An Educated Guess

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:x1= x0+ hy1= y0+ hF(x0,y0)x2= x1+ hy2= y1+ hF(x1,y1) and so on until the final step askedxn= xn-1+ hyn= yn-1+ hF(xn-1,yn-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 Integrating is rather very simple of a task to perform. It usually involves some heavy memorization of several functions in order to facilitate the rather strenuous task of manually revising data in terms of understanding rate of change, accumulation, among other tasks. Most applications of integration involve dealing with traffic flow, waste/resource management, and so on. If and are functions of and have continuous derivatives, then Where u is the simplest piece of the function and dv is the more complicated piece of the function multiplied by the change in the respective variable. This is made in order to simplify the complications most mathematicians and physicist use when trying to deal with multiple equations in their tests. Remembering these integrations rules work in tangent with integration by parts. They will be found on the following page.

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 SeriesA geometric series with a ratio r diverges if |r| >/= 1. If 0 < |r| < 1, then the series converges to the sum ∑ arn = a/(1-r), 0 < |r| < 1nth Term Testif ∑ an converges, then liman= 0 if lim≠ 0, then ∑ an diverges. Integral Test If f is positive, continuous, and decreasing for x>1 and an = f(n), then ∑ an and ∫ f(x) dx either both converge or diverge. p Seriesthe p series ∑ 1/np= 1/1p + 1/2p + 1/3p + 1/4p + …1. converges if p > 1 and2. diverges if 0 < p < 1 Direct ComparisonLet 0 < an < bn for all n1. If ∑ bn converges, then ∑ an converges2. If ∑ an diverges, then ∑ bn converges Alternating SeriesLet an > 0. The alternating series∑ (-1)nan and ∑ (-1)n+1 an converge if the following two conditions are metliman= 0 and an+1 ≤ an, for all n Ratio test∑ an converges absolutely if lim|an+1/an| < 1 ∑ an diverges if lim|an+1/an| > 1 or lim|an+1/an| = ∞ The ratio test is inconclusive if lim|an+1/an| = 1 n→∞ 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. n→∞ n→∞ n→∞ n→∞ n→∞

8. Taylor Polynomials and Expansions Polynomial functions are used to approximate the value of certain function, by choosing a specific value (c,f(c)), where P(c) = f(c) and are measured by their degrees of accuracy to find subsequent values across a never-ending set of data, provided that this never ending set of data is given.In polynomials, the higher the degree of the approximating polynomial, the more accurate the approximation becomes. Polynomial expansions with infinite terms are called power series, because they usually involve an exponent value on a number across an interval of convergence I, something that you test out when you use a Ratio Test on a given series, also represented by (x-c) And give you the indication that at that value c a polynomial can be found giving multiple degrees (how many times a derivative can be taken) of approximation. The definition of a polynomial is as follows: If f has n derivatives at c, then polynomial Pn(x) = f(c) +f’(c)(x-c) +(x-c)2 +(x-c)3 +…+(x-c)n Is a Taylor polynomial centered at point c. Maclaurin Polynomials are defined as Pn(x) = f(c) +f’(c)(x-c) +x2 +x3 +…+xn Where the polynomial function designation is centered around x = 0 Usually polynomials are used in predictions of calculating behaviors of scientific models in physics experiments and engineering. Heat transfer is one of the few examples of using polynomials in the Taylor Series format.

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/ComplexPowerSeriesFormulas_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!