Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Graphing Vectors

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

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

Example Questions

Example Question #951 : Calculus Ii

What is the arclength, from x = 0 to $x=\frac{\pi}{3}$ , of the curve:

$y = ln \left | cos x\right |$

Hint:

$\int (sec (x)) dx= ln \left | sec (x) +tan(x)\right | + C$

Possible Answers:

$\sqrt{2} + 1$

$ln (\sqrt{3} + {2})$

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

$ln (\sqrt{2} + \sqrt3)$

$\sqrt{3} - \sqrt2$

Correct answer:

$ln (\sqrt{3} + {2})$

Explanation:

Arclength is given by the formula:

$Arclength = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}}\right)^2 dx$

We should find dy/dx first, which we find to be:

$\frac{\mathrm{d} }{\mathrm{d} x}(ln(cos(x)) = \frac{1}{cos(x)}\cdot (-sin(x)) = -tan(x)$

Now let's proceed with the integral:

$\int_{0}^{\frac{\pi}{3}} \sqrt{1 + (-tan x)^2}dx = \int_{0}^{\frac{\pi}{3}} \sqrt{(sec^2x)}dx = \int_{0}^{\frac{\pi}{3}} sec(x)dx$

(here we apply the integration described in the hint)

$= \left[ln \left | tan(x) + sec(x)\right |\bigg]_{0}^{\frac{\pi}{3}} = ln (\sqrt{3} + {2})$

which is obtained by evaluating at both boundaries.

Report an Error

Example Question #2 : Graphing Vectors

Find the area of the polar equation:

$r = \sqrt{cos(\theta )}$

Possible Answers:

$\frac{1}{2}$

$\pi$

Correct answer:

Explanation:

When you plot the graph of $r = \sqrt{cos(\theta )}$ , the bounds are between $\theta = -\frac{\pi }{2}$ and $\theta = \frac{\pi }{2}$ .

Use the area formula for polar equations:

$Area = \int_{\theta_{1} }^{\theta_{2}} \frac{1}{2}r^2 d\theta$

And so we find the area to be:

$\frac{1}{2}\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} (\sqrt{cos( \theta)})^2d\theta = \frac{1}{2}\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} cos (\theta)d\theta = \frac{1}{2}[sin \theta ]_{-\frac{\pi }{2}}^{\frac{\pi }{2}}$

$= \frac{1}{2}(1+1) = 1$

Report an Error

Example Question #3 : Graphing Vectors

The graph of the vector valued function

$\vec{r(t)}=<t^3, sin(t^2)>$

looks most like which of the following scalar functions graphed on a Cartesian coordinate system?

Possible Answers:

$y=sin^3({x})$

$y=sin({x}^\frac{3}{2})$

$y=sin({x}^\frac{2}{3})$

$y=x^{3sin(x^2)}$

Correct answer:

$y=sin({x}^\frac{2}{3})$

Explanation:

To determine this, we must convert our parametric form back into Cartesian form

To do this, we will solve for and reverse substitute.

x=t^3

$x^\frac{1}{3}=t$

Since we know the parameterization for y, we substitute in for

$y=sin(t^2)=sin((x^\frac{1}{3})^2)= sin((x^\frac{2}{3}))$

Our answer therefore is

$y=sin(x^\frac{2}{3})$

Report an Error

Example Question #1 : Graphing Vectors

The graph of the vector function $r(t)=\left \langle t+6, \sin(t)\right \rangle$ can also be represented by the graph of which of the following functions in rectangular form?

Possible Answers:

$y=\sin(x-6)$

$y=\sin(x+6)$

$y=-\sin(x-6)$

$y=-\sin(x+6)$

$y=\sin(6-x)$

Correct answer:

$y=\sin(x-6)$

Explanation:

We can find the graph of $r(t)=\left \langle t+6, \sin(t)\right \rangle$ in rectangular form by mapping the parametric coordinates to Cartesian coordinates (x,y) :

x=t+6

t=x-6

We can now use this value to solve for :

$y=\sin(t)$

$y=\sin(x-6)$

Report an Error

Example Question #4 : Graphing Vectors

The graph of the vector function $r(t)=\left \langle t^{2}, \sin(t)\right \rangle$ can also be represented by the graph of which of the following functions in rectangular form?

Possible Answers:

$y=\sin2\sqrt{x}$

$y=\sin\sqrt{\frac{x}2{}}$

$y=-\sin\sqrt{x}$

$y=\sin(\pm\sqrt{x})$

$y=-\sin2\sqrt{x}$

Correct answer:

$y=\sin(\pm\sqrt{x})$

Explanation:

We can find the graph of $r(t)=\left \langle t^{2}, \sin(t)\right \rangle$ in rectangular form by mapping the parametric coordinates to Cartesian coordinates (x,y) :

$x=t^{2}$

$t=\pm\sqrt{x}$

We can now use this value to solve for :

$y=\sin(\pm\sqrt{x})$

Report an Error

Example Question #6 : Graphing Vectors

The graph of the vector function $r(t)=\left \langle \frac{t^{2}}{3}, \cos^{2}(t)\right \rangle$ can also be represented by the graph of which of the following functions in rectangular form?

Possible Answers:

$y=\cos^{2}\left(\sqrt{\frac{x}{3}}\right)$

$y=-\cos^{2}\left(\sqrt{\frac{x}{3}}\right)$

$y=\cos^{2}(\pm\sqrt{3x})$

$y=-\cos^{2}(\sqrt{3x})$

None of the above

Correct answer:

$y=\cos^{2}(\pm\sqrt{3x})$

Explanation:

We can find the graph of $r(t)=\left \langle \frac{t^{2}}{3}, \cos^{2}(t)\right \rangle$ in rectangular form by mapping the parametric coordinates to Cartesian coordinates (x,y) :

$x=\frac{t^{2}}{3}$

$t^{2}=3x$

$t=\pm\sqrt{3x}$

We can now use this value to solve for :

$y=\cos^{2}(\pm\sqrt{3x})$

Report an Error

Example Question #7 : Graphing Vectors

The graph of the vector function $r(t)=\left \langle 4-t,\sin(t)\right \rangle$ can also be represented by the graph of which of the following functions in rectangular form?

Possible Answers:

$y=-\sin(4+x)$

$y=\sin(x-4)$

$y=\sin(4-x)$

$y=-\sin(4-x)$

$y=\sin(4+x)$

Correct answer:

$y=\sin(4-x)$

Explanation:

We can find the graph of $r(t)=\left \langle 4-t,\sin(t)\right \rangle$ in rectangular form by mapping the parametric coordinates to Cartesian coordinates (x,y) :

x=4-t

t=4-x

We can now use this value to solve for :

$y=\sin(t)$

$y=\sin(4-x)$

Report an Error

Example Question #1 : Graphing Vectors

The graph of the vector function $r(t)=\left \langle 4t^{2},\cos(t)\right \rangle$ can also be represented by the graph of which of the following functions in rectangular form?

Possible Answers:

$y=-\cos(\sqrt{\frac{4}{x}})$

$y=\cos(\sqrt{\frac{4}{x}})$

$y=4\cos(\sqrt{x})$

$y=\cos(\sqrt{\frac{x}{4}})$

$y=-\cos(\sqrt{\frac{x}{4}})$

Correct answer:

$y=\cos(\sqrt{\frac{x}{4}})$

Explanation:

We can find the graph of $r(t)=\left \langle 4t^{2},\cos(t)\right \rangle$ in rectangular form by mapping the parametric coordinates to Cartesian coordinates (x,y) :

x=4t^2

$\frac{x}{4}=t^2$

$t=\sqrt{\frac{x}{4}}$

We can now use this value to solve for :

$y=\cos(t)$

$y=\cos(\sqrt{\frac{x}{4}})$

Report an Error

Example Question #1 : Graphing Vectors

The graph of the vector function $r(t)=\left \langle \frac{t^{2}}{3},2\sin(t)\right \rangle$ can also be represented by the graph of which of the following functions in rectangular form?

Possible Answers:

$y=-2\sin(\sqrt{3x})$

$y=2\sin(\sqrt{3x})$

$y=2\sin(x\sqrt{3})$

$y=-2\sin(3\sqrt{x})$

$y=2\sin(3\sqrt{x})$

Correct answer:

$y=2\sin(\sqrt{3x})$

Explanation:

We can find the graph of $r(t)=\left \langle \frac{t^{2}}{3},2\sin(t)\right \rangle$ in rectangular form by mapping the parametric coordinates to Cartesian coordinates (x,y) :

$x=\frac{t^2}{}3$

3x=t^2

$t=\sqrt{3x}$

We can now use this value to solve for :

$y=2\sin(t)$

$y=2\sin(\sqrt{3x})$

Report an Error

Example Question #101 : Vector

The graph of the vector function $r(t)=\left \langle \frac{1}{\sqrt{t}},\sin(t)\right \rangle$ can also be represented by the graph of which of the following functions in rectangular form?

Possible Answers:

$y=-\sin(\frac{1}{x^{2}})$

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

$y=\sin(x^{2})$

$y=\sin(\frac{1}{x^{2}})$

$y=\sin(\frac{1}{x})$

Correct answer:

$y=\sin(\frac{1}{x^{2}})$

Explanation:

We can find the graph of $r(t)=\left \langle \frac{1}{\sqrt{t}},\sin(t)\right \rangle$ in rectangular form by mapping the parametric coordinates to Cartesian coordinates (x,y) :

$x=\frac{1}{\sqrt{t}}$

$\sqrt{t}=\frac{1}{x}$

$t=\frac{1}{x^{2}}$

We can now use this value to solve for :

$y=\sin(t)$

$y=\sin(\frac{1}{x^{2}})$

Report an Error

View Calculus 2 Tutors

Sharif
Certified Tutor

NCCU, Bachelors, Physics and Mathematics.

View Calculus 2 Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

View Calculus 2 Tutors

Alejandro
Certified Tutor

University of Western Ontario, Doctorate (PhD), Mathematics.

All Calculus 2 Resources

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

Popular Subjects

Algebra Tutors in Atlanta, ISEE Tutors in Washington DC, Reading Tutors in Los Angeles, English Tutors in Chicago, French Tutors in Washington DC, Chemistry Tutors in Atlanta, ACT Tutors in New York City, Math Tutors in Seattle, Physics Tutors in San Francisco-Bay Area, Biology Tutors in Denver

Popular Courses & Classes

Spanish Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in New York City, SSAT Courses & Classes in Philadelphia, SAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Miami, SSAT Courses & Classes in Atlanta, ACT Courses & Classes in Washington DC, SSAT Courses & Classes in San Diego, Spanish Courses & Classes in Seattle, ISEE Courses & Classes in San Francisco-Bay Area

Popular Test Prep

LSAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Boston, SSAT Test Prep in Houston, ACT Test Prep in Philadelphia, SAT Test Prep in Los Angeles, GMAT Test Prep in Seattle, ISEE Test Prep in San Francisco-Bay Area, SAT Test Prep in Boston, ACT Test Prep in Atlanta, SSAT Test Prep in Chicago

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 2 : Graphing Vectors

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #951 : Calculus Ii

Example Question #2 : Graphing Vectors

Example Question #3 : Graphing Vectors

Example Question #1 : Graphing Vectors

Example Question #4 : Graphing Vectors

Example Question #6 : Graphing Vectors

Example Question #7 : Graphing Vectors

Example Question #1 : Graphing Vectors

Example Question #1 : Graphing Vectors

Example Question #101 : Vector

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: