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 #32 : Gradient Vector, Tangent Planes, And Normal Lines

Find $\bigtriangledown F$ of the function x^2y+3xy^2z+z^4 .

Possible Answers:

$\left \langle 2x^2+4y^3z,x^3+6xy^2z,3xy^2+4z^2\right \rangle$

$\left \langle 2xy+3y^2z,x^2+6xyz,3xy^2+4z^3\right \rangle$

$\left \langle 2x+3y^2z,6yz,6y\right \rangle$

$\left \langle 3x+6y^3,xyz,4x^2z^3\right \rangle$

Correct answer:

$\left \langle 2xy+3y^2z,x^2+6xyz,3xy^2+4z^3\right \rangle$

Explanation:

The gradient of a function F(x,y,z) is as follows:

$\bigtriangledown F=\left \langle \frac{\mathrm{dF} }{\mathrm{d} x},\frac{\mathrm{dF} }{\mathrm{d} y},\frac{\mathrm{dF} }{\mathrm{d} z} \right \rangle$ .

We compute the derivative of the function with respect to each of the variables and treat the others like constants.

Using the rule $\frac{\mathrm{d} }{\mathrm{d} x}x^n=nx^{n-1}$ , we obtain

$\frac{\mathrm{dF} }{\mathrm{d} x}=2xy+3y^2z$ ,

$\frac{\mathrm{dF} }{\mathrm{d} y}=x^2+6xyz$ ,

and

$\frac{\mathrm{dF} }{\mathrm{d} z}=3xy^2+4z^3$ .

Putting these expressions into the vector completes the problem, and you obtain

$\left \langle 2xy+3y^2z,x^2+6xyz,3xy^2+4z^3\right \rangle$ .

Report an Error

Example Question #33 : Gradient Vector, Tangent Planes, And Normal Lines

Find the tangent plane to the surface given by

f(x, y, z)=x+y^2+z^3

at the point (1, 3, 1)

Possible Answers:

x+6y+3z=22

x+3y+z=22

x+6y+3z=0

x+6y+3z=-22

Correct answer:

x+6y+3z=22

Explanation:

The equation of the tangent plane is given by

f_x(x_0, y_0, z_0)(x-x_0)+f_y(x_0, y_0, z_0)(y-y_0)+f_z(x_0, y_0, z_0)(z-z_0)=0

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

f_x=1

f_y=2y

f_z=3z^2

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Now, we evaluate them at the given point:

f_x=1

f_y=6

f_z=3

Finally, plug in all of our information into the formula and simplify:

x+6y+3z=22

Report an Error

Example Question #34 : Gradient Vector, Tangent Planes, And Normal Lines

Find the tangent plane to the surface given by

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

at the point (0, 2, 5)

Possible Answers:

x+20y+4z=-60

x+20y+4z=60

2y+5z=-60

2y+5z=60

Correct answer:

x+20y+4z=60

Explanation:

The equation of the tangent plane is given by

f_x(x_0, y_0, z_0)(x-x_0)+f_y(x_0, y_0, z_0)(y-y_0)+f_z(x_0, y_0, z_0)(z-z_0)=0

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

f_x=e^x

f_y=2yz

f_z=y^2

The rules used to find the derivatives are

$\frac{\mathrm{d} }{\mathrm{d} x} e^x=e^x$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Evaluated at the given point, the partial derivatives are

f_x=1

f_y=20

f_z=4

Plugging all of this into the above formula, and simplifying, we get

x+20y+4z=60

Report an Error

Example Question #1591 : Calculus 3

Find the tangent plane to the surface given by

f(x, y, z)=x^2+4y^2+5z^2

at the point (1, 3, 3)

Possible Answers:

2x+24y+30z=164

x+3y+3z=164

x+3y+3z=-164

2x+24y+30z=-164

Correct answer:

2x+24y+30z=164

Explanation:

The equation of the tangent plane is given by

f_x(x_0, y_0, z_0)(x-x_0)+f_y(x_0, y_0, z_0)(y-y_0)+f_z(x_0, y_0, z_0)(z-z_0)=0

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

f_x=2x

f_y=8y

f_z=10z

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Next, we evaluate the partial derivatives at the given point:

f_x=2

f_y=24

f_z=30

Plugging in all our information into the formula above, we get

2x+24y+30z=164

Report an Error

Example Question #61 : Applications Of Partial Derivatives

Find $\nabla F$ of the function 4x^2yz+y^3+xz^4

Possible Answers:

$\left \langle 8xz+z^3,4x^2z+y^2,4x^2y\right \rangle$

$\left \langle 8xz+z^4,4x^2+3y^2,4x^2y+4xz^3\right \rangle$

$\left \langle 8xyz+z^4,4x^2z+3y^2,4x^2y+4xz^3\right \rangle$

$\left \langle 8xyz+z^4,4x^2z+3y^2,4x^2y+z^3\right \rangle$

Correct answer:

$\left \langle 8xyz+z^4,4x^2z+3y^2,4x^2y+4xz^3\right \rangle$

Explanation:

To find the gradient vector, you use the following definition: $\bigtriangledown F=\left \langle f_x,f_y,f_z\right \rangle$ . Taking the partial derivatives with respect to x, y, and z, we get $f_x=\frac{\partial }{\partial x}(4x^2yz+y^3+xz^4)=8xyz+z^4$ , $f_y=\frac{\partial }{\partial y}(4x^2yz+y^3+xz^4)=4x^2z+3y^2$ , and $f_z=\frac{\partial }{\partial z}(4x^2yz+y^3+xz^4)=4x^2y+4xz^3$ . Putting these into the vector produces the right answer.

Report an Error

Example Question #62 : Applications Of Partial Derivatives

Find $\bigtriangledown F$ of the function 3x^2y+y^3+z

Possible Answers:

$\left \langle 6xy,3x^2+3y^2,1\right \rangle$

Correct answer:

$\left \langle 6xy,3x^2+3y^2,1\right \rangle$

Explanation:

To find the gradient vector for a function F(x,y,z) , we use the definition

$\bigtriangledown F= \left \langle \frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} \right \rangle$ .

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

$\frac{\partial }{\partial y}(3x^2y+y^3+z)=3x^2+3y^2$

$\frac{\partial }{\partial z}(3x^2y+y^3+z)=1$

Putting these answers into the vector gets us the correct answer

Report an Error

Example Question #63 : Applications Of Partial Derivatives

Find $\bigtriangledown F$ of the function x^3+y^3+z^3

Possible Answers:

$\left \langle 3x,3y,3z\right \rangle$

$\left \langle 3x^2,3y^2,3z^2\right \rangle$

$\left \langle x^4,y^4,z^4\right \rangle$

$\left \langle x,y,z\right \rangle$

Correct answer:

$\left \langle 3x^2,3y^2,3z^2\right \rangle$

Explanation:

To find the gradient vector for a function F(x,y,z) , we use the definition

$\bigtriangledown F= \left \langle \frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} \right \rangle$ .

$\frac{\partial }{\partial x}(x^3+y^3+z^3)=3x^2$

$\frac{\partial }{\partial y}(x^3+y^3+z^3)=3y^2$

$\frac{\partial }{\partial z}(x^3+y^3+z^3)=3z^2$

Putting these answers into the vector gets us the correct answer

Report an Error

Example Question #64 : Applications Of Partial Derivatives

Find $\nabla F$ of the function x^5+2y^2+z

Possible Answers:

$\left \langle 5x^4,4y,1\right \rangle$

$\left \langle 1,4,1\right \rangle$

$\left \langle 2x,3y^2,1\right \rangle$

$\left \langle 5x,4y,1\right \rangle$

Correct answer:

$\left \langle 5x^4,4y,1\right \rangle$

Explanation:

To find the gradient vector for a function F(x,y,z) , we use the definition

$\nabla F= \left \langle \frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z} \right \rangle$ .

$\frac{\partial }{\partial x}(x^5+2y^2+z)=5x^4$

$\frac{\partial }{\partial y}(x^5+2y^2+z)=4y$

$\frac{\partial }{\partial z}(x^5+2y^2+z)=1$

Putting these answers into the vector gets us the correct answer

Report an Error

Example Question #1591 : Calculus 3

Compute the gradient of the following scalar function:

f(x,y,z)=x^4+y^3+z^2

Possible Answers:

$4x^3\mathbf{i}+3y^2\mathbf{j}+2z\mathbf{k}$

$\frac{x^5}{5}\mathbf{i}+\frac{y^4}{4}\mathbf{j}+\frac{z^3}{3}\mathbf{k}$

$4x^5\mathbf{i}+3y^4\mathbf{j}+2z^3\mathbf{k}$

$12x^2\mathbf{i}+6y\mathbf{j}+2\mathbf{k}$

$x^3\mathbf{i}+y^2\mathbf{j}+z\mathbf{k}$

Correct answer:

$4x^3\mathbf{i}+3y^2\mathbf{j}+2z\mathbf{k}$

Explanation:

The gradient of a function is defined as:

$\nabla f(x,y,z)=\frac{\partial f}{\partial x}\mathbf{i}+ \frac{\partial f}{\partial y}\mathbf{j} +\frac{\partial f}{\partial z}\mathbf{k}$

For our function:

$\frac{\partial}{\partial x}(x^4+y^3+z^2)=4x^3$

$\frac{\partial}{\partial y}(x^4+y^3+z^2)=3y^2$

$\frac{\partial}{\partial z}(x^4+y^3+z^2)=2z^2$

Thus, the gradient is:

$\nabla f(x,y,z)=4x^3\mathbf{i}+3y^2\mathbf{j}+2z\mathbf{k}$

Report an Error

Example Question #41 : Gradient Vector, Tangent Planes, And Normal Lines

Compute the gradient of the following scalar function:

f(x,y,z)=x^2y^3z^4

Possible Answers:

$4x^2y^3z^3\mathbf{i}+3x^2y^2z^4\mathbf{j}+2xy^3z^4\mathbf{k}$

$8xy^3z^3\mathbf{i}+6x^2yz^4\mathbf{j}+6xy^3z^3\mathbf{k}$

$6xy^2z^4\mathbf{i}+6x^2yz^4\mathbf{j}+9x^2y^2z^3\mathbf{k}$

$12x^2y^3z^2\mathbf{i}+12x^2y^2z^3\mathbf{j}+8xy^3z^3\mathbf{k}$

$2xy^3z^4\mathbf{i}+3x^2y^2z^4\mathbf{j}+4x^2y^3z^3\mathbf{k}$

Correct answer:

$2xy^3z^4\mathbf{i}+3x^2y^2z^4\mathbf{j}+4x^2y^3z^3\mathbf{k}$

Explanation:

The gradient of a function is defined as:

$\nabla f(x,y,z)=\frac{\partial f}{\partial x}\mathbf{i}+ \frac{\partial f}{\partial y}\mathbf{j} +\frac{\partial f}{\partial z}\mathbf{k}$

For our function:

$\frac{\partial}{\partial x}(x^2y^3z^4)=2xy^3z^4$

$\frac{\partial}{\partial y}(x^2y^3z^4)=3x^2y^2z^4$

$\frac{\partial}{\partial z}(x^2y^3z^4)=4x^2y^3z^3$

Thus, the gradient is:

$\nabla f(x,y,z)=2xy^3z^4\mathbf{i}+3x^2y^2z^4\mathbf{j}+4x^2y^3z^3\mathbf{k}$

Report an Error

View Calculus 3 Tutors

Sumit
Certified Tutor

Calcutta University, Bachelor of Science, Mathematics. Indian Statistical Institute, Master of Science, Mathematics.

View Calculus 3 Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Calculus 3 Tutors

Dilip
Certified Tutor

University of Calcutta, Bachelor of Science, Physics. Indiana Institute of Technology, Master of Science, Physics.

All Calculus 3 Resources

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

Popular Subjects

Chemistry Tutors in Miami, Biology Tutors in Boston, Computer Science Tutors in Phoenix, SAT Tutors in Philadelphia, ISEE Tutors in Atlanta, LSAT Tutors in Los Angeles, English Tutors in New York City, SAT Tutors in Denver, SAT Tutors in Boston, Math Tutors in Washington DC

Popular Courses & Classes

LSAT Courses & Classes in San Diego, SSAT Courses & Classes in Los Angeles, SSAT Courses & Classes in Atlanta, SSAT Courses & Classes in Boston, GMAT Courses & Classes in Denver, SAT Courses & Classes in Atlanta, GMAT Courses & Classes in Miami, ISEE Courses & Classes in San Diego, GRE Courses & Classes in Houston, GRE Courses & Classes in Boston

Popular Test Prep

GRE Test Prep in Los Angeles, GMAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Denver, LSAT Test Prep in Seattle, SAT Test Prep in Phoenix, MCAT Test Prep in Philadelphia, MCAT Test Prep in Miami, GRE Test Prep in Houston, LSAT Test Prep in New York City, GRE Test Prep in Boston

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 #32 : Gradient Vector, Tangent Planes, And Normal Lines

Example Question #33 : Gradient Vector, Tangent Planes, And Normal Lines

Example Question #34 : Gradient Vector, Tangent Planes, And Normal Lines

Example Question #1591 : Calculus 3

Example Question #61 : Applications Of Partial Derivatives

Example Question #62 : Applications Of Partial Derivatives

Example Question #63 : Applications Of Partial Derivatives

Example Question #64 : Applications Of Partial Derivatives

Example Question #1591 : Calculus 3

Example Question #41 : Gradient Vector, Tangent Planes, And Normal Lines

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: