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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Polar Form

Convert from rectangular to polar form:

(2, 2)

Possible Answers:

$(0, \pi)$

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

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

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

Correct answer:

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

Explanation:

To convert from rectangular to polar form, we must use the following formulas:

$x=r\cos(\theta), y=r\sin(\theta)$

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

It is easier to find our angle $\theta$ first, which is done by plugging in our x and y into the second formula:

$\frac{2}{2}=\tan(\theta)=1$

Find the angle by taking the inverse of the function:

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

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

Now find r by plugging in our angle and x and y into the first formula, and solving for r:

$2=r\cos\left(\frac{\pi}{4}\right)$

$r=2\sqrt{2}$

Our final answer is reported in polar coordinate form $(r, \theta)$ :

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

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Example Question #11 : Polar

What is the equation $y=3x^{2}$ in polar form?

Possible Answers:

$r=\tan 3\theta\sec\theta$

$r=\frac{1}{3}\tan \theta\sec\theta$

$r=3\tan \theta\sec\theta$

$r=-3\tan \theta\sec\theta$

$r=-\frac{1}{3}\tan \theta\sec\theta$

Correct answer:

$r=\frac{1}{3}\tan \theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=3x^{2}$ , then:

$r\sin\theta=3(r\cos \theta)^{2}$

$r\sin\theta=3r^{2}\cos^{2} \theta$

Dividing both sides by $r\cos\theta$ , we get:

$\tan \theta=3r\cos\theta$

$\frac{\tan \theta}{\cos\theta}=3r$

$\tan \theta\sec\theta=3r$

$r=\frac{1}{3}\tan \theta\sec\theta$

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Example Question #12 : Functions, Graphs, And Limits

What is the equation $y=-\frac{1}{2}x^{2}$ in polar form?

Possible Answers:

$r=\frac{1}{2}\tan \theta\sec\theta$

$r=-2\tan \theta\sec\theta$

$r=-\tan 2\theta\sec\theta$

$r=2\tan \theta\sec\theta$

$r=-\frac{1}{2}\tan \theta\sec\theta$

Correct answer:

$r=-2\tan \theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=-\frac{1}{2}x^{2}$ , then:

$r\sin\theta=-\frac{1}{2}(r\cos \theta)^{2}$

$r\sin\theta=-\frac{1}{2}r^{2}\cos^{2} \theta$

Dividing both sides by $r\cos\theta$ , we get:

$\tan \theta=-\frac{1}{2}r\cos\theta$

$\frac{\tan \theta}{\cos\theta}=-\frac{1}{2}r$

$\tan \theta\sec\theta=-\frac{1}{2}r$

$r=-2\tan \theta\sec\theta$

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Example Question #12 : Polar

What is the equation $y=\frac{2}{3}x^{2}$ in polar form?

Possible Answers:

$r=-\frac{2}{3}\tan \theta\sec\theta$

$r=\frac{3}{2}\tan \theta\sec\theta$

None of the above

$r=\frac{2}{3}\tan \theta\sec\theta$

$r=-\frac{3}{2}\tan \theta\sec\theta$

Correct answer:

$r=\frac{3}{2}\tan \theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=\frac{2}{3}x^{2}$ , then:

$r\sin\theta=\frac{2}{3}(r\cos \theta)^{2}$

$r\sin\theta=\frac{2}{3}r^{2}\cos^{2} \theta$

Dividing both sides by $r\cos\theta$ , we get:

$\tan \theta=\frac{2}{3}r\cos\theta$

$\frac{\tan \theta}{\cos\theta}=\frac{2}{3}r$

$3\tan \theta\sec\theta=2r$

$r=\frac{3}{2}\tan \theta\sec\theta$

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Example Question #181 : Parametric, Polar, And Vector

What is the polar form of $y=\frac{3}{5}x^{2}$ ?

Possible Answers:

None of the above

$r=-\frac{3}{5}\tan\theta\sec\theta$

$r=\frac{5}{3}\tan\theta\sec\theta$

$r=\frac{3}{5}\tan\theta\sec\theta$

$r=-\frac{5}{3}\tan\theta\sec\theta$

Correct answer:

$r=\frac{5}{3}\tan\theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=\frac{3}{5}x^{2}$ , then:

$r\sin\theta=\frac{3}{5}(r\cos\theta)^{2}$

$r\sin\theta=\frac{3}{5}r^{2}\cos^{2}\theta$

Dividing both sides by $r\cos\theta$ , we get:

$\tan\theta=\frac{3}{5}r\cos\theta$

$\frac{\tan\theta}{\cos\theta}=\frac{3}{5}r$

$\tan\theta\sec\theta=\frac{3}{5}r$

$r=\frac{5}{3}\tan\theta\sec\theta$

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Example Question #3 : Polar Form

What is the polar form of $y=-7x^{2}$ ?

Possible Answers:

$r=-\frac{1}{7}\tan\theta\sec\theta$

$r=\tan7\theta\sec\theta$

$r=7\tan\theta\sec\theta$

$r=\frac{1}{7}\tan\theta\sec\theta$

$r=-7\tan\theta\sec\theta$

Correct answer:

$r=-\frac{1}{7}\tan\theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=-7x^{2}$ , then:

$r\sin\theta=-7(r\cos\theta)^{2}$

$r\sin\theta=-7r^{2}\cos^{2}\theta$

Dividing both sides by $r\cos\theta$ , we get:

$\tan\theta=-7r\cos\theta$

$\frac{\tan\theta}{\cos\theta}=-7r$

$\tan\theta\sec\theta=-7r$

$r=-\frac{1}{7}\tan\theta\sec\theta$

Report an Error

Example Question #12 : Polar Form

What is the polar form of $y=\frac{1}{2}x^{2}$ ?

Possible Answers:

$r=-2\tan\theta\sec\theta$

$r=\frac{1}{2}\tan\theta\sec\theta$

None of the above

$r=-\frac{1}{2}\tan\theta\sec\theta$

$r=2\tan\theta\sec\theta$

Correct answer:

$r=2\tan\theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=\frac{1}{2}x^{2}$ , then:

$r\sin\theta=\frac{1}{2}(r\cos\theta)^{2}$

$r\sin\theta=\frac{1}2{}\cos^{2}\theta$

Dividing both sides by $r\cos\theta$ , we get:

$\tan\theta=\frac{1}{2}r\cos\theta$

$\frac{\tan\theta}{\cos\theta}=\frac{1}{2}r$

$\tan\theta\sec\theta=\frac{1}{2}r$

$r=2\tan\theta\sec\theta$

Report an Error

Example Question #182 : Parametric, Polar, And Vector

What is the polar form of $y=4x^{2}$ ?

Possible Answers:

$r=-4\tan\theta\sec\theta$

$r=-\frac{1}{4}\tan\theta\sec\theta$

$r=\frac{1}{4}\tan\theta\sec\theta$

$r=-4\tan\theta\sec\theta$

$r=\tan4\theta\sec\theta$

Correct answer:

$r=\frac{1}{4}\tan\theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=4x^{2}$ , then:

$r\sin\theta=4(r\cos\theta)^{2}$

Dividing both sides by $r\cos\theta$ , we get:

$\tan\theta=4r\cos\theta$

$\frac{\tan\theta}{\cos\theta}=4r$

$\tan\theta\sec\theta=4r$

$r=\frac{1}{4}\tan\theta\sec\theta$

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Example Question #14 : Functions, Graphs, And Limits

What is the polar form of $y=-x^{2}$ ?

Possible Answers:

$r=\frac{\tan\theta}{\sec\theta}$

None of the above

$r=-\tan\theta\sec\theta$

$r=-\frac{\tan\theta}{\sec\theta}$

$r=\tan\theta\sec\theta$

Correct answer:

$r=-\tan\theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=-x^{2}$ , then:

$r\sin\theta=-(r\cos\theta)^{2}$

Dividing both sides by $r\cos\theta$ , we get:

$\tan\theta=-r\cos\theta$

$\frac{\tan\theta}{\cos\theta}=-r$

$\tan\theta\sec\theta=-r$

$r=-\tan\theta\sec\theta$

Report an Error

Example Question #13 : Polar Form

What is the polar form of $y=\frac{1}{6}x^{2}$ ?

Possible Answers:

$r=\frac{\tan\theta}{6\sec\theta}$

$r=\frac{1}{6}\tan\theta\sec\theta$

$r=-6\tan\theta\sec\theta$

$r=-\frac{1}{6}\tan\theta\sec\theta$

$r=6\tan\theta\sec\theta$

Correct answer:

$r=6\tan\theta\sec\theta$

Explanation:

We can convert from rectangular to polar form by using the following trigonometric identities: $y=r\sin\theta$ and $x=r\cos\theta$ . Given $y=\frac{1}{6}x^{2}$ , then:

$r\sin\theta=\frac{1}{6}(r\cos\theta)^{2}$

Dividing both sides by $r\cos\theta$ , we get:

$\tan\theta=\frac{1}{6}r\cos\theta$

$\frac{\tan\theta}{\cos\theta}=\frac{1}{6}r$

$\tan\theta\sec\theta=\frac{1}{6}r$

$r=6\tan\theta\sec\theta$

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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Polar Form

Example Question #11 : Polar

Example Question #12 : Functions, Graphs, And Limits

Example Question #12 : Polar

Example Question #181 : Parametric, Polar, And Vector

Example Question #3 : Polar Form

Example Question #12 : Polar Form

Example Question #182 : Parametric, Polar, And Vector

Example Question #14 : Functions, Graphs, And Limits

Example Question #13 : Polar Form

All Calculus 2 Resources

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