Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus BC : AP Calculus BC

Study concepts, example questions & explanations for AP Calculus BC

Create An Account

All AP Calculus BC Resources

2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 3 4 5 6 7 8 9 … 49 50 Next →

Example Question #1 : Maclaurin Series

Use the first 3 terms of the Maclaurin series to approximate $\sin (\frac{\pi}{8})$

Possible Answers:

0.3827

0.3928

Correct answer:

0.3827

Explanation:

The Maclaurin series for sine is $\sum_{n=0}^{\infty } \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5! } - ...$

Plugging in $\frac{\pi }{ 8 }$ for x gives:

.3927 - .0101 + .0001 = .3827

Report an Error

Example Question #31 : Polynomial Approximations And Series

Use the first 4 terms of the Maclaurin series to approximate $e^{0.78}$

Possible Answers:

2.16329

1.78

1.3042

2.18147

Correct answer:

2.16329

Explanation:

The Maclaurin series for e^x is $\sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2 }{2! } + \frac{x^3}{3!} + ...$

Plugging in 0.78 for x gives:

1 + .78 + .3042 + .079092 = 2.163292

Report an Error

Example Question #1 : Maclaurin Series

Use the first four terms of the Maclaurin series to approximate $e^{0.38}$ to 6 decimal places.

Possible Answers:

1.461345

1.460782

1.462285

1.469385

Correct answer:

1.461345

Explanation:

The Maclaurin series for e^x is $\sum_{n=0}^{\infty } \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$

Plugging in x = 0.38 gives:

$1 + 0.38 + .0722 + .09145 \overline{3} = 1.461345$

Report an Error

Example Question #1 : Maclaurin Series

$\begin{align*}&\text{Approximate }f(1.2) \\&\text{Where }f(x)=-cos(x)\\&\text{By using a Maclaurin expansion to }3\text{ terms}\end{align*}$

Possible Answers:

-0.07

-0.28

2.52

1.40

Correct answer:

-0.28

Explanation:

$\begin{align*}&\text{The Maclaurin series is a special case of the Taylor series.}\\&\text{Like the Taylor series, it is used to approximate an unknown}\\&\text{value of a function at a given point, if values of the function}\\&\text{and its derivatives are known at another point. In the case}\\&\text{of the Maclaurin series, this latter point has the value of}\\&\text{zero (note that for many functions, such as the trigonometric}\\&\text{functions, f(0) is readily known):}\\&f(x)\approx\sum_{i=0}^{n}\frac{f^{(i)}(0)}{i!}x^i\approx f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+...+\frac{f^{(n)}(0)}{n!}x^n\\&\text{For the function }-cos(x)\text{ at }x=1.2\\&\text{This expansion becomes:}\\&f(1.2)\approx \frac{-cos((0))}{1}(1.2)^{0}+\frac{sin((0))}{1}(1.2)^{1}+\frac{cos((0))}{2}(1.2)^{2}\\&f(1.2)\approx-0.28\end{align*}$

Report an Error

Example Question #31 : Taylor Series

$\begin{align*}&\text{Approximate }f(4) \\&\text{Where }f(x)=-2^{(3x)}cos(9x)\\&\text{With a Taylor series centered at zero, using }3\text{ terms}\end{align*}$

Possible Answers:

151.02

-4228.63

604.09

-4832.72

Correct answer:

604.09

Explanation:

$\begin{align*}&\text{The Maclaurin series is a special case of the Taylor series.}\\&\text{Like the Taylor series, it is used to approximate an unknown}\\&\text{value of a function at a given point, if values of the function}\\&\text{and its derivatives are known at another point. In the case}\\&\text{of the Maclaurin series, this latter point has the value of}\\&\text{zero (note that for many functions, such as the trigonometric}\\&\text{functions, f(0) is readily known):}\\&f(x)\approx\sum_{i=0}^{n}\frac{f^{(i)}(0)}{i!}x^i\approx f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+...+\frac{f^{(n)}(0)}{n!}x^n\\&\text{For the function }-2^{(3x)}cos(9x)\text{ at }x=4\\&\text{This expansion becomes:}\\&f(4)\approx \frac{-2^{(3(0))}cos(9(0))}{1}(4)^{0}+\frac{9\cdot 2^{(3(0))}sin(9(0)) - 3\cdot 2^{(3(0))}cos(9(0))ln(2)}{1}(4)^{1}+\frac{81\cdot 2^{(3(0))}cos(9(0)) + 54\cdot 2^{(3(0))}sin(9(0))ln(2) - 9\cdot 2^{(3(0))}cos(9(0))ln(2)^{2}}{2}(4)^{2}\\&f(4)\approx604.09\end{align*}$

Report an Error

Example Question #31 : Polynomial Approximations And Series

$\begin{align*}&\text{Approximate }f(2.5) \\&\text{Where }f(x)=-cos(6x)e^{(3x)}\\&\text{With a Taylor series centered at zero, using }3\text{ terms}\end{align*}$

Possible Answers:

10.84

-607.00

-151.75

75.88

Correct answer:

75.88

Explanation:

$\begin{align*}&\text{The Maclaurin series is a special case of the Taylor series.}\\&\text{Like the Taylor series, it is used to approximate an unknown}\\&\text{value of a function at a given point, if values of the function}\\&\text{and its derivatives are known at another point. In the case}\\&\text{of the Maclaurin series, this latter point has the value of}\\&\text{zero (note that for many functions, such as the trigonometric}\\&\text{functions, f(0) is readily known):}\\&f(x)\approx\sum_{i=0}^{n}\frac{f^{(i)}(0)}{i!}x^i\approx f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+...+\frac{f^{(n)}(0)}{n!}x^n\\&\text{For the function }-cos(6x)e^{(3x)}\text{ at }x=2.5\\&\text{This expansion becomes:}\\&f(2.5)\approx \frac{-cos(6(0))e^{(3(0))}}{1}(2.5)^{0}+\frac{6sin(6(0))e^{(3(0))} - 3cos(6(0))e^{(3(0))}}{1}(2.5)^{1}+\frac{27cos(6(0))e^{(3(0))} + 36sin(6(0))e^{(3(0))}}{2}(2.5)^{2}\\&f(2.5)\approx75.88\end{align*}$

Report an Error

Example Question #31 : Polynomial Approximations And Series

$\begin{align*}&\text{Approximate }f(2.9) \\&\text{Where }f(x)=-3^{(4x)}cos(9x)\\&\text{With a Taylor series centered at zero, using }3\text{ terms}\end{align*}$

Possible Answers:

1228.29

245.66

736.97

2210.92

Correct answer:

245.66

Explanation:

$\begin{align*}&\text{The Maclaurin series is a special case of the Taylor series.}\\&\text{Similarly, it is used to approximate an unknown function value,}\\&\text{if values of the function and its derivatives are known at a}\\&\text{particular point. In the case of the Maclaurin series, this}\\&\text{ point has the value of zero (for many functions f(0) is readily}\\&\text{known):}\\&f(x)\approx\sum_{i=0}^{n}\frac{f^{(i)}(0)}{i!}x^i\approx f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+...+\frac{f^{(n)}(0)}{n!}x^n\\&\text{For the function }-3^{(4x)}cos(9x)\text{ at }x=2.9\\&\text{This expansion becomes:}\\&f(2.9)\approx (-3^{(4(0))}cos(9(0)))(2.9)^{0}+(9\cdot 3^{(4(0))}sin(9(0)) - 4\cdot 3^{(4(0))}cos(9(0))ln(3))(2.9)^{1}+\frac{81\cdot 3^{(4(0))}cos(9(0)) + 72\cdot 3^{(4(0))}sin(9(0))ln(3) - 16\cdot 3^{(4(0))}cos(9(0))ln(3)^{2}}{2}(2.9)^{2}\\&f(2.9)\approx245.66\end{align*}$

Report an Error

Example Question #31 : Taylor Series

$\begin{align*}&\text{For the function: }f(x)=cos(4x)sin(x)\\&\text{Approximate }f(3.1)\\&\text{Using a Maclaurin expansion to }3\text{ terms}\end{align*}$

Possible Answers:

0.52

-9.30

24.80

3.10

Correct answer:

3.10

Explanation:

$\begin{align*}&\text{The Maclaurin series is a special case of the Taylor series.}\\&\text{Similarly, it is used to approximate an unknown function value,}\\&\text{if values of the function and its derivatives are known at a}\\&\text{particular point. In the case of the Maclaurin series, this}\\&\text{ point has the value of zero (for many functions f(0) is readily}\\&\text{known):}\\&f(x)\approx\sum_{i=0}^{n}\frac{f^{(i)}(0)}{i!}x^i\approx f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+...+\frac{f^{(n)}(0)}{n!}x^n\\&\text{For the function }cos(4x)sin(x)\text{ at }x=3.1\\&\text{This expansion becomes:}\\&f(3.1)\approx (cos(4(0))sin((0)))(3.1)^{0}+(cos(4(0))cos((0)) - 4sin(4(0))sin((0)))(3.1)^{1}+\frac{- 17cos(4(0))sin((0)) - 8sin(4(0))cos((0))}{2}(3.1)^{2}\\&f(3.1)\approx3.10\end{align*}$

Report an Error

Example Question #35 : Taylor Series

$\begin{align*}&\text{Approximate }f(1.8) \\&\text{Where }f(x)=-cos(5x)e^{(-3x)}\\&\text{By using a Maclaurin expansion to }3\text{ terms}\end{align*}$

Possible Answers:

30.32

-15.16

-151.60

3.37

Correct answer:

30.32

Explanation:

$\begin{align*}&\text{The Maclaurin series is a special case of the Taylor series.}\\&\text{Similarly, it is used to approximate an unknown function value,}\\&\text{if values of the function and its derivatives are known at a}\\&\text{particular point. In the case of the Maclaurin series, this}\\&\text{ point has the value of zero (for many functions f(0) is readily}\\&\text{known):}\\&f(x)\approx\sum_{i=0}^{n}\frac{f^{(i)}(0)}{i!}x^i\approx f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+...+\frac{f^{(n)}(0)}{n!}x^n\\&\text{For the function }-cos(5x)e^{(-3x)}\text{ at }x=1.8\\&\text{This expansion becomes:}\\&f(1.8)\approx (-cos(5(0))e^{(-3(0))})(1.8)^{0}+(3cos(5(0))e^{(-3(0))} + 5sin(5(0))e^{(-3(0))})(1.8)^{1}+\frac{16cos(5(0))e^{(-3(0))} - 30sin(5(0))e^{(-3(0))}}{2}(1.8)^{2}\\&f(1.8)\approx30.32\end{align*}$

Report an Error

Example Question #31 : Taylor Series

$\begin{align*}&\text{Approximate }f(3.4) \\&\text{Where }f(x)=-sin(5x)sin(6x)\\&\text{By using a Maclaurin expansion to }3\text{ terms}\end{align*}$

Possible Answers:

173.40

2774.40

-346.80

-1734.00

Correct answer:

-346.80

Explanation:

$\begin{align*}&\text{The Maclaurin series is a special case of the Taylor series.}\\&\text{Similarly, it is used to approximate an unknown function value,}\\&\text{if values of the function and its derivatives are known at a}\\&\text{particular point. In the case of the Maclaurin series, this}\\&\text{ point has the value of zero (for many functions f(0) is readily}\\&\text{known):}\\&f(x)\approx\sum_{i=0}^{n}\frac{f^{(i)}(0)}{i!}x^i\approx f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+...+\frac{f^{(n)}(0)}{n!}x^n\\&\text{For the function }-sin(5x)sin(6x)\text{ at }x=3.4\\&\text{This expansion becomes:}\\&f(3.4)\approx (-sin(5(0))sin(6(0)))(3.4)^{0}+(- 5cos(5(0))sin(6(0)) - 6cos(6(0))sin(5(0)))(3.4)^{1}+\frac{61sin(5(0))sin(6(0)) - 60cos(5(0))cos(6(0))}{2}(3.4)^{2}\\&f(3.4)\approx-346.80\end{align*}$

Report an Error

← Previous 1 2 3 4 5 6 7 8 9 … 49 50 Next →

View AP Calculus BC Tutors

Pankaj
Certified Tutor

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

View AP Calculus BC Tutors

Harrison
Certified Tutor

Hamilton College, Bachelor in Arts, Mathematics. University of Connecticut, Master of Science, Applied Mathematics.

View AP Calculus BC Tutors

Kira
Certified Tutor

University of Arizona, Bachelor of Science, Deaf Studies.

All AP Calculus BC Resources

2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GRE Tutors in San Francisco-Bay Area, Spanish Tutors in Seattle, English Tutors in Miami, ACT Tutors in Chicago, Math Tutors in Miami, LSAT Tutors in Phoenix, Physics Tutors in New York City, Physics Tutors in San Diego, MCAT Tutors in San Francisco-Bay Area, Reading Tutors in Phoenix

Popular Courses & Classes

SSAT Courses & Classes in Boston, SSAT Courses & Classes in Houston, GRE Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Washington DC, GMAT Courses & Classes in Washington DC, ACT Courses & Classes in Miami, GRE Courses & Classes in Seattle, LSAT Courses & Classes in Philadelphia

Popular Test Prep

SSAT Test Prep in Denver, SSAT Test Prep in Miami, SAT Test Prep in Miami, GMAT Test Prep in San Diego, GMAT Test Prep in Philadelphia, SAT Test Prep in Chicago, GRE Test Prep in Seattle, ISEE Test Prep in San Francisco-Bay Area, SSAT Test Prep in New York City, GMAT Test Prep in Seattle

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

AP Calculus BC : AP Calculus BC

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #1 : Maclaurin Series

Example Question #31 : Polynomial Approximations And Series

Example Question #1 : Maclaurin Series

Example Question #1 : Maclaurin Series

Example Question #31 : Taylor Series

Example Question #31 : Polynomial Approximations And Series

Example Question #31 : Polynomial Approximations And Series

Example Question #31 : Taylor Series

Example Question #35 : Taylor Series

Example Question #31 : Taylor Series

All AP Calculus BC Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: