Call Now to Set Up Tutoring:

(888) 888-0446

GRE Subject Test: Math : Taylor's Theorem

Study concepts, example questions & explanations for GRE Subject Test: Math

Create An Account

All GRE Subject Test: Math Resources

2 Diagnostic Tests 148 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Taylor And Maclaurin Series

For which of the following functions can the Maclaurin series representation be expressed in four or fewer non-zero terms?

Possible Answers:

$f(x)= \frac{x^{5}}{3}+2$

$f(x)=\ln|x+3|$

$f(x)=x+3\sin(x)$

$f(x)= x^{2}+\sqrt{x}+1$

Correct answer:

$f(x)= \frac{x^{5}}{3}+2$

Explanation:

Recall the Maclaurin series formula:

$f(x)=\sum_{n=0}^{\infty}\frac{f^n(0)}{n!}x^n$

$=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+...$

Despite being a 5th degree polynomial recall that the Maclaurin series for any polynomial is just the polynomial itself, so this function's Taylor series is identical to itself with two non-zero terms.

The only function that has four or fewer terms is $f(x)= \frac{x^{5}}{3}+2$ as its Maclaurin series is $2+\frac{1}{3}x^5$ .

Report an Error

Example Question #1 : Taylor And Maclaurin Series

Let $f(x)=(1+\frac{x}{2})^{1/3}$

Find the the first three terms of the Taylor Series for centered at .

Possible Answers:

$T_3=1+\frac{x}{3}+4x^2$

$T_{3}=0+0+0$

$T_{2}=0+\frac{x}{6} -\frac{x^2}{18}$

$T_{2}(x)=1+\frac{x}{6}+\frac{x^2}{36}$

$T_{2}(x)=1+\frac{x}{6}-\frac{x^2}{36}$

Correct answer:

$T_{2}(x)=1+\frac{x}{6}-\frac{x^2}{36}$

Explanation:

Using the formula of a binomial series centered at 0:

$(1+x)^k =\sum_{n=0}^{\infty } \binom{k}{n}x^n$ ,

where we replace with $\frac{x}{2}$ and $k=\frac{1}{3}$ , we get:

$\sum_{n=0}^{2}\binom{\frac{1}{3}}{n}\left(\frac{x}{2}\right)^n$ for the first 3 terms.

Then, we find the terms where,

$T_2(x)=\binom{\frac{1}{3}}{0}(\frac{x}{2})^0 + \binom{\frac{1}{3}}{1}(\frac{x}{2})^1+\binom{\frac{1}{3}}{2}(\frac{x}{2})^2=1+\frac{x}{6}- \frac{x^2}{36}$

Report an Error

Example Question #1 : Taylor's Theorem

Determine the convergence of the Taylor Series for at x=5 where $f(x)=\sum_{n=0}^{\infty }\frac{x^n}{n!}$ .

Possible Answers:

Conditionally Convergent.

Inconclusive.

Divergent.

Does not exist.

Absolutely Convergent.

Correct answer:

Absolutely Convergent.

Explanation:

By the ratio test, the series $\sum_{n=0}^{\infty }\frac{5^n}{n!}$ converges absolutely:

$\lim_{n\rightarrow \infty }\left | \frac{a_{n+1}}{a_n} \right |=\lim_{n\rightarrow \infty }\left |\frac{5^{n+1}}{(n+1)!} \cdot \frac{n!}{5^n} \right |=\lim_{n\rightarrow \infty }\left | \frac{5}{n+1} \right |=0<1$

Report an Error

Example Question #1 : Taylor Series

Find the interval of convergence for of the Taylor Series $\sum_{n=0}^{\infty }n!(x-5)^n$ .

Possible Answers:

$\left ( \frac{-1}{5}, \frac{1}{5} \right )$

x=0

$\left ( -5, 5\right )$

$(-\infty, \infty)$

x=5

Correct answer:

x=5

Explanation:

Using the root test

$\lim_{n\rightarrow \infty }\left | \frac{a_{n+1}}{a_n} \right |=\lim_{n\rightarrow \infty }\left |\frac{(n+1)!(x-5)^{n+1}}{n!(x-5)^n} \right |=\left |x-5 \right |\lim_{n\rightarrow \infty }\left | n+1 \right |$

and

$\lim_{n\rightarrow \infty }\left | n+1 \right |=\infty$ . T

herefore, the series only converges when it is equal to zero.

This occurs when x=5.

Report an Error

Example Question #1 : Taylor And Maclaurin Series

Suppose that the derivative of a function, denoted f'(x) , can be approximated by the third degree Taylor polynomial, centered at :

$f'(x) \approx 1 + (x-2) + \frac{4}{3} (x-2)^2 + \frac{8}{9}(x-2)^3$

If f(2) = 7 , find the third degree Taylor polynomial for centered at .

Possible Answers:

$7 + x + (x-2)^2 + \frac{4}{9}(x-2)^3$

$(x-2) + (x-2)^2 + \frac{4}{9}(x-2)^3 + \frac{8}{27}(x-2)^4$

$7 + (x-2) + (x-2)^2 + \frac{4}{9}(x-2)^3$

$(x-2) + 2(x-2)^2 + {4}(x-2)^3$

7 + (x-2) + 2(x-2)^2 + 4(x-2)^4

Correct answer:

$7 + (x-2) + (x-2)^2 + \frac{4}{9}(x-2)^3$

Explanation:

To get f(x) , we need to find the antiderivative of f'(x) by integrating the third degree polynomial term by term.

We only want up to a third degree polynomial, so we can disregard the fourth order term:

$f(x) = c + (x-2) + (x-2)^2 + \frac{4}{9}(x-2)^3$

Since f(2) = 7 , substitute c=7 for the final f(x) .

$f(x)=7 + (x-2) + (x-2)^2 + \frac{4}{9}(x-2)^3$

Report an Error

Example Question #1 : Taylor Series

Write out the first four terms of the Taylor series about a=1 for the following function:

f(x)=x^2+2x+1

Possible Answers:

4+4(x-1)+(x-1)^2+0

0+4+4(x-1)+(x-1)^2

4+4(x-1)^2+(x-1)^3+0

0+0+0+0

Correct answer:

4+4(x-1)+(x-1)^2+0

Explanation:

The Taylor series about x=a of any function is given by the following:

$\sum_{n=0}^{\infty }f^{(n)}(a)\frac{(x-a)^n}{n!}$

So, we must find the zeroth, first, second, and third derivatives of the function (for n=0, 1, 2, and 3 which makes the first four terms):

$f^{0}(x)=f(x)$

f'(x)=2x+2

f''(x)=2

f'''(x)=0

The derivatives were found using the following rule:

$\frac{d}{dx}(x^n)=nx^{n-1}$

Now, evaluated at x=a=1, and plugging in the correct n where appropriate, we get the following:

$\frac{4(x-1)^0}{0!}+\frac{4(x-1)^1}{1!}+\frac{2(x-1)^2}{2!}+\frac{0(x-1)^3}{3!}$

which when simplified is equal to

4+4(x-1)+(x-1)^2+0 .

Report an Error

Example Question #5 : Taylor And Maclaurin Series

Find the first two terms of the Taylor series about a=3 for the following function:

f(x)=2x+5

Possible Answers:

11+7(x-3)

11+2(x-3)

$\infty +2(x-3)$

11-2(x-3)

Correct answer:

11+2(x-3)

Explanation:

The general formula for the Taylor series about x=a for a function is

$\sum_{n=0}^{\infty }f^{(n)}(a)\frac{(x-a)^n}{n!}$

First, we must find the zeroth and first derivative of the function.

The zeroth derivative of a function is just the function itself, so we only have to find the first derivative:

f'(x)=2

The derivative was found using the following rule:

$\frac{d}{dx}(x^n)=nx^{n-1}$

Now, write the first two terms of the sequence (n=0 and n=1):

$\frac{[2(3)+5](x-3)^0}{0!}+\frac{2(x-3)^1}{1!}=11+2(x-3)$

Report an Error

Example Question #6 : Taylor And Maclaurin Series

Write out the first three terms of the Taylor series for the following function about a=1 :

f(x)=2e^x+2

Possible Answers:

0+0+0

(2e^1+2)+2e^1(x-1)+e^1(x-1)^2

$0+(x-1)+\frac{(x-1)^2}{2}$

0+2e^1(x-1)+2e^1

Correct answer:

(2e^1+2)+2e^1(x-1)+e^1(x-1)^2

Explanation:

The general formula for the Taylor series of a given function about x=a is

$\sum_{n=0}^{\infty }f^{(n)}(a)\frac{(x-a)^n}{n!}$ .

We were asked to find the first three terms, which correspond to n=0, 1, and 2. So first, we need to find the zeroth, first, and second derivative of the given function. The zeroth derivative is just the function itself.

f'(x)=2e^x

f''(x)=2e^x

The derivatives were found using the following rules:

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

Now use the above formula to write out the first three terms:

$\frac{(2e^1+2)(x-1)^0}{0!}+\frac{(2e^1)(x-1)^1}{1!}+\frac{(2e^1)(x-1)^2}{2!}$

Simplified, this becomes

(2e^1+2)+2e^1(x-1)+e^1(x-1)^2

Report an Error

View Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

View Tutors

Samarth
Certified Tutor

Maharaja Sayajirao University of Baroda, Bachelor of Engineering, Mechanical Engineering. Concordia University-Seward, Master...

View Tutors

Max
Certified Tutor

SHAHiDBEHESHTiUNiVERSiTY, Bachelor of Science, Mathematics. SHAHiDBEHESHTiUNiVERSiTY, Master of Science, Mathematics.

All GRE Subject Test: Math Resources

2 Diagnostic Tests 148 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

MCAT Tutors in Miami, Spanish Tutors in Philadelphia, Algebra Tutors in Philadelphia, Chemistry Tutors in Denver, MCAT Tutors in Houston, Math Tutors in Washington DC, Spanish Tutors in Miami, Math Tutors in Atlanta, Spanish Tutors in San Diego, LSAT Tutors in Miami

Popular Courses & Classes

Spanish Courses & Classes in Los Angeles, SSAT Courses & Classes in Washington DC, LSAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Miami, GRE Courses & Classes in Denver, GMAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in San Diego, SSAT Courses & Classes in Seattle, SSAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Los Angeles

Popular Test Prep

GRE Test Prep in Dallas Fort Worth, SAT Test Prep in New York City, MCAT Test Prep in Dallas Fort Worth, MCAT Test Prep in San Diego, GMAT Test Prep in Dallas Fort Worth, GRE Test Prep in Washington DC, SSAT Test Prep in Dallas Fort Worth, SAT Test Prep in Atlanta, ACT 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

GRE Subject Test: Math : Taylor's Theorem

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Taylor And Maclaurin Series

Example Question #1 : Taylor And Maclaurin Series

Example Question #1 : Taylor's Theorem

Example Question #1 : Taylor Series

Example Question #1 : Taylor And Maclaurin Series

Example Question #1 : Taylor Series

Example Question #5 : Taylor And Maclaurin Series

Example Question #6 : Taylor And Maclaurin Series

All GRE Subject Test: Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: