We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : How to find prediction models

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 #1 : Prediction Models

Suppose you are a banker and set up a very unique function for your interest rate over time given by

$y=sec(\pi x)$

However, you find your computer incapable of calculating the interest rate at x=2.01 . Estimate the value of the interest rate at x=2.01 by using a linear approximation, using the slope of the function at x=2 .

Possible Answers:

Undefined

Correct answer:

Explanation:

To do a linear approximation, we're going to create a function

y_1=mz+b , that approximates our situation. In our case, m will be the slope of the function $sec(\pi x)$ at x=2 , while b will be the value of the function $sec(\pi x)$ at x=2 . The z will be distance from our starting position (x=2) to our end position (x=2.01) , which is 0.01 .

Firstly, we need to find the derivative of $sec(\pi x)$ with respect to x to determine slope.

By the power rule:

$\frac{\mathrm{d} }{\mathrm{d} x} (cos(\pi x))^{-1}= -(cos(\pi x))^{-2}*-\pi sin(\pi x)=(cos(\pi x))^{-2}*\pi sin(\pi x)$

The slope at x=2 will therefore be 0 since $sin(2\pi)=0$ .

Since this is the case, the approximate value of our interest rate will be identical to the value of the original function at x=2, which is $sec(2\pi)=1$ .

y_1=0*0.01+1=1

1 is our final answer.

Report an Error

Example Question #2684 : Functions

Approximate the value at x=0.02 of the function $y=3e^{-x}+sec(x)$ ,with a linear approximation using the slope of the function at x=0 .

Possible Answers:

3.94

-3.94

Correct answer:

3.94

Explanation:

To do this, we must determine the slope of the function at x=0 , which we will call , and the initial value of the function at x=0 , which we will call , and since x=0.02 is only away from x=0 , our linear approximation will look like:

y=m(.02)+b

To determine slope, we take the derivative of the function with respect to x and find its value at x=0 , which in our case is:

$\frac{\mathrm{d} }{\mathrm{d} x}[3e^{-x}+sec(x)]= -3e^{-x}-sec(x)*tan(x).$

At x=0 , our value for is

To determine , we need to determine the value of the original equation at x=0

$y=3e^{-x}+sec(x)$

At x=0 , our value for b is

Since y=m(.02)+b , y=3.94

Report an Error

Example Question #2 : Prediction Models

Determine the tangent line to $y=3e^{-x}$ at $x=ln(e^{-2})$ , and use the tangent line to approximate the value at x=ln(e^3) .

Possible Answers:

-12e^2

-6e^2

e^2

12e^2

Correct answer:

-12e^2

Explanation:

First recall that

$ln(e^{-2})=-2$

To find the tangent line of $y=3e^{-x}$ at x=-2 , we first determine the slope of $y=3e^{-x}$ . To do so, we must find its derivative.

Recall that derivatives of exponential functions involving are given as:

$\frac{\mathrm{d} }{\mathrm{d} x}Ae^{g(x)}= A*g'(x)e^{g(x)}$ , where is a constant and g(x) is any function of

In our case, A=3, g(x)=-x, g'(x)=-1 ,.

$\frac{d}{dx}3e^{-x}= -3e^{-x}$

At x=-2 ,

$\frac{\mathrm{d} y}{\mathrm{d} x}=-3e^{-(-2)}}= -3e^2=m$ , where is the slope of the tangent line.

To use point-slope form, we need to know the value of the original function at x=-2 ,

$y=3e^{-x}$

$y(-2)=3e^{-(-2)}=3e^2$

Therefore,

$(y_{appx}-3e^2)=-3e^2(x+2)$

At x=ln(e^3)=3 ,

$(y_{appx}-3e^2)=-3e^2(3+2)$

$y_{appx}=-15e^2+3e^2=-12e^2$

Report an Error

View Calculus Tutors

Eugene
Certified Tutor

San Jacinto Community College, Associate in Arts, Mathematics. University of Houston, Bachelor of Science, Mechanical Enginee...

View Calculus Tutors

Steven
Certified Tutor

University of Louisville, Bachelor of Science, Mechanical Engineering. University of Phoenix-Louisville Campus, Master of Eng...

View Calculus Tutors

Andrew
Certified Tutor

University of Wisconsin-Stevens Point, Bachelor of Science, Mathematics. University of Wisconsin-Oshkosh, Master of Science, ...

All Calculus 1 Resources

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

Popular Subjects

ACT Tutors in Phoenix, ACT Tutors in New York City, Reading Tutors in Dallas Fort Worth, Biology Tutors in New York City, Spanish Tutors in San Francisco-Bay Area, LSAT Tutors in Los Angeles, ACT Tutors in Boston, Biology Tutors in Dallas Fort Worth, Spanish Tutors in Dallas Fort Worth, SAT Tutors in Phoenix

Popular Courses & Classes

ISEE Courses & Classes in Chicago, GRE Courses & Classes in Miami, SAT Courses & Classes in Phoenix, ACT Courses & Classes in Philadelphia, Spanish Courses & Classes in Phoenix, LSAT Courses & Classes in Atlanta, LSAT Courses & Classes in San Diego, SSAT Courses & Classes in New York City, ISEE Courses & Classes in Denver, Spanish Courses & Classes in San Francisco-Bay Area

Popular Test Prep

SSAT Test Prep in Houston, GMAT Test Prep in Atlanta, MCAT Test Prep in Seattle, SAT Test Prep in Seattle, MCAT Test Prep in Atlanta, ISEE Test Prep in Dallas Fort Worth, SSAT Test Prep in Atlanta, ISEE Test Prep in Denver, GRE Test Prep in Los Angeles, MCAT Test Prep in Dallas Fort Worth

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 : How to find prediction models

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : Prediction Models

Example Question #2684 : Functions

Example Question #2 : Prediction Models

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: