Olym-Power Series

An Instructional Text of Olympic Proportions

Table of ContentsPage 1 - Title PagePage 2 - Table of ContentsPage 3- IntroductionPage 4 - Apollo-lytical Example: Generating a Power SeriesPage 5 - Zeusical ChoicePage 6 - Zeuslution to Zeusical ChoicePage 7 - Hermesoning Reasoning to Zeusical ChoicePage 8 - Conceptual ExampzeusPage 9 - Graphing Calcul♥ver Pr♥blemPage 10 - AP Level Free Res-poseidon Page 11 - Zeuslution to AP Level Free Res-Poseidon Method #1Page 12 - Zeuslution to AP Level Free Res-Poseidon Method #2Page 13 - Real World ApplicationPage 14 - Leonhard Euler’s Contribution

Courtesy of www.cute-wallpaper.com

Welcome to our Textbook Demigods!

As part of your training to become a full time god, you must study the art of calculus! This text includes teachings on the wonders of Power Series! Don’t be just another half-mortal; use this knowledge to become a GOD of Calculus!

Enjoy and Good Luck, The Gods

PS: Zeus lost his beloved flavor of Chobani. Search through the text and help him find it! (There is one on each page!)

Courtesy of www.disneyscreencaps.com

Apollo-lytical Example: Generating a Power Series

Find a Power Series for f(x)= centaured at zero.

= Achieve a 1 in the denominator

Convert to sum of geometric series .

a=1 r= Identify “a” and “r”.

Insert “a” and “r” into the sum of a geometric series equation (provided by Athena)

Apollo’s Words of Wisdom:Remember that

Courtesy of vectors123.com

Cour

tesy

of

ww

w.s

anta

s.ne

t

Zeusical Choice

If is a Taylor series that converges to f’(x) for all real x, then f(3)=

(A) (B) (C) (D) (E)

This problem separates the

Demigods from the mortals!

Courtesy of Disney

Zueslution Courtesy of www.12voltcustoms.com

Courtesy of www.vectorfree.com

Remember, although this problem looks a little strange, 5never forget the chain rule.

Hermesoning Reasoning(A) The demi-god installed 3 into f’(x) before integrating to find f(x).

(B) This answer is correct, except for then incorrect index. A demi-god would only select this if he or she cursorily viewed the answers instead of thoroughly analyzing each answer choice.

(C) Rather than integrating f’(x), the demi-god differentiated, therefore arriving at f’’(x) instead of f(x).

(D) This answer is correct.

(E) This answer is clearly incorrect.

Conceptual Exampzeus

Find a power series for .

Differentiate.

Identify “a” and “r”

Convert to powers series.

Integrate.

Simplify.

Notice this looks like the sum of a geometric series!

Courtesy of Disney

Graphing Calcul♥ver Pr♥blemUse a p♥wer series to appr♥ximate using the first f♥ur terms ♥f the series. C♥mpare your appr♥ximati♥n with the graphing utility’s. Remember: Plug in for Integrate first

four terms 1st Fundamental

Theorem of Calculus

Graphing utility approximation

Aphrodite’s Kissable Snidbits:You are now able to approximate a

series using the same method as your calculator! Congratulations! Xoxo

Courtesy of Disney

AP Level Free Res-Posiedon

Try-Dent sea if you can do

this!

The function is defined by the power series:

Find and . Determine whether is concave up or concave down at . Justify.

Courtesy of Disney

www.clker.com

Poseidon’s Sea-cret to Success:

Always expand the series a couple of terms to

make integration and differentiation simpler.

Zeuslution Method #1

Expand

Take the derivative of

Plug in zero for

Take the derivative of

Plug in one for x is concave up at . which is a positive value. A positive value given by the second derivative indicates upward concavity.

Zeuslution Method #2

Rewrite the problem if you like

Take the derivative of

… Expand

Plug zero into

Take the derivative of

… Expand

Plug zero into

is concave up at . which is a positive value. A positive value given by the second derivative indicates upward concavity.

Courtesy of www.Chobani .com

Real World Application

Calculators have always seemed like enigmas, finding answers to obscure problems instantly without showing any trace of work or effort. Unlocking this riddle has been something many demi-

gods have struggled to accomplish, until now. Power Series, seemingly worthless, are actually the way calculators solve sin and cosine functions. By using the power series of the Taylor

approximations for these functions and utilizing a very high “n” value, calculators can generate an approximation with little

error.

Courtesy of www.sporcle.com

Leonhard Euler’s contribution

Leonhard Euler discovered many ways to represent logarithmic functions as power series. He furthered his contribution by defining the arctan function as a powers

series:

He is most famous for the Basel Problem. This problem defined:

In essence, this defined as a power series. (Just multiply the sum by 6 and

take the square root) Courtesy of www.micro.magnet.fsu.eduand www.lakoniaoliveoil.blogspot.com

Bibli-GOD-raphy-Class notes taken from Mr. Groden’s lesson as a reference for the apollolytical examples-Information from slide 14 courtesy of

http://en.wikipedia.org/wiki/Leonhard_Euler-The Larson, Hostetler and Edwards Calculus Textbook, Eighth Edition as a reference for the analytical examples-The College Board website as an aid for the AP level Exampzues

The Exerc-AresWelcome to the

Exerc-Ares section of the

textbook!

If you wanna be buff like me, complete these

problems and become a POWER(series)HOUSE!

Exercises: Generating a Power Series

Generate a Power Series centaured at the given value “c”.1. , 2. , 3. ,

4. ,

Differentiating and Integrating Power Series

Differentiate or Integrate the following power series.1. 2. Find . Find .

3. 4. Find Find .

Understanding Concepts1. Find a Power Series for .

2. Find a Power Series for .

AP Level Zeusical Choice Practice

1. Which of these statements is incorrect if ?

(A) (B) (C) (D) (E)

AP Level Zeusical Choice Practice

2. =

A) B) C) D) E)

AP Level Zeusical Choice Practice3. If , what is the index of ?

A) B) C) D) E) does not exist because diverges for all real x

AP Level Zeusical Choice Practice

4.Allow to be defined as . Evaluate .

A) -3 B) 0 C) 1 D) 2 E) 3

AP Level Zeusical Choice Practice5. If , what is the power series representation of ?

A)

B)

C)

D)

E)

AP Free ResposiedonLet be the function given by .a) Write the first three nonzero terms and the

general term of the Maclaurin series.b) Does the series found in part (a), when

evaluated at , converge to ? Explain why or why not?

c) Find the power series representation of .