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 #1431 : Partial Derivatives

Find the total differential, , of the following function

z=xy+2x+2y

Possible Answers:

dz=(x+2)dx+(y+2)dy

dz=(y+2)dx+(x+2)dy

dz=(2xy)dx+(2xy)dy

Correct answer:

dz=(y+2)dx+(x+2)dy

Explanation:

The total differential is defined as

$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$

For the function

z=xy+2x+2y

We first find

$\frac{\partial z}{\partial x}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial x}=y+2$

We then find

$\frac{\partial z}{\partial y}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial y}=x+2$

We then substitute these partial derivatives into the first equation to get the total differential

dz=(y+2)dx+(x+2)dy

Report an Error

Example Question #11 : Differentials

Find the total differential, , of the following function

$z=x^2y^2+e^x+cos\,y$

Possible Answers:

$dz=(x^2y^2+e^x)dx+(x^2y^2+cos\,y)dy$

$dz=(2xy^2+e^x)dx+(2x^2y-sin\,y)dy$

dz=(2xy^2)dx+(2x^2y)dy

Correct answer:

$dz=(2xy^2+e^x)dx+(2x^2y-sin\,y)dy$

Explanation:

The total differential is defined as

$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$

For the function $z=x^2y^2+e^x+cos\,y$

We first find

$\frac{\partial z}{\partial x}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial x}=2xy^2+e^x$

We then find

$\frac{\partial z}{\partial y}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial y}=2x^2y-sin\,y$

We then substitute these partial derivatives into the first equation to get the total differential

$dz=(2xy^2+e^x)dx+(2x^2y-sin\,y)dy$

Report an Error

Example Question #16 : Differentials

Find the total differential, , of the following function

$z=x+sin\,y+cos\,x+y$

Possible Answers:

dz=dx+dy

$dz=(1+sin\,x+y)dx+(x-cos\,y+1)dy$

$dz=(1-sin\,x)dx+(cos\,y+1)dy$

Correct answer:

$dz=(1-sin\,x)dx+(cos\,y+1)dy$

Explanation:

The total differential is defined as

$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$

For the function $z=x+sin\,y+cos\,x+y$

We first find

$\frac{\partial z}{\partial x}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial x}=1-sin\,x$

We then find

$\frac{\partial z}{\partial y}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial y}=cos\,y+1$

We then substitute these partial derivatives into the first equation to get the total differential

$dz=(1-sin\,x)dx+(cos\,y+1)dy$

Report an Error

Example Question #17 : Differentials

If f(x,y) = xe^y , calculate the differential when moving from (x_0,y_0)=(1,0) to (0,1) .

Possible Answers:

e-1

e+1

Correct answer:

Explanation:

The equation for is

dz = f_x(x_0,y_0)dx +f_y(x_0,y_0)dy .

Evaluating partial derivatives and substituting, we get

$f_x(x_0,y_0) = f_x(1,0) = e^{(0)} =1$

$f_y(x_0,y_0) = f_y(1,0) = (1)e^{(0)}=1$

$dx = \Delta x = -1, dy=\Delta y = 1$

Plugging in, we get

dz = (1)(-1)+(1)(1)=0 .

Report an Error

Example Question #18 : Differentials

If f(x,y) = x^2-y^2 , calculate the differential when moving from the point (x_0,y_0)=(1,1) to the point (2,3) .

Possible Answers:

-1.5

-.5

Correct answer:

Explanation:

The equation for is

dz = f_x(x_0,y_0)dx +f_y(x_0,y_0)dy .

Evaluating partial derivatives and substituting, we get

f_x(x_0,y_0) = f_x(1,1) = 2(1) =2

f_y(x_0,y_0) = f_y(1,1) = -2(1) =-2

$dx = \Delta x = 1, dy=\Delta y = 2$

Plugging in, we get

dz = (2)(1)+(-2)(2) =-2 .

Report an Error

Example Question #19 : Differentials

If $f(x,y) = \ln(xy)$ , calculate the differential when moving from the point (x_0,y_0)=(1,1) to the point (e,e) .

Possible Answers:

e-1

-e-1

2(e-1)

3e-2

Correct answer:

2(e-1)

Explanation:

The equation for is

dz = f_x(x_0,y_0)dx +f_y(x_0,y_0)dy .

Evaluating partial derivatives and substituting, we get

$f_x(x_0,y_0) = f_x(1,1) = \frac{1}{(1)} =1$

$f_y(x_0,y_0) = f_y(1,1) = \frac{1}{(1)}=1$

$dx = \Delta x = e-1, dy=\Delta y = e-1$

Plugging in, we get

dz = (1)(e-1)+(1)(e-1) =2e-2=2(e-1).

Report an Error

Example Question #1431 : Partial Derivatives

Find the differential of the following function:

g=y^4+13xz^2+3e^x

Possible Answers:

(13z^2+3e^x)dx+4y^3dy+13xzdz

13z^2+3e^x+4y^3+26xz

(z^2+3e^x)dx+4y^3dy+26xzdz

(13z^2+3e^x)dx+4y^3dy+26xzdz

Correct answer:

(13z^2+3e^x)dx+4y^3dy+26xzdz

Explanation:

The differential of the function is given by

f_xdx+f_ydy+f_zdz

The partial derivatives are

f_x=13z^2+3e^x , f_y=4y^3dy , f_z=26xzdz

Report an Error

Example Question #21 : Differentials

Find the differential of the following function:

$h=z\cos(xyz)+y^2\ln(xz^2)$

Possible Answers:

$-yz^2\sin(xyz)+\frac{y^2}{x}-xz^2\sin(xyz)+2y\ln(xz^2)+\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z}$

$(-yz^2\sin(xyz)+\frac{y^2}{x})dx+(-xz^2\sin(xyz)+2y\ln(xz^2))dy+(\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z})dz$

$(-yz^2\sin(xyz)+\frac{y^2}{x})dx+(xz^2\sin(xyz)+2y\ln(xz^2))dy+(\cos(xyz)+xyz\sin(xyz)+\frac{2y^2}{z})dz$

$(yz^2\sin(xyz)+\frac{y^2}{x})dx+(-xz^2\sin(xyz)+2y\ln(xz^2))dy+(\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z})dz$

Correct answer:

$(-yz^2\sin(xyz)+\frac{y^2}{x})dx+(-xz^2\sin(xyz)+2y\ln(xz^2))dy+(\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z})dz$

Explanation:

The differential of the function is given by

f_xdx+f_ydy+f_zdz

The partial derivatives are

$f_x=-yz^2\sin(xyz)+\frac{y^2}{x}$ , $f_y=-xz^2\sin(xyz)+2y\ln(xz^2)$ , $f_z=\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z}$

Report an Error

Example Question #22 : Differentials

Find the differential of the following function:

$f(x,y,z)=\cos^2(xy)+\sin(z)$

Possible Answers:

$-2y\cos(xy)\sin(xy)dx-2x\cos(xy)\sin(xy)dy-\cos(z)dz$

$-y\cos(xy)\sin(xy)dx-x\cos(xy)\sin(xy)dy+\cos(z)dz$

$2y\cos(xy)\sin(xy)dx+2x\cos(xy)\sin(xy)dy+\cos(z)dz$

$-2y\cos(xy)\sin(xy)dx-2x\cos(xy)\sin(xy)dy+\cos(z)dz$

$-2x\cos(xy)\sin(xy)dx-2y\cos(xy)\sin(xy)dy+\cos(z)dz$

Correct answer:

$-2y\cos(xy)\sin(xy)dx-2x\cos(xy)\sin(xy)dy+\cos(z)dz$

Explanation:

The differential of a function f(x,y,z) is given by

f_xdx+f_ydy+f_zdz

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

$f_x=-2y\cos(xy)\sin(xy)$

$f_y=-2x\cos(xy)\sin(xy)$

$f_z=\cos(z)$

Report an Error

Example Question #3803 : Calculus 3

Find the differential of the following function:

$f(x,y,z)=7xy+6x^2y^3e^{z^2}$

Possible Answers:

$(7x+12xy^2e^{z^2})dx+(7y+18x^2y^2e^{z^2})dy+(12x^2y^3ze^{z^2})dz$

$(7y+12xy^2e^{z^2})dx+(7x+12x^2y^2e^{z^2})dy+(12x^2y^3ze^{z^2})dz$

$(7y+12xy^2e^{z^2})dx+(7x+18x^2y^2e^{z^2})dy+(6x^2y^3ze^{z^2})dz$

$(7y+12xy^2e^{z^2})dx+(7x+18x^2y^2e^{z^2})dy+(12x^2y^3ze^{z^2})dz$

$(7y+12xy^2e^{z^2})dx+(7x+18x^2y^2e^{z^2})dy+(12x^2y^3z^2e^{z^2})dz$

Correct answer:

$(7y+12xy^2e^{z^2})dx+(7x+18x^2y^2e^{z^2})dy+(12x^2y^3ze^{z^2})dz$

Explanation:

The differential of a function f(x,y,z) is given by

f_xdx+f_ydy+f_zdz

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

$f_x=7y+12xy^2e^{z^2}$

$f_y=7x+18x^2y^2e^{z^2}$

$f_z=12x^2y^3ze^{z^2}$

Report an Error

View Calculus 3 Tutors

Bahaeddine
Certified Tutor

Sorbonne University, Bachelor of Science, Mathematics. Sorbonne University, Master of Science, Statistics.

View Calculus 3 Tutors

Garth
Certified Tutor

View Calculus 3 Tutors

Satweka
Certified Tutor

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

All Calculus 3 Resources

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

Popular Subjects

SAT Tutors in Washington DC, Reading Tutors in Denver, Algebra Tutors in San Francisco-Bay Area, LSAT Tutors in Los Angeles, Reading Tutors in Los Angeles, Chemistry Tutors in Washington DC, Calculus Tutors in Philadelphia, Algebra Tutors in Seattle, ACT Tutors in Miami, GRE Tutors in Los Angeles

Popular Courses & Classes

ISEE Courses & Classes in San Diego, GMAT Courses & Classes in Boston, MCAT Courses & Classes in New York City, GMAT Courses & Classes in New York City, GMAT Courses & Classes in Seattle, SSAT Courses & Classes in Phoenix, GRE Courses & Classes in Houston, Spanish Courses & Classes in Atlanta, LSAT Courses & Classes in Seattle, MCAT Courses & Classes in Atlanta

Popular Test Prep

GRE Test Prep in Dallas Fort Worth, ISEE Test Prep in Boston, SSAT Test Prep in San Diego, ISEE Test Prep in Seattle, SSAT Test Prep in Houston, ISEE Test Prep in Miami, SAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Chicago, GRE Test Prep in Miami, LSAT 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 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1431 : Partial Derivatives

Example Question #11 : Differentials

Example Question #16 : Differentials

Example Question #17 : Differentials

Example Question #18 : Differentials

Example Question #19 : Differentials

Example Question #1431 : Partial Derivatives

Example Question #21 : Differentials

Example Question #22 : Differentials

Example Question #3803 : Calculus 3

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: