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Olym-Power Series

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

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Olym-Power Series

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  1. Olym-Power Series An Instructional Text of Olympic Proportions

  2. 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 Page 7 - Hermesoning Reasoning to Zeusical Choice Page 8 - Conceptual Exampzeus Page 9 - Graphing Calcul♥ver Pr♥blem Page 10 - AP Level Free Res-poseidon Page 11 - Zeuslution to AP Level Free Res-Poseidon Method #1 Page 12 - Zeuslution to AP Level Free Res-Poseidon Method #2 Page 13 - Real World Application Page 14 - Leonhard Euler’s Contribution Courtesy of www.cute-wallpaper.com

  3. 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

  4. Apollo-lytical Example: Generating a Power Series Apollo’s Words of Wisdom: Remember that 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) Courtesy of www.santas.net Courtesy of vectors123.com

  5. 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

  6. Zueslution Courtesy of www.12voltcustoms.com Remember, although this problem looks a little strange, 5never forget the chain rule. Courtesy of www.vectorfree.com

  7. 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), thedemi-god differentiated, therefore arriving at f’’(x) instead of f(x). (D) This answer is correct. (E) This answer is clearly incorrect.

  8. 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

  9. www1.macys.com Graphing Calcul♥ver Pr♥blem Use 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 Courtesy of sreejzmod.blogspot.com

  10. AP Level Free Res-Posiedon The function is defined by the power series: Find and . Determine whether is concave up or concave down at . Justify. Try-Dent sea if you can do this! www.clker.com Poseidon’s Sea-cret to Success: Always expand the series a couple of terms to make integration and differentiation simpler. Courtesy of Disney

  11. ZeuslutionMethod #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.

  12. 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

  13. 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

  14. 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.edu and www.lakoniaoliveoil.blogspot.com

  15. 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

  16. The Exerc-Ares Welcome to the Exerc-Ares section of the textbook! If you wanna be buff like me, complete these problems and become a POWER(series)HOUSE!

  17. Exercises: Generating a Power Series Generate a Power Series centaured at the given value “c”. 1. , 2. , 3. , 4. ,

  18. Differentiating and Integrating Power Series Differentiate or Integrate the following power series. 1. 2. Find . Find . 3. 4. Find Find .

  19. Understanding Concepts 1. Find a Power Series for . 2. Find a Power Series for .

  20. AP Level Zeusical Choice Practice 1. Which of these statements is incorrect if ? (A) (B) (C) (D) (E)

  21. AP Level Zeusical Choice Practice 2. = A) B) C) D) E)

  22. AP Level Zeusical Choice Practice 3. If , what is the index of ? A) B) C) D)E) does not exist because diverges for all real x

  23. AP Level Zeusical Choice Practice 4.Allow to be defined as . Evaluate . • -3 B) 0 C) 1 D) 2 E) 3

  24. AP Level Zeusical Choice Practice 5. If , what is the power series representation of ? A) B) C) D) E)

  25. AP Free Resposiedon Let be the function given by . • Write the first three nonzero terms and the general term of the Maclaurin series. • Does the series found in part (a), when evaluated at , converge to ? Explain why or why not? • Find the power series representation of .

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