Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Derivatives of 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

← Previous 1 2 3 4 Next →

Example Question #11 : Derivatives Of Vectors

We define the vector $\vec{u}=(t^{n}, e^{nt}, 1 , 1 , 1 , 1 , 1 ,1 )$ .

What is the $n^{th}$ derivative of $\vec{u}$ ?

Possible Answers:

$\vec{u}=(n!, n^ne^{nt}, 0, 0 , 0 , 0 , 0 ,0 )$

$\vec{u}=(n, n^ne^{nt}, 0, 0 , 0 , 0 , 0 ,0 )$

$\vec{u}=(n, n^ne^{n}, 0, 0 , 0 , 0 , 0 ,0 )$

$\vec{u}=(n!, n^ne^{t}, 0, 0 , 0 , 0 , 0 ,0 )$

$\vec{u}=(n!, ne^{nt}, 0, 0 , 0 , 0 , 0 ,0 )$

Correct answer:

$\vec{u}=(n!, n^ne^{nt}, 0, 0 , 0 , 0 , 0 ,0 )$

Explanation:

To obtain the derivative we need to compute the derivative of each component. Since we are looking for the derivative of order n we have to differentiate each component n times to obtain the required result.

We know that :

$\{t^{n}\}^{[n]}=n!$

$\{e^{nt}\}^{[n]}=n\times n\cdots (e^{nt}) (ntimes)=n^{n}e^{nt}$

The nth derivatives of the constant is 0.

Thereore we obtain:

$\vec{u}^{[n]}=(n!, n^ne^{nt}, 0 , 0 , 0, 0 , 0 ,0 )$

Report an Error

Example Question #12 : Derivatives Of Vectors

Let $\vec{u}=(sin(2t),cos(2t),t^3)$ .

What is the third derivative of $\vec{u}$ ?

Possible Answers:

(-8cos(2x),8sin(2x),2t^2)

(-8cos(2x),8sin(2x),3)

(-8cos(2x),8sin(2x),6)

(-8cos(2x),8sin(2x),t)

(-8cos(x),8sin(x),3t)

Correct answer:

(-8cos(2x),8sin(2x),6)

Explanation:

To obtain the derivative, we simply differentiate 3 times each of the components.

Using the Chain Rule and the Power Rule we have :

$f(x)=sin(2t)\rightarrow f'(x)=2cos(2t)\rightarrow f''(x)=-4sin(2t)$

$\rightarrow cos(2x)^{[3]}=8sin(2x)$

$g(x)=cos(2t)\rightarrow g'(x)=-2sin(2t)\rightarrow g''(x)=-4cos(2t)$

$\rightarrow sin(2x)^{[3]}=-8cos(2x)$

$h(x)=t^3 \rightarrow h'(x)=3t^2 \rightarrow h''(x)=6t$

$\rightarrow t^3^{[3]}=6$

Report an Error

Example Question #13 : Derivatives Of Vectors

$\newline {\vec{r}(t)} = (t^{2},3t,5t)\newline \textnormal{Calculate } \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)}$

Possible Answers:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2,3,5)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2t,0,0)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2t,3t,5t)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2t,3,5)$

Correct answer:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2t,3,5)$

Explanation:

$\newline \textnormal{In general:}\newline \textnormal{If} \ {\vec{r}(t)} = (f(t),g(t),z(t))\newline \textnormal{Then} \ \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}f(t), \frac{\mathrm{d} }{\mathrm{d} t}g(t),\frac{\mathrm{d} }{\mathrm{d} t}z(t)\right)$

$\newline \textnormal{Derivative of a constant is always 0. For example, } \frac{\mathrm{d} }{\mathrm{d} t} 5 = 0\newline \textnormal{Power Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}\newline \textnomal{Sum Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t}(t^n+10) = \frac{\mathrm{d} }{\mathrm{d} t}t^n + \frac{\mathrm{d} }{\mathrm{d} t}10 = n*t^{n-1}\newline \textnormal{Use the power rule to differentiate each component of the vector.}\newline \textnormal{For example} \ \frac{\mathrm{d} }{\mathrm{d} t} t^2 = 2 * t ^ {2-1} = 2*t$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}t^{2},\frac{\mathrm{d} }{\mathrm{d} t}3t,\frac{\mathrm{d} }{\mathrm{d} t}5t\right) = (2t,3,5)$

Report an Error

Example Question #14 : Derivatives Of Vectors

$\newline {\vec{r}(t)} = (4t,8t,10t^{3})\newline \textnormal{Calculate } \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)}$

Possible Answers:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (0,8,10t)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4,8,30t)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4,8,30t^{2})$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4,8,10t^{2})$

Correct answer:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4,8,30t^{2})$

Explanation:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}4t,\frac{\mathrm{d} }{\mathrm{d} t}8t,\frac{\mathrm{d} }{\mathrm{d} t}10t^{3}\right) = (4,8,30t^{2})$

Report an Error

Example Question #11 : Derivatives Of Vectors

$\newline {\vec{r}(t)} = (4t^{4},9t^{2},4t)\newline \textnormal{Calculate } \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)}$

Possible Answers:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (16t^3,9t,4)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (12,9,4)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (24,9t,4)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (16^{3},18t,4)$

Correct answer:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (16^{3},18t,4)$

Explanation:

$\newline \textnormal{Derivative of a constant is always 0. For example, } \frac{\mathrm{d} }{\mathrm{d} t} 5 = 0\newline \textnormal{Power Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1} \newline \textnormal{Use the power rule to differentiate each component of the vector.}\newline \textnormal{For example} \ \frac{\mathrm{d} }{\mathrm{d} t}4t^4 = 4*4*t^{4-1} = 16*t^3$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}4t^{4},\frac{\mathrm{d} }{\mathrm{d} t}9t^{2},\frac{\mathrm{d} }{\mathrm{d} t}4t\right) = (16t^{3},18t,4)$

Report an Error

Example Question #16 : Derivatives Of Vectors

$\newline {\vec{r}(t)} = (11t,6t,3t)\newline \textnormal{Calculate } \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)}$

Possible Answers:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (11,6,3)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (11,0,0)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (11,6,0)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (0,0,0)$

Correct answer:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (11,6,3)$

Explanation:

$\newline \textnormal{Derivative of a constant is always 0. For example, } \frac{\mathrm{d} }{\mathrm{d} t} 5 = 0\newline \textnormal{Power Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1} \newline \textnormal{Use the power rule to differentiate each component of the vector.}\newline \textnormal{For example} \ \frac{\mathrm{d} }{\mathrm{d} t}11t = 11*1*t^{1-1} = 11$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}11t,\frac{\mathrm{d} }{\mathrm{d} t}6t,\frac{\mathrm{d} }{\mathrm{d} t}3t\right) = (11,6,3)$

Report an Error

Example Question #17 : Derivatives Of Vectors

$\newline {\vec{r}(t)} = (3t^3,10t+1,t)\newline \textnormal{Calculate } \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)}$

Possible Answers:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (9t,10,1)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (9t^2,10,1)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (9t^2,0,0)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (9t^3,11,1)$

Correct answer:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (9t^2,10,1)$

Explanation:

$\newline \textnormal{Derivative of a constant is always 0. For example, } \frac{\mathrm{d} }{\mathrm{d} t} 5 = 0\newline \textnormal{Power Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}\newline \textnomal{Sum Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t}(t^n+10) = \frac{\mathrm{d} }{\mathrm{d} t}t^n + \frac{\mathrm{d} }{\mathrm{d} t}10 = n*t^{n-1}\newline \textnormal{Use the power rule to differentiate each component of the vector.}\newline \textnormal{For example} \ \ \frac{\mathrm{d} }{\mathrm{d} t}3t^3 = 3*3*t^{3-1} = 9*t^2$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}3t^3,\frac{\mathrm{d} }{\mathrm{d} t}(10t+1),\frac{\mathrm{d} }{\mathrm{d} t}t\right) = (9t^2,10,1)$

Report an Error

Example Question #11 : Derivatives Of Vectors

$\newline {\vec{r}(t)} = (2t,4t,t^3+3)\newline \textnormal{Calculate } \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)}$

Possible Answers:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2,4,3)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2,4,6)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2,4,3t)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2,4,3t^2)$

Correct answer:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (2,4,3t^2)$

Explanation:

$\newline \textnormal{Derivative of a constant is always 0. For example, } \frac{\mathrm{d} }{\mathrm{d} t} 5 = 0\newline \textnormal{Power Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}\newline \textnormal{Use the power rule to differentiate each component of the vector.}\newline \textnormal{For example} \ \frac{\mathrm{d} }{\mathrm{d} t} t ^3 = 3 * t^{3-1} = 3*t^2$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}2t,\frac{\mathrm{d} }{\mathrm{d} t}4t,\frac{\mathrm{d} }{\mathrm{d} t}(t^3+3)\right) = (2,4,3t^2)$

Report an Error

Example Question #19 : Derivatives Of Vectors

$\newline {\vec{r}(t)} = (t^4+1,t^2+t,4t+3)\newline \textnormal{Calculate } \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)}$

Possible Answers:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4t^3,2t,0)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4t^3,2t+1,4)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4t,2t,4)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4t^3,2,4)$

Correct answer:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (4t^3,2t+1,4)$

Explanation:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}(t^4+1),\frac{\mathrm{d} }{\mathrm{d} t}(t^2+t),\frac{\mathrm{d} }{\mathrm{d} t}(4t+3)\right) = (4t^3,2t+1,4)$

Report an Error

Example Question #20 : Derivatives Of Vectors

$\newline {\vec{r}(t)} = (cos(3t)+1,t^2+t,3t+11)\newline \textnormal{Calculate } \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)}$

Possible Answers:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (-3sin(3t),2t,3)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (3cos(3t),2t,3)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (-3sin(3t),2t+1,3)$

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (-3sin(3t),2,3)$

Correct answer:

$\frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = (-3sin(3t),2t+1,3)$

Explanation:

$\newline \textnormal{Derivative of a constant is always 0. For example, } \frac{\mathrm{d} }{\mathrm{d} t} 5 = 0\newline \textnormal{Power Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}\newline \textnomal{Sum Rule:} \ \frac{\mathrm{d} }{\mathrm{d} t}(t^n+10) = \frac{\mathrm{d} }{\mathrm{d} t}t^n + \frac{\mathrm{d} }{\mathrm{d} t}10 = n*t^{n-1}\newline \textnormal{Use the power rule to differentiate each component of the vector.}\newline \textnormal{For example} \frac{\mathrm{d} }{\mathrm{d} t}(cos(3t) + 1) = \frac{\mathrm{d} }{\mathrm{d} t}cos(3t) + \frac{\mathrm{d} }{\mathrm{d} t}1 = -3sin(3t)$

$\\ \frac{\mathrm{d} }{\mathrm{d} t}{\vec{r}(t)} = \left(\frac{\mathrm{d} }{\mathrm{d} t}(cos(3t)+1),\frac{\mathrm{d} }{\mathrm{d} t}(t^2+t),\frac{\mathrm{d} }{\mathrm{d} t}(3t+11)\right) = (-3sin(3t),2t+1,3)$

Report an Error

← Previous 1 2 3 4 Next →

View Calculus 2 Tutors

Yoonsik
Certified Tutor

Seoul National University, Bachelor of Science, Physics. University of Pennsylvania, Doctor of Philosophy, Atomic and Molecul...

View Calculus 2 Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Calculus 2 Tutors

Filiberto
Certified Tutor

University of Arizona, Bachelor of Science, Electrical Engineering.

All Calculus 2 Resources

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

Popular Subjects

ISEE Tutors in San Francisco-Bay Area, Math Tutors in Boston, SAT Tutors in Washington DC, SAT Tutors in Dallas Fort Worth, MCAT Tutors in New York City, Reading Tutors in Seattle, Biology Tutors in Houston, Spanish Tutors in New York City, Reading Tutors in Houston, English Tutors in Phoenix

Popular Courses & Classes

ACT Courses & Classes in Chicago, Spanish Courses & Classes in Denver, ACT Courses & Classes in Washington DC, LSAT Courses & Classes in Philadelphia, ISEE Courses & Classes in Philadelphia, MCAT Courses & Classes in Washington DC, LSAT Courses & Classes in Seattle, MCAT Courses & Classes in Houston, MCAT Courses & Classes in Phoenix, LSAT Courses & Classes in Chicago

Popular Test Prep

ISEE Test Prep in New York City, ISEE Test Prep in Phoenix, ACT Test Prep in Chicago, ACT Test Prep in Phoenix, MCAT Test Prep in Houston, LSAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Los Angeles, ISEE Test Prep in Dallas Fort Worth, SAT Test Prep in Phoenix, SAT Test Prep in Atlanta

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 : Derivatives of Vectors

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #11 : Derivatives Of Vectors

Example Question #12 : Derivatives Of Vectors

Example Question #13 : Derivatives Of Vectors

Example Question #14 : Derivatives Of Vectors

Example Question #11 : Derivatives Of Vectors

Example Question #16 : Derivatives Of Vectors

Example Question #17 : Derivatives Of Vectors

Example Question #11 : Derivatives Of Vectors

Example Question #19 : Derivatives Of Vectors

Example Question #20 : Derivatives Of Vectors

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: