View
30
Download
0
Category
Tags:
Preview:
DESCRIPTION
Olym-Power Series. An Instructional Text of Olympic Proportions. Table of Contents. Page 1 - Title Page Page 2 - Table of Contents Page 3- Introduction Page 4 - Apollo- lytical Example: Generating a Power Series Page 5 - Zeusical Choice Page 6 - Zeuslution to Zeusical Choice - PowerPoint PPT Presentation
Citation preview
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 .
Recommended