We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Rate of Change

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #461 : Rate Of Change

Find the slope at x=0 given the following function: f(x)= (x^4 + ln(x) - 3)^2

Possible Answers:

254.5

890.1

45.8

24.8

Answer not listed

Correct answer:

890.1

Explanation:

In order to find the slope of a certain point, you first find the derivative of the following function: f(x)= (x^4 + ln(x) - 3)^2

The derivative is: $f'(x)= 2(x^4 + ln(x) - 3) (4x^{3} + \frac{1}{x} )$

Then, plug x= 2 into the derivative: $f'(2)= 2({\color{Blue} 2}^4 + ln({\color{Blue} 2}) - 3) (4({\color{Blue} 2})^{3} + \frac{1}{{\color{Blue} 2}} )$

Therefore, the slope is: 890.1

Report an Error

Example Question #462 : Rate Of Change

Find the slope at x=0 given the following function: $f(x)= \sin(2x^4 - e^{2x})$

Possible Answers:

1.476

23.7

-.004

Answer not listed

3.41

Correct answer:

Answer not listed

Explanation:

In order to find the slope of a certain point, you first find the derivative of the following function: $f(x)= \sin(2x^4 - e^{2x})$

The derivative is: $f'(x)= \cos(2x^4 - e^{2x}) (8x^{3} - 2e^{2x})$

Then, plug x= 0 into the derivative: $f'(0)= \cos(2({\color{Blue} 0})^4 - e^{2({\color{Blue} 0})}) (8({\color{Blue} 0})^{3} - 2e^{2({\color{Blue} 0})})$

Therefore, the slope is: -1.08

Report an Error

Example Question #463 : Rate Of Change

Find the slope at x=1 given the following function: $f(x)= e^{4x^5 + ln(x)}$

Possible Answers:

23.597

9.245

513.5

1146.6

Answer not listed

Correct answer:

1146.6

Explanation:

In order to find the slope of a certain point, you first find the derivative of the following function: $f(x)= e^{4x^5 + ln(x)}$

The derivative is: $f'(x)= e^{4x^5 + ln(x)} (20x^{4} + \frac{1}{x})$

Then, plug x= 1 into the derivative: $f'(x)= e^{4({\color{Blue} 1})^5 + ln({\color{Blue} 1})} (20({\color{Blue} 1})^{4} + \frac{1}{{\color{Blue} 1}})$

Therefore, the slope is: 1146.6

Report an Error

Example Question #464 : Rate Of Change

Find the slope at x=0 given the following function: $f(x)= -\sin( 2x+ e^{2x})$

Possible Answers:

4.94

9.24

-7.81

3.96

Answer not listed

Correct answer:

3.96

Explanation:

In order to find the slope of a certain point, you first find the derivative of the following function: $f(x)= -\sin( 2x+ e^{2x})$

The derivative is: $f'(x)= -\cos( 2x+ e^{2x}) (2 + 2e^{2x})$

Then, plug x= 0 into the derivative: $f'(0)= -\cos( 2({\color{Blue} 0})+ e^{2({\color{Blue} 0})}) (2 + 2e^{2({\color{Blue} 0})})$

Therefore, the slope is: 3.96

Report an Error

Example Question #465 : Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's surface area to the rate of growth of its diagonal when its sides have length $7\sqrt{2}$ ?

Possible Answers:

$28\sqrt{2}$

$7\sqrt{6}$

$14\sqrt{2}$

$14\sqrt{3}$

$28\sqrt{6}$

Correct answer:

$28\sqrt{6}$

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area and diagonal in terms of the length of its sides:

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(7\sqrt{2})\sqrt{3}=28\sqrt{6}$

Report an Error

Example Question #466 : Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's surface area to the rate of growth of its diagonal when its sides have length 11?

Possible Answers:

$44\sqrt{3}$

$11\sqrt{3}$

$11\sqrt{2}$

$44\sqrt{6}$

$22\sqrt{6}$

Correct answer:

$44\sqrt{3}$

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area and diagonal in terms of the length of its sides:

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(11)\sqrt{3}=44\sqrt{3}$

Report an Error

Example Question #467 : Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's surface area to the rate of growth of its diagonal when its sides have length $5\sqrt{6}$ ?

Possible Answers:

$30\sqrt{3}$

$20\sqrt{6}$

$30\sqrt{2}$

$60\sqrt{3}$

$60\sqrt{2}$

Correct answer:

$60\sqrt{2}$

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area and diagonal in terms of the length of its sides:

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(5\sqrt{6})\sqrt{3}=60\sqrt{2}$

Report an Error

Example Question #468 : Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its diagonal when its sides have length $7\sqrt{3}$ ?

Possible Answers:

$84\sqrt{3}$

$28\sqrt{3}$

$28\sqrt{6}$

Correct answer:

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area and diagonal in terms of the length of its sides:

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(7\sqrt{3})\sqrt{3}=84$

Report an Error

Example Question #469 : Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its diagonal when its sides have length $\frac{\sqrt{3}}{2}$ ?

Possible Answers:

$\frac{3}{2}$

$2\sqrt{3}$

Correct answer:

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area and diagonal in terms of the length of its sides:

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(\frac{\sqrt{3}}{2})\sqrt{3}=6$

Report an Error

Example Question #470 : Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its diagonal when its sides have length 103?

Possible Answers:

$412\sqrt{6}$

$412\sqrt{3}$

$412\sqrt{2}$

$206\sqrt{3}$

$206\sqrt{6}$

Correct answer:

$412\sqrt{3}$

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area and diagonal in terms of the length of its sides:

A=6s^2

$d=s\sqrt{3}$

The rates of change of these can be found by taking the derivative of each side of the equations with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=4s\sqrt{3}$

$\phi =4(103)\sqrt{3}=412\sqrt{3}$

Report an Error

View Calculus Tutors

Dalton
Certified Tutor

University of Massachusetts Amherst, Doctor of Philosophy, Mathematics.

View Calculus Tutors

Vicky
Certified Tutor

Clemson University, Doctor of Science, Civil Engineering.

View Calculus Tutors

Mehak
Certified Tutor

Guru Nanak Dev University, Bachelor of Science, Chemistry. Guru Nanak Dev University, Master of Science, Chemistry.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Reading Tutors in New York City, Math Tutors in Chicago, Computer Science Tutors in Washington DC, GMAT Tutors in San Diego, MCAT Tutors in Atlanta, Statistics Tutors in Boston, ACT Tutors in Miami, SAT Tutors in Dallas Fort Worth, Computer Science Tutors in Chicago, GMAT Tutors in New York City

Popular Courses & Classes

Spanish Courses & Classes in Chicago, ACT Courses & Classes in Miami, GRE Courses & Classes in Phoenix, MCAT Courses & Classes in Phoenix, LSAT Courses & Classes in Denver, SSAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Denver, SSAT Courses & Classes in Boston, LSAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Miami

Popular Test Prep

LSAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Denver, ACT Test Prep in Denver, ISEE Test Prep in Boston, ISEE Test Prep in Houston, SSAT Test Prep in Chicago, ACT Test Prep in Washington DC, GRE Test Prep in Phoenix, MCAT Test Prep in Boston, MCAT Test Prep in Philadelphia

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 1 : Rate of Change

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #461 : Rate Of Change

Example Question #462 : Rate Of Change

Example Question #463 : Rate Of Change

Example Question #464 : Rate Of Change

Example Question #465 : Rate Of Change

Example Question #466 : Rate Of Change

Example Question #467 : Rate Of Change

Example Question #468 : Rate Of Change

Example Question #469 : Rate Of Change

Example Question #470 : Rate Of Change

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: