Call Now to Set Up Tutoring:

(888) 888-0446

High School Math : Taylor and Maclaurin Series

Study concepts, example questions & explanations for High School Math

Create An Account

All High School Math Resources

8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2243 : High School Math

Give the $x^{2}$ term of the Maclaurin series of the function $f \left ( x\right ) = \frac{1}{x-2}$

Possible Answers:

$-\frac{1}{4} x^{2}$

$-\frac{1}{12} x^{2}$

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

$-\frac{1}{6} x^{2}$

$-\frac{1}{8} x^{2}$

Correct answer:

$-\frac{1}{8} x^{2}$

Explanation:

The $x^{2}$ term of the Maclaurin series of a function has coefficient $\frac{f '' (0) }{2!} = \frac{f '' (0) }{2}$

The second derivative of can be found as follows:

$f \left ( x\right ) = \frac{1}{x-2} = (x-2) ^{-1}$

$f' \left ( x\right ) = -1\cdot (x-2)^{-2} = -1(x-2)^{-2}$

$f'' \left ( x\right ) = (-2) ( -1(x-2)^{-3}) = 2(x-2)^{-3} = \frac{2}{(x-2)^{3}}$

$f'' \left ( 0 \right ) =\frac{2}{(0-2)^{3}} = -\frac{1}{4}$

The coeficient of $x^{2}$ in the Maclaurin series is therefore

$\frac{f '' (0) }{2} = \frac{-\frac{1}{4}}{2} = - \frac{1}{8}$

Report an Error

Example Question #2242 : High School Math

Give the (x-1)^2 term of the Taylor series expansion of the function $f (x) = \log_5 x$ about x = 1 .

Possible Answers:

$- \frac{1}{ \ln 25 } (x-1)^{2}$

$\frac{\ln 5}{ 2 } (x-1)^{2}$

$\frac{2}{ \ln 5 } (x-1)^{2}$

$\frac{1}{ \ln 25 } (x-1)^{2}$

$\ln 25\cdot (x-1)^{2}$

Correct answer:

$- \frac{1}{ \ln 25 } (x-1)^{2}$

Explanation:

The $(x-1)^{2}$ term of a Taylor series expansion about x = 1 is

$\frac{f ' ' (1)}{2!} (x-1)^{2}= \frac{f ' ' (1)}{2}(x-1)^{2}$ .

We can find f''(x) by differentiating twice in succession:

$f (x) = \log_5 x$

$f'(x) = \frac{1}{x \ln 5}$

$f'(x) =\frac{1}{ \ln 5} \cdot x^{-1}$

$f''(x) = \frac{\mathrm{d} }{\mathrm{d} x}\left (\frac{1}{ \ln 5} \cdot x^{-1} \right )$

$f''(x) =\frac{1}{ \ln 5}\cdot \frac{\mathrm{d} }{\mathrm{d} x}\left ( x^{-1} \right )$

$f''(x) =\frac{1}{ \ln 5} \cdot \left ( -1 \cdot x^{-2} \right )$

$f''(x) =- \frac{1}{ \ln 5 \cdot x^{2}}$

$f''(1) =- \frac{1}{ \ln 5 \cdot 1^{2}} = - \frac{1}{ \ln 5 }$

so the (x-1)^2 term is $\frac{f''(1) }{2} \; (x-1)^{2} = - \frac{1}{2 \ln 5 } (x-1)^{2} = - \frac{1}{ \ln 25 } (x-1)^{2}$

Report an Error

Example Question #2243 : High School Math

Give the $x^{4}$ term of the Maclaurin series expansion of the function $f(x) = x \tan^{-1} x$ .

Possible Answers:

$- \frac{1}{12} x^{4 }$

$- \frac{1}{3} x^{4 }$

$- \frac{3}{4} x^{4 }$

$- \frac{1}{4} x^{4 }$

Correct answer:

$- \frac{1}{3} x^{4 }$

Explanation:

This can most easily be answered by recalling that the Maclaurin series for $\tan^{-1} x$ is

$\tan^{-1} x = x - \frac{1}{3} x^{3 } + \frac{1}{5} x^{5 }...$

Multiply by to get:

$x\tan^{-1} x = x\cdot x - x\cdot \frac{1}{3} x^{3 } + x\cdot \frac{1}{5} x^{5 }...$

$x \tan^{-1} x=x^{2 } - \frac{1}{3} x^{4 } + \frac{1}{5} x^{6 }...$

The $x^{4}$ term is therefore $- \frac{1}{3} x^{4 }$ .

Report an Error

Example Question #2244 : High School Math

Give the $x^ {3}$ term of the Maclaurin series of the function $f(x) = \sinh 3x$ .

Possible Answers:

$-9x^{3}$

$-\frac{9}{2} x^{3}$

$\frac{9}{2} x^{3}$

$9x^{3}$

Correct answer:

$\frac{9}{2} x^{3}$

Explanation:

The $x^ {3}$ term of a Maclaurin series expansion has coefficient

$\frac{f ' '' (0)}{3!} = \frac{f ' '' (0)}{6}$ .

We can find f'''(x) by differentiating three times in succession:

$f(x) = \sinh 3x$

$f'(x) = 3 \cosh 3x$

$f''(x) = 3\cdot 3\sinh 3x = 9\sinh 3x$

$f'''(x) = 9 \cdot 3\cosh 3x = 27 \cosh 3x$

$f'''(0) = 27 \cosh \left (3 \cdot 0 \right ) = 27 \cosh 0 = 27\cdot 1 = 27$

The term we want is therefore

$\frac{f ' '' (0)}{6} \cdot x^{3} = \frac{27}{6} \cdot x^{3} = \frac{9}{2} x^{3}$

Report an Error

Example Question #1 : Taylor And Maclaurin Series

Give the $x^{2}$ term of the Maclaurin series expansion of the function $f (x) = 7 ^{x}$ .

Possible Answers:

$\frac{7}{2}x^{2}$

$\left (2 \ln 7 \right )x^{2}$

$\frac{e^{7}}{2}x^{2}$

$\frac{\left ( \ln 7 \right )^{2}}{2} x^{2}$

$\frac{ \ln 7}{2}x^{2}$

Correct answer:

$\frac{\left ( \ln 7 \right )^{2}}{2} x^{2}$

Explanation:

The $x^{2}$ term of a Maclaurin series expansion has coefficient

$\frac{f ' ' (0)}{2!} = \frac{f ' ' (0)}{2}$ .

We can find f''(x) by differentiating twice in succession:

$f (x) = 7^{x}$

$f ' (x) = \frac{\mathrm{d} }{\mathrm{d} x} \left ( 7^{x} \right )= \ln 7 \cdot 7^{x}$

$f ' '(x) = \frac{\mathrm{d} }{\mathrm{d} x} \left (\ln 7 \cdot 7^{x} \right )$

$f ' '(x)= \ln 7\cdot \frac{\mathrm{d} }{\mathrm{d} x} \left ( 7^{x} \right )$

$f ' '(x)= \ln 7\cdot \ln 7\cdot 7^{x}$

$f ' '(x)=\left ( \ln 7 \right )^{2} \cdot 7^{x}$

$f ' '(0)=\left ( \ln 7 \right )^{2} \cdot 7^{0} = \left ( \ln 7 \right )^{2} \cdot 1 = \left ( \ln 7 \right )^{2}$

The coefficient we want is

$\frac{f ' ' (0)}{2} =\frac{\left ( \ln 7 \right )^{2}}{2}$ ,

so the corresponding term is $\frac{\left ( \ln 7 \right )^{2}}{2} x^{2}$ .

Report an Error

View Tutors

Sydni
Certified Tutor

Louisiana Tech University, Bachelor of Science, Finance. Louisiana Tech University, Master of Arts Teaching, Mathematics.

View Tutors

Jeff
Certified Tutor

Cleveland State University, Bachelor of Science, Applied Mathematics. Cleveland State University, Master of Science, Mathemat...

View Tutors

Jean Pierre
Certified Tutor

Mountain View College, Associate in Arts, Arts, General.

All High School Math Resources

8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

French Tutors in Houston, French Tutors in Philadelphia, Chemistry Tutors in Houston, Physics Tutors in Chicago, SSAT Tutors in Philadelphia, Calculus Tutors in Miami, MCAT Tutors in San Diego, Reading Tutors in Miami, SSAT Tutors in Houston, ISEE Tutors in Philadelphia

Popular Courses & Classes

GRE Courses & Classes in Chicago, SAT Courses & Classes in San Diego, LSAT Courses & Classes in Houston, MCAT Courses & Classes in Chicago, GRE Courses & Classes in Houston, ISEE Courses & Classes in Seattle, GRE Courses & Classes in Seattle, SSAT Courses & Classes in Houston, MCAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

SAT Test Prep in Atlanta, MCAT Test Prep in San Diego, GRE Test Prep in Seattle, GMAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Houston, SSAT Test Prep in New York City, MCAT Test Prep in Seattle, MCAT Test Prep in New York City, SSAT Test Prep in Houston, GMAT 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

High School Math : Taylor and Maclaurin Series

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #2243 : High School Math

Example Question #2242 : High School Math

Example Question #2243 : High School Math

Example Question #2244 : High School Math

Example Question #1 : Taylor And Maclaurin Series

All High School Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: