1. AN EDUCATED GUESS JOHANN ROSARIO KAMIL PROSTKO

2. TABLE OF CONTENTS Open letter Euler’s Method Basic Integration Rules and Integrating By Parts Series and their Tests Taylor Polynomials and Expansions About the Authors Sources

3. TO THE READER Dear Reader, Calculus is amazing. If it were not for calculus, the world would not work the way it does now. Because of calculus, we can get our computers to work more efficiently thanks to the engineers and mechanics who deal with them everyday. Physics would not be in the level of sophistication it is in now without calculus, since there would be no way to organize and integrate all of the variables and information that we have into formal equations and functions. Most importantly, if it were not for calculus we would not be able to do what we do many times in our lives, and that is guess. Making random guesses is one thing, but using Calculus is another. In some cases, using calculus can help people create substantial guesses as to regulating all sorts of parts of everyday life. Hopefully this mini book will be able to bring to light just how wonderful calculus can be and make it a educational read in the process Johann and Kamil

4. EULER’S METHOD Euler's method is a numerical approach to approximating the particular solution of the differentiable equation y’ = F(x,y) where y’ is the derivative function that is given and where the first x and y values are provided as well. Using h to represent a small “step” taken each time, we can calculate the approximation with the formulas: x 1 = x 0 + h y 1 = y 0 + hF(x 0,y 0 ) x 2 = x 1 + hy 2 = y 1 + hF(x 1,y 1 ) and so on until the final step asked x n = x n-1 + hy n = y n-1 + hF(x n- 1,y n-1 ) Euler’s Method can be used for economics to predict frequently occurring phenomena such as national or global debt, the growth of a certain new product, and so on.

5. BASIC INTEGRATION RULES AND INTEGRATION BY PARTS

6. This table is EXTREMELY helpful in going through all sorts of problems as efficiently as possible. We had to memorize these for all of our tests and most likely, they will need to be memorized for easy calculation in college.

7. SERIES AND THEIR TESTS There are many ways to find the convergence of series. The convergence of series may be found using one of these rules and formulas depending on which requirements it meets. Geometric Series A geometric series with a ratio r diverges if |r| >/= 1. If 0 < |r| < 1, then the series converges to the sum ∑ ar n = a/(1-r), 0 < |r| < 1 nth Term Test if ∑ a n converges, then lim a n = 0 if lim ≠ 0, then ∑ a n diverges. Integral Test If f is positive, continuous, and decreasing for x>1 and a n = f(n), then ∑ a n and ∫ f(x) dx either both converge or diverge. p Series the p series ∑ 1/n p = 1/1 p + 1/2 p + 1/3 p + 1/4 p + … 1. converges if p > 1 and 2. diverges if 0 0. The alternating series ∑ (-1) n a n and ∑ (-1) n+1 a n converge if the following two conditions are met lim a n = 0 and a n+1 ≤ a n, for all n Ratio test ∑ a n converges absolutely if lim |a n+1 /a n | < 1 ∑ a n diverges if lim |a n+1 /a n | > 1 or lim |a n+1 /a n | = ∞ The ratio test is inconclusive if lim |a n+1 /a n | = 1 n→∞ Series are mostly used in electronics and magnetics in the Physics field. These are also used to predict the accuracy of several behavioral models in mathematics.

8. TAYLOR POLYNOMIALS AND EXPANSIONS

9. A quick look at the common Power series Taylor Expansions. Be sure to replace z with x if the x is the variable designate.

10. ABOUT THE AUTHORS Johann Rosario was born in New York, United States. After having had struggled with math in his early years in kindergarten, Johann’s stayed determined to do his best in math, and has grown passionate about the subject as he learned its role in science and computers, which was an incentive for joining the AP Calculus AB and BC courses. On his off time, Johann loves gaming and drawing. He can also be seen listening to music and playing violin, building techniques and musical adeptness. Kamil Prostko was born in Grajewo, Poland. He came to America in 2006 with no English knowledge whatsoever. He always had a big interest in math and logical puzzles. That is why he joined both Calculus AB and BC; He spent his high school years focused on school work and was challenged by Calculus courses. In his free time, he likes to pass time by creating models in blender and play strategy games.

11. SOURCES http://math.fullerton.edu/mathews/c2003/complexpowerseries /ComplexPowerSeriesFormulas/Images/ComplexPowerSeries Formulas_gr_1.gif https://excel.ucf.edu/classes/2009/Spring/appsII/Chapter7.pdf Google “Applications of Euler’s Method” Calculus of a Single Variable 8e - Larson and Hostetler

12. THE END Hope you enjoyed reading this as much as we enjoyed making it!