We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Taylor Series

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #41 : Taylor And Maclaurin Series

Write the first three terms of the Taylor series about a= $2\pi$ for the following function:

$f(x)=e^{3x}+\cos(x)$

Possible Answers:

$0+(e^{6\pi}+1)+3e^{6\pi}(x-2\pi)$

$(e^{6\pi}+1)+3e^{6\pi}(x-2\pi)+\frac{(9e^{6\pi}-1)(x-2\pi)^2}{2}$

$(e^{2\pi}+1)+3e^{2\pi}(x-2\pi)+\frac{(9e^{2\pi}-1)(x-2\pi)^2}{2}$

$(e^{6\pi}+1)+3e^{6\pi}(x-2\pi)+(9e^{6\pi}-1)(x-2\pi)^2$

Correct answer:

$(e^{6\pi}+1)+3e^{6\pi}(x-2\pi)+\frac{(9e^{6\pi}-1)(x-2\pi)^2}{2}$

Explanation:

The Taylor series about x=a for a function is given by

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

So, for the first three terms (n=0, 1, 2), we must find the zeroth, first, and second derivatives. The zeroth derivative is just the function itself.

$f'(x)=3e^{3x}-\sin(x)$

$f''(x)=9e^{3x}-\cos(x)$

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} e^{ax}=ae^{ax}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ $\frac{\mathrm{d} }{\mathrm{d} x} \sin(x)=\cos(x)$

Now, using the above formula, we can write out the first three terms:

$(e^{6\pi}+1)\frac{(x-2\pi)^0}{0!}+3e^{6\pi}\frac{(x-2\pi)^1}{1!}+(9e^{6\pi}-1)\frac{(x-2\pi)^2}{2!}$

which simplifies to

$(e^{6\pi}+1)+3e^{6\pi}(x-2\pi)+\frac{(9e^{6\pi}-1)(x-2\pi)^2}{2}$

Report an Error

Example Question #287 : Series In Calculus

Find the third degree Maclaurin Polynomial of the function

$f(x)=e^{3x}$

Possible Answers:

P(x)=1-3x+9x^2-27x^3

$P(x)=3+6x+\frac{9x^2}{2}+\frac{12x^3}{3!}$

$P(x)=1-3x+\frac{9x^2}{2}-\frac{27x^3}{3!}$

$P(x)=1+3x+\frac{9x^2}{2}+\frac{27x^3}{3!}$

Correct answer:

$P(x)=1+3x+\frac{9x^2}{2}+\frac{27x^3}{3!}$

Explanation:

The formula for the Maclaurin Series of a function is defined as follows

$P(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)x^n}{n!}$ where $f^{(n)}(0)$ is the n-th derivative when x=0

For the third degree polynomial we solve find the sum with the upper bound n=3

$\sum_{n=0}^{3}\frac{f^{(n)}(0)x^n}{n!}$

First we evaluate

$f(0)=e^{3(0)}=1$

We must also solve for the first, second, and third derivative of f(x)

$f'(x)=3e^{3x}$

$f''(x)=9e^{3x}$

$f^{(3)}(x)=27e^{3x}$

When x=0 we find the derivative values of

$f'(0)=3e^{3(0)}=3$

$f''(0)=9e^{3(0)}=9$

$f^{(3)}(0)=27e^{3(0)}=27$

Using these values we find the third degree Maclaurin Polynomial to be

$P(x)=1+3x+\frac{9x^2}{2}+\frac{27x^3}{3!}$

Report an Error

Example Question #288 : Series In Calculus

Find the Taylor series about a=7 for the following function:

$f(x)=x\ln x$

Possible Answers:

$7\ln 7+\ln7 +1 + \sum_{n=0}^{\infty} (-1)^n(n-2)(n-1)(7)^{-n+1}(x-7)^n$

$7\ln 7+\ln7 +1 + \sum_{n=0}^{\infty} (-1)^n(n-2)(n-1)(7)^{-n+1}(x-7)^{n+1}$

$0+0+0+ \sum_{n=0}^{\infty} (-1)^n(n-2)(n-1)(7)^{-n+1}(x-7)^n$

$\sum_{n=0}^{\infty} (-1)^n(n-2)(n-1)(7)^{-n+1}(x-7)^n$

Correct answer:

$7\ln 7+\ln7 +1 + \sum_{n=0}^{\infty} (-1)^n(n-2)(n-1)(7)^{-n+1}(x-7)^n$

Explanation:

The Taylor series about x=a for any function is given by

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

We must take the nth derivative of the given function and determine the trend as n goes to infinity. To determine this, we start at n=0 (the zeroth derivative, or the function itself), and go further:

$f^0(x)=x\ln x$

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

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

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

$f^{(4)}(x)=2x^{-3}$

$f^{(5)}(x)=-6x^{-4}$

The derivatives were found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)\cdot f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x}\ln(x)=\frac{1}{x}$

We must now find a pattern for the derivatives we took. The zeroth and first derivatives do not follow a pattern, but the derivatives after do follow a pattern: the sign alternates, the power of x decreases by 1, and the coefficient is the factorial of (n-2) starting at the n=2 derivative.

The derivative can then be expressed as

$f^{(n)}(a)=(-1)^n(n-2)!(x)^{-n+1}$

for n greater than or equal to 2.

For the Taylor series itself, we must evaluate the derivative at x=a=7. When we do this, and write the pattern elements for the derivatives past n=1, we get

$7\ln 7+\ln7 +1 + \sum_{n=0}^{\infty} (-1)^n(n-2)(n-1)(7)^{-n+1}(x-7)^n$

Report an Error

Example Question #41 : Taylor Series

Find an expression for the Taylor Series of $e^{2x}$ .

Possible Answers:

$\sum_{n=0}^\infty 2^n\frac{x^{2n}}{n!}$

$\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}$

$\sum_{n=0}^\infty 2\frac{x^{2n}}{n!}$

$\sum_{n=0}^\infty 2^n\frac{x^{2}}{n!}$

$\sum_{n=0}^\infty \frac{x^{2n}}{2(n!)}$

Correct answer:

$\sum_{n=0}^\infty 2^n\frac{x^{2n}}{n!}$

Explanation:

To obtain the Taylor Series for $e^{2x}$ , we start with the Taylor Series for $e^{x} = \sum_{n=0}^\infty \frac{x^n}{n!}$ , and substitute on both sides with , and simplify.

$e^{2x} = \sum_{n=0}^\infty \frac{(2x)^n}{n!} = \sum_{n =0}^\infty \frac{2^nx^n}{n!}=\sum_{n=0}^\infty2^n\frac{x^n}{n!}$ .

Report an Error

Example Question #3071 : Calculus Ii

Find the expression for the Taylor Series of $x^2\sin(x^2)$ .

Possible Answers:

$\sum_{n=0}^\infty\frac{(-1)^nx^{n+3}}{n!}$

$\sum_{n=0}^\infty\frac{2^{n+1}x^{n+3}}{(n+1)!}$

$\sum_{n=0}^\infty\frac{(-1)^n2^{n+1}x^{n+1}}{(n+1)!}$

$\sum_{n=0}^\infty\frac{(-1)^n2^{n+1}x^{n+3}}{n+1}$

$\sum_{n=0}^\infty\frac{(-1)^nx^{2n+4}}{(n+1)!}$

Correct answer:

$\sum_{n=0}^\infty\frac{(-1)^nx^{2n+4}}{(n+1)!}$

Explanation:

To obtain the Taylor Series for $x^2\sin(x^2)$ , we start with the Taylor Series for $\sin(x) = \sum_{n=0}^\infty \frac{(-1)^nx^{n+1}}{(n+1)!}$ .

$\sin(x^2) = \sum_{n=0}^\infty \frac{(-1)^n(x^2)^{n+1}}{(n+1)!}$ . (Substitute with x^2 on both sides)

$x^2\sin(x^2) = x^2\sum_{n=0}^\infty \frac{(-1)^n(x^2)^{n+1}}{(n+1)!}$ . (Multiply both sides by x^2 ,

$= \sum_{n=0}^\infty x^2\frac{(-1)^nx^{2n+2}}{(n+1)!} =\sum_{n=0}^\infty\frac{(-1)^nx^{2n+4}}{(n+1)!}$ .

Report an Error

Example Question #42 : Taylor Series

Use Taylor Expansion of around x=0 and determine the value of y''

Possible Answers:

$y''=\sum_{2}^{\infty} (n)(n-1)\frac{y^{(n)}(0)}{n!}(x)^{n-2}$

$y''=\sum_{0}^{\infty} (n)(n-1)\frac{y^{(n)}(0)}{n!}(x)^{n-2}$

$y''=\sum_{2}^{\infty} (n)(n-2)\frac{y^{(n)}(0)}{n!}(x)^{n-2}$

$y''=\sum_{-2}^{\infty} (n)(n-1)\frac{y^{(n)}(0)}{n!}(x)^{n+2}$

$y''=\sum_{1}^{\infty} (n)(n-1)\frac{y^{(n)}(0)}{n!}(x)^{n-1}$

Correct answer:

$y''=\sum_{2}^{\infty} (n)(n-1)\frac{y^{(n)}(0)}{n!}(x)^{n-2}$

Explanation:

We can write as a Mclaurin series by saying that:

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

Since y is just a polynomial expression:

$y''=\sum_{2}^{\infty} (n)(n-1)\frac{y^{(n)}(0)}{n!}(x)^{n-2}$

The reason we must bring the lower bound to 2 is because Taylor Series can ONLY be written as a summation of polynomials and no rational components.

Report an Error

View Calculus 2 Tutors

Mohammad
Certified Tutor

Rutgers University-New Brunswick, Bachelor of Science, Mathematics. Rutgers University-New Brunswick, Master of Science, Stat...

View Calculus 2 Tutors

Jason
Certified Tutor

Brigham Young University, Bachelor, Neuroscience.

View Calculus 2 Tutors

Reza
Certified Tutor

University of Sherbrooke, Doctor of Philosophy, Mathematics. University of Manitoba, Master of Science, Mathematics.

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

French Tutors in Philadelphia, Physics Tutors in Los Angeles, Math Tutors in Los Angeles, GMAT Tutors in New York City, SAT Tutors in Houston, SAT Tutors in San Francisco-Bay Area, SAT Tutors in New York City, Computer Science Tutors in Los Angeles, ACT Tutors in Los Angeles, LSAT Tutors in Atlanta

Popular Courses & Classes

GMAT Courses & Classes in Houston, ISEE Courses & Classes in Chicago, SAT Courses & Classes in Philadelphia, Spanish Courses & Classes in Los Angeles, GMAT Courses & Classes in Seattle, MCAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Houston, ISEE Courses & Classes in Atlanta, SSAT Courses & Classes in Boston, GRE Courses & Classes in San Francisco-Bay Area

Popular Test Prep

GMAT Test Prep in Chicago, GMAT Test Prep in Atlanta, GMAT Test Prep in Boston, MCAT Test Prep in San Diego, LSAT Test Prep in San Diego, ISEE Test Prep in Chicago, SSAT Test Prep in Chicago, MCAT Test Prep in Philadelphia, ACT Test Prep in Los Angeles, GRE Test Prep in Boston

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 2 : Taylor Series

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #41 : Taylor And Maclaurin Series

Example Question #287 : Series In Calculus

Example Question #288 : Series In Calculus

Example Question #41 : Taylor Series

Example Question #3071 : Calculus Ii

Example Question #42 : Taylor Series

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: