Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

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

Example Questions

Example Question #1 : Calculus 3

Suppose the vectors ci+10j+ck and ci-2j-k are orthogonal. Find all real values of .

Possible Answers:

$-3\ \textup{and}\ 3$

$-4\ \textup{and}\5$

$2\ \textup{and}\ 10$

$4\ \textup{and}\ -5$

Correct answer:

$-4\ \textup{and}\5$

Explanation:

$-4\ \textup{and}\ 5$

Report an Error

Example Question #2 : Calculus 3

The vector-valued function r(t) paremeterizes a curve , where $r(t)=\left \langle e^{2t}, 3t+1, t^{2} \right \rangle$ .

Find a tangent vector to at the point $\left ( e^{10}, 16, 25 \right )$ .

Possible Answers:

$\left \langle e^{10}, 16, 25 \right \rangle$

$\left \langle -2e^{10}, -3, -10\right \rangle$

$\left \langle e, 0, 1 \right \rangle$

$\left \langle 2e^{10}, 3, 10 \right \rangle$

$\left \langle 1,1,1 \right \rangle$

Correct answer:

$\left \langle 2e^{10}, 3, 10 \right \rangle$

Explanation:

$\left \langle 2e^{10}, 3, 10 \right \rangle$

Report an Error

Example Question #2 : Calculus 3

Let $f\left ( x,y,z \right )=\sqrt{3x^{2}y+z^{3}}$ .

Find $f_{x}\left ( x,y,z \right )$ .

Possible Answers:

$\frac{1}{\sqrt{3x^{2}y+z^{3}}}$

$\frac{3xy}{\sqrt{3x^{2}y+z^{3}}}$

$\frac{-3xy}{\sqrt{3x^{2}y-z^{3}}}$

$\sqrt{3x^{2}y+z^{3}}$

3xy

Correct answer:

$\frac{3xy}{\sqrt{3x^{2}y+z^{3}}}$

Explanation:

$\frac{3xy}{\sqrt{3x^{2}y+z^{3}}}$

Report an Error

Example Question #4 : Calculus 3

Let $f(x,y)=e^{3x+y+1}$ .

Find its linear approximation at $\left ( 0,0 \right )$ .

Possible Answers:

$L\left ( x,y \right )=3ex$

$L\left ( x,y \right )=ey$

$L\left ( x,y \right )=e+3ex+ey$

$L\left ( x,y \right )=e-3ex-ey$

$L\left ( x,y \right )=3ex-ey$

Hint: Use Taylor's formula.

Correct answer:

$L\left ( x,y \right )=e+3ex+ey$

Explanation:

$L\left ( x,y \right )=e+3ex+ey$

Report an Error

Example Question #1 : Calculus 3

Let $f(x)=\ln (1-x^2)$ . Which of the following is equal to $f'(x)$ ?

Possible Answers:

$f'(x)=x^2\ln x-\ln x$

$f'(x)=\frac{1}{1-x^2}$

$f'(x)=2x\ln (1-x^2)$

$f'(x)=\frac{\ln x}{1-x^2}$

$f'(x)=\frac{2x}{x^2-1}$

Correct answer:

$f'(x)=\frac{2x}{x^2-1}$

Explanation:

Notice that $f(x)$ can be expressed as a composite function, i.e. a function within a function. If we let $g(x)=1-x^2$ and $h(x)=\ln (x)$ , then $f(x)=h(g(x))$ . In order to differentiate $f(x)$ , we will need to apply the Chain Rule, as shown below:

$f'(x)=h'(g(x))\cdot g'(x)$

First, we need to find $h'(x)$ , which equals $\frac{1}{x}$ .

Then, we need to find $g'(x)$ by applying the Product Rule.

$g'(x)=-2x$

$f'(x)=h'(g(x))\cdot g'(x)$ $=\frac{1}{1-x^2}\cdot (-2x)=\frac{2x}{x^2-1}$

The answer is $f'(x)=\frac{2x}{x^2-1}$ .

Report an Error

Example Question #12 : Integrals

Evaluate:

$\sum_{n=1}^{\infty }4(\frac{-1}{2})^{n-1}$

Possible Answers:

cannot be determined

$\infty$

$\frac{8}{3}$

Correct answer:

$\frac{8}{3}$

Explanation:

First, we can write out the first few terms of the sequence $a_{n}=$ $4(\frac{-1}{2})^{n-1}$ , where ranges from 1 to 3.

$a_{1}=4(\frac{-1}{2})^{1-1}=4$

$a_{2}=4(\frac{-1}{2})^{2-1}=-2$

$a_{3}=4(\frac{-1}{2})^{3-1}=1$

Notice that each term $\left ( n>1 \right )$ , is found by multiplying the previous term by $-\frac{1}{2}$ . Therefore, this sequence is a geometric sequence with a common ratio of $-\frac{1}{2}$ . We can find the sum of the terms in an infinite geometric sequence, provided that |r|<1 , where is the common ratio between the terms. Because $r=-\frac{1}{2}$ in this problem, $\left | r \right |$ is indeed less than 1. Therefore, we can use the following formula to find the sum, , of an infinite geometric series.

$S=\frac{a_{1}}{1-r}=\frac{4}{1-(\frac{-1}{2})}=\frac{4}{3/2}=\frac{8}{3}$

The answer is $\frac{8}{3}$ .

Report an Error

Example Question #4 : Calculus 3

Find $\frac{d^{2}y}{dx^{2}}$ if y^2-x^2 = 4 .

Possible Answers:

$\frac{x^2-4}{\sqrt{x^2+4}}$

$\frac{x^2}{(x^2-4)^\frac{3}{2}}$

$\frac{x^2+4}{(x^2-4)^\frac{3}{2}}$

$\frac{4}{(x^{2}+4)^{\frac{3}{2}}}$

$\frac{x}{\sqrt{4+x^2}}$

Correct answer:

$\frac{4}{(x^{2}+4)^{\frac{3}{2}}}$

Explanation:

We will have to find the first derivative of with respect to using implicit differentiation. Then, we can find $\frac{d^{2}y}{dx^{2}}$ , which is the second derivative of with respect to .

$\frac{d}{dx}[y^2-x^2]=\frac{d}{dx}[4]$

We will apply the chain rule on the left side.

$\frac{d}{dx}[y^2]\cdot \frac{dy}{dx}-\frac{d}{dx}[x^2]=0$

$2y\cdot \frac{dy}{dx}-2x=0$

We now solve for the first derivative with respect to .

$\frac{dy}{dx}=\frac{x}{y}$

In order to get the second derivative, we will differentiate $\frac{x}{y}$ with respect to . This will require us to employ the Quotient Rule.

$\frac{d}{dx}\frac{dy}{dx}=\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}[\frac{x}{y}]=\frac{y\cdot \frac{d}{dx}[x]-x\cdot \frac{d}{dx}[y]}{y^{2}} =\frac{y\cdot 1-x\cdot \frac{dy}{dx}}{y^{2}}$

We will replace $\frac{\mathrm{d}y }{\mathrm{d} x}$ with $\frac{x}{y}$ .

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

But, from the original equation, $y^{2}-x^{2}=4$ . Also, if we solve for , we obtain $y = (4+x^{2})^{\frac{1}{2}}$ .

$\frac{d^{2}y}{dx^{2}}=\frac{y^{2}-x^{2}}{y^{3}}=\frac{4}{(x^{2}+4)^{\frac{3}{2}}}$

The answer is $\frac{4}{(x^{2}+4)^{\frac{3}{2}}}$ .

Report an Error

Example Question #5 : Interpretations And Properties Of Definite Integrals

Which of the following represents the graph of the polar function $r = 4\cos\theta$ in Cartestian coordinates?

Possible Answers:

(x-2)^2 +(y-2)^2 =4

x^2 +y^2 =4

x^2+(y-2)^2 =4

(x-2)^2 +y^2 =4

(x-1)^2 +(y-2)^2 =4

Correct answer:

(x-2)^2 +y^2 =4

Explanation:

$r = 4\cos\theta$

First, mulitply both sides by r.

$r^2 =4r\cos\theta$

Then, use the identities x^2+y^2 =r^2 and $x = r\cos\theta$ .

x^2+y^2=4x

x^2-4x+y^2=0

x^2 -4x + 4 +y^2 =4

(x-2)^2 +y^2 =4

The answer is (x-2)^2 +y^2 =4 .

Report an Error

Example Question #4 : Calculus 3

$\int_{-1}^{0}e^{1-t}dt =$

Possible Answers:

$1-e^{2}$

$e^{2}-e$

$undefined$

$e+1$

$e^{2}-1$

Correct answer:

$e^{2}-e$

Explanation:

We can use the substitution technique to evaluate this integral.

Let u=1-t .

We will differentiate with respect to .

$\frac{\mathrm{d}u }{\mathrm{d} t}=-1$ , which means that ${\mathrm{d} u}=-{\mathrm{d} t}$ .

We can solve for ${\mathrm{d}t }$ in terms of ${\mathrm{d} u}$ , which gives us ${\mathrm{d}t }=-{\mathrm{d} u}$ .

We will also need to change the bounds of the integral. When t=-1 , u=1-(-1) , and when t=0 , u=1-0=1 .

We will now substitute in for the 1-t , and we will substitute $-{\mathrm{d}u }$ for ${\mathrm{d} t}$ .

$\int_{2}^{1}-e^{u}du$

$\int_{2}^{1}-e^{u}du = -e^{u}|_{2}^{1}=-e^{1}-(-e^{2})=e^{2}-e^{1}$

The answer is $e^{2}-e$ .

Report an Error

Example Question #5 : Fundamental Theorem Of Calculus

Evaluate the following limit:

$\lim_{x\rightarrow \infty }\frac{1-x^4}{x^2-4x^4}$

Possible Answers:

$\frac{1}{4}$

$\infty$

$\frac{1}{2}$

Does not exist.

Correct answer:

$\frac{1}{4}$

Explanation:

First, let's multiply the numerator and denominator of the fraction in the limit by $\frac{1}{x^{4}}$ .

$\lim_{x\rightarrow \infty }\frac{1-x^4}{x^2-4x^4}$ $=\lim_{x\rightarrow \infty }\frac{(\frac{1}{x^4})(1-x^4)}{\frac{1}{x^{4}}(x^2-4x^4)}$

$=\lim_{x\rightarrow \infty }\frac{\frac{1}{x^4}-1}{\frac{1}{x^2}-4}$

As becomes increasingly large the $\frac{1}{x^{4}}$ and $\frac{1}{x^{2}} ^{}$ terms will tend to zero. This leaves us with the limit of $\frac{1}{4}$ .

$\lim_{x\rightarrow \infty }\frac{-1}{-4}=\frac{1}{4}$ .

The answer is $\frac{1}{4}$ .

Report an Error

View Calculus 3 Tutors

Siddharth
Certified Tutor

University of Petroleum and Energy Studies and, Masters in Business Administration, Aviation Management.

View Calculus 3 Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View Calculus 3 Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

All Calculus 3 Resources

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

Popular Subjects

SSAT Tutors in Los Angeles, Computer Science Tutors in Denver, Physics Tutors in Washington DC, Statistics Tutors in Houston, French Tutors in San Francisco-Bay Area, Spanish Tutors in Miami, Statistics Tutors in Atlanta, Statistics Tutors in Miami, Statistics Tutors in Phoenix, ISEE Tutors in Miami

Popular Courses & Classes

MCAT Courses & Classes in Los Angeles, SSAT Courses & Classes in San Diego, SSAT Courses & Classes in Chicago, Spanish Courses & Classes in Los Angeles, SAT Courses & Classes in Washington DC, ACT Courses & Classes in Washington DC, ACT Courses & Classes in New York City, SAT Courses & Classes in Chicago, SSAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Miami

Popular Test Prep

SSAT Test Prep in Boston, GMAT Test Prep in Seattle, GMAT Test Prep in Denver, GRE Test Prep in Phoenix, GMAT Test Prep in San Diego, SSAT Test Prep in Denver, ACT Test Prep in Phoenix, GRE Test Prep in New York City, ACT Test Prep in New York City, ISEE Test Prep in Washington DC

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 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Calculus 3

Example Question #2 : Calculus 3

Example Question #2 : Calculus 3

Example Question #4 : Calculus 3

Example Question #1 : Calculus 3

Example Question #12 : Integrals

Example Question #4 : Calculus 3

Example Question #5 : Interpretations And Properties Of Definite Integrals

Example Question #4 : Calculus 3

Example Question #5 : Fundamental Theorem Of Calculus

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: