We are open Saturday and Sunday!

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 #1053 : Partial Derivatives

Calculate the partial derivative with respect to of f(x,y,z)=x^2y^4z^5 at the point (3,1,-2) .

Possible Answers:

720

1620

864

Correct answer:

720

Explanation:

When calculating the partial derivative with respect to the variable of a function of more than one variable, apply the standard rules for differentiating a function f(x) of a single variable, and treat the other variables as constants. In this case, we are given f(x,y,z)=x^2y^4z^5 , and its partial derivative with respect to can be calculated by treating and and constants and differentiating z^5 :

$\frac{\partial f}{\partial z}=5x^2y^4z^4$ .

Now substitute the values for the point (3,1,-2) into $\frac{\partial f}{\partial z}$ to calculate its value at that point:

$F_z(x,y,z)=5(3)^2(1)^4(-2)^4=5\times9\times1\times16=720$ .

Report an Error

Example Question #1054 : Partial Derivatives

Calculate the partial derivative with respect to of the following function:

$f(x,y,z)=7e^x^z\ln(y)$

Possible Answers:

$\frac{\partial f}{\partial y}=\frac{7e^x^z}{y}$

$\frac{\partial f}{\partial y}=\frac{7e^x^z}{\ln(y)}$

$\frac{\partial f}{\partial y}=7xze^x^z\ln(y)$

$\frac{\partial f}{\partial y}=7ye^x^z\ln(y)$

$\frac{\partial f}{\partial y}=\frac{7xze^x^z}{y}$

Correct answer:

$\frac{\partial f}{\partial y}=\frac{7e^x^z}{y}$

Explanation:

When calculating the partial derivative with respect to the variable of a function of more than one variable, apply the standard rules for differentiating a function f(y) of a single variable, and treat the other variables as constants. In this case, we have:

$f(x,y,z)=7e^x^z\ln(y)$

We are being asked to differentiate f(x,y,z) with respect to , so we treat the variables and as constants, recognize that the term 7e^x^z is now just a constant, and apply the rule of differentiation for the natural logarithm to find the partial derivative $\frac{\partial f}{\partial y}$ , as shown:

$\frac{\partial f}{\partial y}=7e^x^z(\ln(y))'$

$=7e^x^z(\frac{1}{y})$

$=\frac{7e^x^z}{y}$

Report an Error

Example Question #1055 : Partial Derivatives

Find $f_{xy}$ of the following function:

$f(x,y)=e^{x^2y^2}+x\tan(y)$

Possible Answers:

$4xye^{x^2y^2}+4x^3y^3e^{x^2y^2}+\sec(y)$

$4xye^{x^2y^2}+4x^2y^2e^{x^2y^2}+\sec^2(y)$

$4xye^{x^2y^2}+4x^3y^3e^{x^2y^2}+\sec^2(y)$

$2xy^2e^{x^2y^2}+\tan(y)$

Correct answer:

$4xye^{x^2y^2}+4x^3y^3e^{x^2y^2}+\sec^2(y)$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we find the partial derivative of the function with respect to x:

$f_x=2xy^2e^{x^2y^2}+\tan(y)$

Finally, we take the partial derivative of the function with respect to y:

$f_{xy}=4xye^{x^2y^2}+4x^3y^3e^{x^2y^2}+\sec^2(y)$

Report an Error

Example Question #1056 : Partial Derivatives

Find $f_{xx}$ of the following function:

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

Possible Answers:

$-y^4\cos(xy)$

$-y^3\sin(xy)$

$y^4\cos(xy)$

Correct answer:

$-y^4\cos(xy)$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

First, we find the partial derivative of the function with respect to x:

$f_x=-y^3\sin(xy)$

Next, we take the partial derivative of this function with respect to x:

$f_{xx}=-y^4\cos(xy)$

Report an Error

Example Question #1052 : Partial Derivatives

Find f_x of the function $f(x,y,z)=xy+z\cos(x)$

Possible Answers:

$x-z\sin(x)$

$y-z\sin(x)$

$y+z\sin(x)$

$y-z\cos(y)$

Correct answer:

$y-z\sin(x)$

Explanation:

To find f_x of the function, you take the partial derivative of the function with respect to x

$\frac{\partial }{\partial x}(xy+z\cos(x))=y-z\sin(x)$

Report an Error

Example Question #1058 : Partial Derivatives

Find $f_{yy}$ of the function $f(x,y,z)=e^{2xy}+2xy^3$

Possible Answers:

$4x^2e^{2xy}+12xy$

$e^{5xy}+10xy$

$x^3e^{2xy}+12xy$

$4x^2e^{2xy}+12x^2$

Correct answer:

$4x^2e^{2xy}+12xy$

Explanation:

To find $f_{yy}$ of the function, you must take two consecutive partial derivatives

$\frac{\partial }{\partial y}(e^{2xy}+2xy^3)=2xe^{2xy}+6xy^2$

$\frac{\partial }{\partial y}(2xe^{2xy}+6xy^2)=4x^2e^{2xy}+12xy$

Report an Error

Example Question #1059 : Partial Derivatives

Find $(f_{xy})^2$ of the following function:

f(x,y)=y^2x+x^2e^y

Possible Answers:

$4y^2+8xye^y+4x^2e^{2y}$

2y+2xe^y

$y^2+2xye^y+x^2e^{2y}$

$4y^2+4x^2e^{2y}$

Correct answer:

$4y^2+8xye^y+4x^2e^{2y}$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we take the partial derivative of the function with respect to x:

f_x=y^2+2xe^y

Next, we take the partial derivative of this function with respect to y:

$f_{xy}=2y+2xe^y$

Finally, we square this:

$(f_{xy})^2=(2y+2xe^y)^2=4y^2+8xye^y+4x^2e^{2y}$

Report an Error

Example Question #1060 : Partial Derivatives

Find $f_{xx}$ for the following function:

$f(x,y)=\frac{x^2}{y^2}+\ln(xy)$

Possible Answers:

$\frac{2}{y^2}-\frac{1}{x^2}$

$\frac{2x}{y^2}+\frac{y}{x}$

$\frac{2}{y^2}-\frac{y}{x^2}$

Correct answer:

$\frac{2}{y^2}-\frac{y}{x^2}$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

First, we take the partial derivative of the function with respect to x:

$f_x=\frac{2x}{y^2}+\frac{y}{x}$

Finally, we take the partial derivative of this function with respect to x:

$f_{xx}=\frac{2}{y^2}-\frac{y}{x^2}$

Report an Error

Example Question #1061 : Partial Derivatives

Find $f_{xy}$ of the function $f(x,y,z)=3y\sin(x)+8x^2yz^3$

Possible Answers:

$5\cos(x)+16x^2z^4$

$3\cos(x)+16xz^3$

$3\sin(x)+16xz^3$

$3\cos(x)+14xz^3$

Correct answer:

$3\cos(x)+16xz^3$

Explanation:

To find $f_{xy}$ of the function, you take two consecutive partial derivatives:

$\frac{\partial }{\partial x}(3y\sin(x)+8x^2yz^3)=3y\cos(x)+16xyz^3$

$\frac{\partial }{\partial y}(3y\cos(x)+16xyz^3)=3\cos(x)+16xz^3$

Report an Error

Example Question #1062 : Partial Derivatives

Find $f_{xy}$ of the function $f(x,y,z)=e^{xy}+7xz^3$

Possible Answers:

$e^{xy}+xye^{xy}$

$e^{xy}+ye^{xy}$

$e^{y}+xye^{x}$

$e^{xy}+xe^{xy}$

Correct answer:

$e^{xy}+xye^{xy}$

Explanation:

To find $f_{xy}$ of the function, you take two consecutive partial derivatives:

$\frac{\partial }{\partial x}(e^{xy}+7xz^3)=ye^{xy}+7z^3$

$\frac{\partial }{\partial y}(ye^{xy}+7z^3)=e^{xy}+xye^{xy}$

Report an Error

View Calculus 3 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Calculus 3 Tutors

Colton
Certified Tutor

Texas State University-San Marcos, Bachelor of Science, Mathematics.

View Calculus 3 Tutors

MrG
Certified Tutor

Long Island University-Brooklyn Campus, Master of Science, Applied Mathematics.

All Calculus 3 Resources

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

Popular Subjects

Computer Science Tutors in Miami, Reading Tutors in New York City, ISEE Tutors in Dallas Fort Worth, Math Tutors in Chicago, MCAT Tutors in Seattle, English Tutors in Seattle, Chemistry Tutors in Seattle, ISEE Tutors in Chicago, Math Tutors in Boston, SAT Tutors in Washington DC

Popular Courses & Classes

GRE Courses & Classes in Seattle, Spanish Courses & Classes in Houston, GRE Courses & Classes in Boston, SAT Courses & Classes in Boston, ACT Courses & Classes in Los Angeles, Spanish Courses & Classes in New York City, SSAT Courses & Classes in Chicago, ISEE Courses & Classes in Denver, SSAT Courses & Classes in Phoenix, SSAT Courses & Classes in New York City

Popular Test Prep

GRE Test Prep in Philadelphia, ACT Test Prep in Philadelphia, LSAT Test Prep in New York City, GRE Test Prep in Boston, MCAT Test Prep in Philadelphia, ISEE Test Prep in Houston, GRE Test Prep in Seattle, SAT Test Prep in Philadelphia, SSAT Test Prep in Denver, ACT 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 #1053 : Partial Derivatives

Example Question #1054 : Partial Derivatives

Example Question #1055 : Partial Derivatives

Example Question #1056 : Partial Derivatives

Example Question #1052 : Partial Derivatives

Example Question #1058 : Partial Derivatives

Example Question #1059 : Partial Derivatives

Example Question #1060 : Partial Derivatives

Example Question #1061 : Partial Derivatives

Example Question #1062 : Partial Derivatives

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: