Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : 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 : How To Find 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 #1 : How To Find Prediction Models

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 : How To Find 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:

-6e^2

12e^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

James
Certified Tutor

St. Mary of the Plains College, Bachelors, Mathematics. Baker University, Masters, Business Administration and Management.

View Calculus Tutors

Satweka
Certified Tutor

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

View Calculus Tutors

Anjali
Certified Tutor

Delhi University, Bachelor of Science, Physics. Kumuan University, Master of Science, Physics.

All Calculus 1 Resources

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

Popular Subjects

GRE Tutors in Boston, Computer Science Tutors in San Diego, MCAT Tutors in Phoenix, Math Tutors in San Francisco-Bay Area, Biology Tutors in San Francisco-Bay Area, GRE Tutors in Seattle, GRE Tutors in New York City, LSAT Tutors in Seattle, SSAT Tutors in Denver, Reading Tutors in New York City

Popular Courses & Classes

SSAT Courses & Classes in Washington DC, GMAT Courses & Classes in Miami, Spanish Courses & Classes in Washington DC, LSAT Courses & Classes in Boston, MCAT Courses & Classes in Miami, Spanish Courses & Classes in Los Angeles, SAT Courses & Classes in Seattle, SAT Courses & Classes in Boston, LSAT Courses & Classes in Philadelphia, MCAT Courses & Classes in Denver

Popular Test Prep

SAT Test Prep in Seattle, MCAT Test Prep in Denver, MCAT Test Prep in Washington DC, SSAT Test Prep in Washington DC, LSAT Test Prep in New York City, ISEE Test Prep in Seattle, MCAT Test Prep in Phoenix, SAT Test Prep in Houston, SAT Test Prep in New York City, MCAT Test Prep in New York City

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 : Prediction Models

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : How To Find Prediction Models

Example Question #1 : How To Find Prediction Models

Example Question #2 : How To Find Prediction Models

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: