)

$r=\sqrt{x^2+y^2}\rightarrow5.1$

$\theta=arctan(\frac{y}{x})\rightarrow -78.69^{\circ}$ (Bearing in mind sign convention)

z=-6

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Example Question #113 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(5,90^{\circ},7)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(5,5,7)

(5,0,7)

(0,5,7)

(0,-5,7)

(-5,5,7)

Correct answer:

(0,5,7)

Explanation:

When given cylindrical coordinates of the form $(r,\theta,z)$ and converting to Cartesian coordinates of the form (x,y,z) , the relationship between , , , and $\theta$ will be of use:

$x=rcos(\theta)$

$y=rsin(\theta)$

z=z (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates

The Cartesian coordinates are $(5,90^{\circ},7)$

$x=rcos(\theta)\rightarrow0$

$y=rsin(\theta)\rightarrow5$

$z=z\rightarrow7$

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Example Question #114 : 3 Dimensional Space

A point in space is located, in cylindrical coordinates, at $(6,45^{\circ},2)$ . What is the position of this point in Cartesian coordinates?

Possible Answers:

(-4.24,4.24,2)

(4.24,-4.24,2)

(-2,-2,2)

(4.24,4.24,2)

(2,2,2)

Correct answer:

(4.24,4.24,2)

Explanation:

When given cylindrical coordinates of the form $(r,\theta,z)$ and converting to Cartesian coordinates of the form (x,y,z) , the relationship between , , , and $\theta$ will be of use:

$x=rcos(\theta)$

$y=rsin(\theta)$

z=z (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates $(6,45^{\circ},2)$

The Cartesian coordinates are

$x=rcos(\theta)\rightarrow4.24$

$y=rsin(\theta)\rightarrow4.24$

$z=z\rightarrow2$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #105 : 3 Dimensional Space

Example Question #106 : 3 Dimensional Space

Example Question #21 : Cylindrical Coordinates

Example Question #107 : 3 Dimensional Space

Example Question #1771 : Calculus 3

Example Question #22 : Cylindrical Coordinates

Example Question #111 : 3 Dimensional Space

Example Question #112 : 3 Dimensional Space

Example Question #113 : 3 Dimensional Space

Example Question #114 : 3 Dimensional Space

All Calculus 3 Resources

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