LIMITS AND CONTINUITY 1. The concept of limit x2 − 4 . Examine the behavior of f (x) as x approaches 2. Example 1.1. Let f (x) = x−2 Solution. Let us compute some values of f (x) for x close to 2‚ as in the tables below. We see from the first table that f (x) is getting closer and closer to 4 as x approaches 2 from the left side. We express this by saying that “the limit of f (x) as x approaches 2 from left is 4”‚ and write x→2− lim f (x) = 4. Similarly‚ by looking at the second table
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Unit 1: Introduction to Polynomial Functions Activity 4: Factor and Remainder Theorem Content In the last activity‚ you practiced the sketching of a polynomial graph‚ if you were given the Factored Form of the function statement. In this activity‚ you will learn a process for developing the Factored Form of a polynomial function‚ if given the General Form of the function. Review A polynomial function is a function whose equation can be expressed in the form of: f(x) = anxn + an-1xn-1 +
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INTRODUCTION TO CRYPTOGRAPHY INTRODUCTION Cryptography is the branch of science that deals with data security relating to encryption and decryption. This science has been in existence about 4000 years ago with its initial and limited use by the Egyptians‚ to the twentieth century where it played a crucial role in the outcome of both world wars. Cryptography was used as a tool to protect national secrets and strategies. DES‚ the Data Encryption Standard‚ is an adoption with the Federal Information
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Chidambaram Ramanujam | | | Top of FormBottom of Form | | | | | Before we look at the life and work of Chidambaram Padmanabhan Ramanujam we must warn the reader that this article is on Ramanujam‚ NOT Ramanujan the number theorist who worked with G H Hardy (there is only a difference of one letter in their names!). Ramanujam’s father was C S Padmanabhan who was an advocate working in Madras‚ India‚ at the High Court. C P Ramanujam was educated in Madras‚ first at Ewart’s School‚ where
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Running Head: History of Trigonometry History of Trigonometry Rome Fiedler History of Mathematics 501 University of Akron April 29‚ 2012 History of Trigonometry: An Introduction Trigonometry is useful in our world. By exploring where these concepts come from provides an understanding in putting this mathematics to use. The term Trigonometry comes from the Greek word trigon‚ meaning triangle and the Greek word meatria meaning measurement. However it
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common divisor of two numbers. Book eight looked at numbers in geometrical progression. Book X -- Incommensurables Book ten covered mainly the work of Theaetetus and dealt with the theory of irrational numbers. Euclid altered the proofs of several theorems in this book so that they fitted the new definition of proportion given by
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Socrates and Aristotle were both Greek philosophers who contributed philosophies. Socrates believed that all people contained real knowledge within them and that self critical examination was needed to bring this knowledge out. Socrates once stated‚ “The unexamined life is not worth living.” In this philosophical idea‚ Socrates is suggesting that an individual‚ who chooses to not think about their own actions‚ does not truly care about their own life. Aristotle believed in the concept of examining
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European Journal of Operational Research 154 (2004) 345–362 www.elsevier.com/locate/dsw Returns to scale in different DEA models Rajiv D. Banker a‚ William W. Cooper b‚ Lawrence M. Seiford c‚ Robert M. Thrall d‚ Joe Zhu e‚* c School of Management‚ The University of Texas at Dallas‚ Richardson‚ TX 75083-0658‚ USA Graduate School of Business‚ The University of Texas at Austin‚ Austin‚ TX 78712-1174‚ USA Department of Industrial and Operations Engineering‚ University of Michigan‚ Ann Arbor‚ MI
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polygon with 17 sides could be drawn using just a compass and a straight edge. The first theorem he proved was‚ “The Fundamental Theorem of Algebra.” This theorem says that every algebraic equation has at least one root or solution. Later he published a book called‚ “Disquisitions Arithmetic” which he published in 1801. It showed the study of the number theory and also provided proof for‚ “The Fundamental Theorem of Algebra.” Also he calculated the orbit of asteroid Ceres‚ and was able to predict its
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* Deductive Reasoning – * making a specific conclusion based on a collection of generally accepted assumptions. * There are no counterexamples * Premises – undefined terms‚ definitions and postulates (or previously proven theorems) * Fallacy: A conclusion that does not necessarily follow from the premises. * Proof by Negation – Indirect Proofs – * Start with the Givens * Assume the negation of the Conclusion/Proof * The conclusion of the
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