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Calculus 3 : Calculus 3

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

Example Questions

Example Question #331 : 3 Dimensional Space

Find a parametric representation of the circle 3x^2 + 9 y ^2 = 27 .

Possible Answers:

$\vec{r}(t)= \left< \cos 3t, \sqrt{3} \sin 3t \right>$

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \tan t \right>$

$\vec{r}(t)= \left< \cos t, \sin t \right>$

Correct answer:

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

Explanation:

We can begin by rewriting the equation for a circle as

$3x^2 + 9y^2 = 27 \Rightarrow \left(\frac{x}{3} \right )^2 + \left(\frac{y}{\sqrt{3}} \right )^2=1$ .

This directly tells us that $x = 3 \cos t, y = \sqrt{3} \sin t$ . This allows us to write our final expression for the parametric representation as

$\vec{r}(t)= \left< 3 \cos t, \sqrt{3} \sin t \right>$

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Example Question #242 : Cylindrical Coordinates

Convert the following vector in Cartesian coordinates into cylindrical coordinates.

(3xy, x^2z, y)

Possible Answers:

$(3r\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

$(3r^2\sin\theta \cos\theta, rz\sin^2 \theta, r\cos \theta )$

$(3r^2\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

$(3r^2\sin\theta \cos\theta, r\sin^2 \theta, \cos \theta )$

$(3r^2\sin\theta \cos\theta, r\sin^2 \theta, r\cos \theta )$

Correct answer:

$(3r^2\cos\theta \sin \theta, rz\cos^2 \theta, r\sin \theta )$

Explanation:

The conversion from Cartesian to cylindrical coordinates is as follows:

$x=r\cos \theta$

$y=r \sin \theta$

z=z

The three components of the vector then become:

$3xy\rightarrow3r\cos \theta \:r\sin \theta=3r^2\cos\theta \sin\theta$

$x^2z\rightarrow r^2 \cos^2\theta z$

$y\rightarrow r\sin \theta$

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Example Question #251 : Cylindrical Coordinates

Express the three-dimensional (x,y,z) Cartesian coordinates as cylindrical coordinates (r, θ, z):

$\left ( 2,1, -2\right )$

Possible Answers:

$\left ( \sqrt{5}, \tan^{-1}2, -2 \right )$

$\left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, -2 \right )$

$\left ( 5, \tan^{-1}\frac{1}{2}, -2 \right )$

$\left ( 5, \tan^{-1}2, -2 \right )$

$\left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, 1 \right )$

Correct answer:

$\left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, -2 \right )$

Explanation:

The coordinates (2, 1, -2) corresponds to: x = 2, y = 1, z = -2, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

$\\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z$

So, filling in for x, y, z:

$\\ r = {\sqrt{2^{2}+1^{2}}} = \sqrt{5} \\ \tan(\theta) = \frac{1}{2} \Rightarrow \theta = \tan^{-1}\frac{1}{2} \\ z = -2$

Then the cylindrical coordinates are represented as:

$\left ( \sqrt{5}, \tan^{-1}\frac{1}{2}, -2 \right )$

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Example Question #252 : Cylindrical Coordinates

Express the three-dimensional (x,y,z) Cartesian coordinates as cylindrical coordinates (r, θ, z):

$\left (0, 3, 4\right )$

Possible Answers:

$\left (9, \frac{3}{0}, 4\right )$

$\left (3, \frac{3}{0}, 4\right )$

$\left (3, \frac{1}{2}\pi, 4\right )$

$\left (3, \pi, 4\right )$

$\left (9, \frac{1}{2}\pi, 4\right )$

Correct answer:

$\left (3, \frac{1}{2}\pi, 4\right )$

Explanation:

The coordinates (0, 3, 4) corresponds to: x = 0, y = 3, z = 4, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

$\\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z$

So, filling in for x, y, z:

$\\ r = {\sqrt{0^{2}+3^{2}}} = \sqrt{9} = 3 \\ \tan(\theta) = \frac{3}{0} \Rightarrow \theta = \frac{1}{2}\pi \\ z = 4$

Then the cylindrical coordinates are represented as:

$\left (3, \frac{1}{2}\pi, 4 \right )$

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Example Question #253 : Cylindrical Coordinates

Express the three-dimensional (x,y,z) Cartesian coordinates as cylindrical coordinates (r, θ, z):

$\left (\sqrt{2}, 1, 1\right )$

Possible Answers:

$\left (9, \tan^{-1}\frac{\sqrt{2}}{2}, 1\right )$

$\left (3, \frac{\sqrt{2}}{2}, 1\right )$

$\left ( \sqrt{3}, \tan^{-1}\frac{\sqrt{2}}{2}, 1\right )$

$\left ( \sqrt{3}, \frac{\sqrt{2}}{2}, 1\right )$

$\left (3, \tan^{-1}\frac{\sqrt{2}}{2}, \sqrt{2}\right )$

Correct answer:

$\left ( \sqrt{3}, \tan^{-1}\frac{\sqrt{2}}{2}, 1\right )$

Explanation:

The coordinates (√2, 1, 1) corresponds to: x = √2, y = 1, z = 1, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

$\\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z$

So, filling in for x, y, z:

$\\ r = {\sqrt{\sqrt{2}^{2}+1^{2}}} = \sqrt{2+1} = \sqrt{3} \\ \tan(\theta) = \frac{1}{\sqrt{2}} \Rightarrow \theta = \tan^{-1}\frac{\sqrt{2}}{2} \\ z = 1$

Then the cylindrical coordinates are represented as:

$\left (\sqrt{3}, \tan^{-1}\frac{\sqrt{2}}{2}, 1 \right )$

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Example Question #254 : Cylindrical Coordinates

Express the three-dimensional cylindrical coordinates (r, θ, z) as three-dimensional (x,y,z) Cartesian coordinates:

$\left (3, \frac{1}{3}\pi, -4 \right)$

Possible Answers:

$\left (\frac{3}{2}, \frac{3\sqrt{3}}{2}, -4 \right)$

$\left (3\cos^{-1}\frac{1}{2}, 3\sin^{-1}\frac{\sqrt{3}}{2}, -4 \right)$

$\left (\frac{1}{2}, \frac{\sqrt{3}}{2}, -4 \right)$

$\left (3\pi, 3\pi, -4 \right)$

$\left (\frac{3}{2}, \frac{\sqrt{3}}{2}, -4 \right)$

Correct answer:

$\left (\frac{3}{2}, \frac{3\sqrt{3}}{2}, -4 \right)$

Explanation:

The coordinates (3, π/3, -4) corresponds to: r = 3, θ = π/3, z = -4, and are to be converted to the Cartesian coordinates in form of (x, y, z), where:

$\\ x = r\cos\theta \\ y = r\sin\theta \\ z = z$

So, filling in for r, θ, z:

$\\ x = 3\cos{\frac{1}{3}\pi} = 3\cdot \frac{1}{2} = \frac{3}{2} \\ y = 3\sin{\frac{1}{3}\pi} = 3\cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{2}}{2} \\ z = -4$

Then the Cartesian coordinates are represented as:

$\left (\frac{3}{2}, \frac{3\sqrt{3}}{2}, -4 \right)$

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Example Question #251 : Cylindrical Coordinates

Express the three-dimensional Cartesian coordinates (x,y,z) as three-dimensional cylindrical coordinates (r, θ, z):

$\left ( -2, 2, 3\right )$

Possible Answers:

$\left ( 8,-1, 3\right )$

$\left ( 2\sqrt{2}, -1, 3\right )$

$\left ( 2\sqrt{2}, -1, \sqrt{3}\right )$

$\left ( 8, \frac{3}{4}\pi, \sqrt{3}\right )$

$\left ( 2\sqrt{2}, \frac{3}{4}\pi, 3\right )$

Correct answer:

$\left ( 2\sqrt{2}, \frac{3}{4}\pi, 3\right )$

Explanation:

The coordinates (-2, 2, 3) corresponds to: x = -2, y = 2, z = 3, and are to be converted to the cylindrical coordinates in form of (r, θ, z), where:

$\\ r = \sqrt{x^{2}+y^{2}} \\ \tan(\theta) = \frac{y}{x} \\ z = z$

So, filling in for x, y, z:

$\\ r = {\sqrt{\left ( -2\right )^{2}+2^{2}}} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2} \\ \tan(\theta) = \frac{2}{-2} \Rightarrow \theta = \tan^{-1}\ \left ( -1\right ) \Rightarrow \theta = \frac{3}{4}\pi \\ z = 3$

Then the cylindrical coordinates are represented as:

$\left ( 2\sqrt{2}, \frac{3}{4}\pi, 3\right )$

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Example Question #252 : Cylindrical Coordinates

Express the three-dimensional cylindrical coordinates (r, θ, z) as three-dimensional (x,y,z) Cartesian coordinates:

$\left ( 1, 45^{\circ}, 1\right )$

Possible Answers:

$\left ( \frac{\sqrt{2}}{2}, 1,\frac{\sqrt{2}}{2}\right )$

$\left ( \sqrt{2},\sqrt{2}, 1\right )$

$\left ( \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}, 1\right )$

$\left ( \cos^{-1}\frac{\sqrt{2}}{2},\sin^{-1}\frac{\sqrt{2}}{2}, 1\right )$

$\left ( \frac{45\sqrt{2}}{2}, 1,\frac{45\sqrt{2}}{2}\right )$

Correct answer:

$\left ( \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}, 1\right )$

Explanation:

The coordinates (1, 45°, 1) corresponds to: r = 1, θ = 45°, z = 1, and are to be converted to the Cartesian coordinates in form of (x, y, z), where:

$\\ x = r\cos\theta \\ y = r\sin\theta \\ z = z$

So, filling in for r, θ, z:

$\\ x = 1\cos{45^{\circ}} = 1\cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \\ y = 1\sin{45^{\circ}} = 1\cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \\ z = 1$

Then the Cartesian coordinates are represented as:

$\left ( \frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}, 1\right )$

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Example Question #253 : Cylindrical Coordinates

Express the three-dimensional cylindrical coordinates (r, θ, z) as three-dimensional (x,y,z) Cartesian coordinates:

$\left ( 2, \frac{1}{2}\pi, -4\right )$

Possible Answers:

$\left ( 2,0,-4\right )$

$\left (\cos^{-1}0,\sin^{-1}2,-4\right )$

$\left ( 2,0,16\right )$

$\left (\cos^{-1}0,\sin^{-1}2,16\right )$

$\left ( 0,2,-4\right )$

Correct answer:

$\left ( 0,2,-4\right )$

Explanation:

The coordinates (2, π/2, -4) corresponds to: r = 2, θ = π/2, z = -4, and are to be converted to the Cartesian coordinates in form of (x, y, z), where:

$\\ x = r\cos\theta \\ y = r\sin\theta \\ z = z$

So, filling in for r, θ, z:

$\\ x = 2\cos{\frac{1}{2}\pi} = 2\cdot 0 = 0 \\ y = 2\sin{\frac{1}{2}\pi} = 2\cdot 1 = 2 \\ z = -4$

Then the Cartesian coordinates are represented as:

$\left ( 0,2,-4\right )$

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Example Question #1 : Spherical Coordinates

Convert the following into spherical coordinates.

$x=\sin(t)$

$y=\cos(t)$

z=1

Possible Answers:

$\rho=\sqrt{2}$

$\phi=0$

$\theta=\tan(\cot(t))$

$\rho=2$

$\phi=-\frac{\pi}{4}$

$\theta=\tan^{-1}(\cot(t))$

$\rho=\sqrt{2}$

$\phi=\frac{\pi}{4}$

$\theta=\tan^{-1}(\cot(t))$

$\rho=\sqrt{2}$

$\phi=\frac{\pi}{4}$

$\theta=\tan^{-1}(\cot(t))$

$\rho=1$

$\phi=\frac{\pi}{2}$

$\theta=\tan(\cot(t))$

Correct answer:

$\rho=\sqrt{2}$

$\phi=\frac{\pi}{4}$

$\theta=\tan^{-1}(\cot(t))$

Explanation:

In order to convert to spherical coordinates , we need to remember the conversion equations.

$\rho=\sqrt{x^2+y^2+z^2}$

$\theta=\tan^{-1}\Big(\frac{y}{x}\Big)$

$\phi=\tan^{-1}\Big(\frac{\sqrt{x^2+y^2}}{z}\Big)$

Now lets apply this to our problem.

$\rho=\sqrt{(\sin(t))^2+(\cos(t))^2+1^2}=\sqrt{1+1}=\sqrt{2}$

$\phi=\tan^{-1}\Big(\frac{\sqrt{(\sin{t})^2+(\cos{t})^2}}{1}\Big)=\tan^{-1}(1)=\frac{\pi}{4}$

$\theta=\tan^{-1}\Big(\frac{\cos(t)}{\sin(t)}\Big)=\tan^{-1}(\cot(t))$

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #331 : 3 Dimensional Space

Example Question #242 : Cylindrical Coordinates

Example Question #251 : Cylindrical Coordinates

Example Question #252 : Cylindrical Coordinates

Example Question #253 : Cylindrical Coordinates

Example Question #254 : Cylindrical Coordinates

Example Question #251 : Cylindrical Coordinates

Example Question #252 : Cylindrical Coordinates

Example Question #253 : Cylindrical Coordinates

Example Question #1 : Spherical Coordinates

All Calculus 3 Resources

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