Surname Centre No. Candidate No. Paper Reference(s) Initial(s) Paper Reference Signature 6 6 6 4 6664/01 0 1 Examiner’s use only Edexcel GCE Core Mathematics C2 Advanced Subsidiary Thursday 24 May 2012 – Morning Time: 1 hour 30 minutes Team Leader’s use only Question Leave Number Blank 1 2 3 4 Materials required for examination Mathematical Formulae (Pink) Items included with question papers Nil 5 6 7 8 9 Candidates may use any calculator allowed by the regulations of the Joint Council
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Radical Expressions ) II. To solve a quadratic equation arranged in the form ax2+ bx=0. Strategy: To factor the binomial using the greatest common factor (GCF)‚ set the monomial factor and the binomial factor equal to zero‚ and solve. Ex. 2) 12x2- 18x=0 6x2x-3= 0 Factor using the GCF 6x=0 2x-3=0 Set the monomial and binomial equal to zero x=0 x= 32 Solutions * In some cases‚ the GCF is simply the variable with
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Optimization methods in portfolio management and option hedging ∗ Huyˆn PHAM e Laboratoire de Probabilit´s et e Mod`les Al´atoires e e CNRS‚ UMR 7599 Universit´ Paris 7 e e-mail: pham@math.jussieu.fr and Institut Universitaire de France April 24‚ 2007 Abstract These lecture notes give an introduction to modern‚ continuous-time portfolio management and option hedging. We present the stochastic control method to portfolio optimization‚ which covers Merton’s pioneering work. The
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FINANCIAL MODELING The materials in this book are intended for instructional and educational purposes‚ to illustrate situations similar to those encountered in the real world. The reader will understand that MIT Press and its authors do not guarantee the accuracy or completeness of any information published in this book. Neither MIT Press nor its authors is responsible for the consequences of the implementation of models or information presented in this book. FINANCIAL MODELING Simon Benninga
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1440: Nicolas Cusanus‚ Earth is in motion 1440: Nicolas Cusanus‚ infinite universe 1450: Johann Gutenberg‚ first printing press in Europe 1472: Johannes Regiomontanus‚ observation of Halley’s comet 1480: Leonardo de Vinci‚ description of parachute 1480: Leonardo de Vinci‚ compares reflection of light to reflection of sound waves 1490: Leonardo de Vinci‚ capillary action 1492: Leonardo de Vinci‚ foresees flying machines 1494: Leonardo de Vinci‚ foresees pendulum clock 1514: Nicolaus Copernicus
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McGRAW-HILL EDITION LIST OF SYMBOLS TOPIC LOGIC SYMBOL ""’p p 1\ q P vq P (J) q p-+q p ++ q p=.q MEANING PAGE T F P(xJ‚ ... ‚ x n) VxP(x) 3xP(x) 3!xP(x) p{S}q SETS negation of p conjunction of p and q disjunction of p and q exclusive or of p and q the implication p implies q biconditional of p and q equivalence of p and q tautology contradiction propositional function universal quantification of P(x) existential quantification of P(x) uniqueness quantification of P(x) therefore
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effectively used in some of these areas‚ but it can often be harmful in gaining insight into some others. Mathematics is one area where reason plays an integral part. Reason is the basis on which mathematics is founded. Before any mathematical theorem can be taken as true‚ it must be backed by a reasonable mathematical proof that shows‚ beyond a doubt‚ that the answer arrived at is correct. This type of empirical‚ reasonable proof shows that of all the areas of knowledge‚ Mathematics uses reason
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HISTORY OF MATHEMATICS The history of mathematics is nearly as old as humanity itself. Since antiquity‚ mathematics has been fundamental to advances in science‚ engineering‚ and philosophy. It has evolved from simple counting‚ measurement and calculation‚ and the systematic study of the shapes and motions of physical objects‚ through the application of abstraction‚ imagination and logic‚ to the broad‚ complex and often abstract discipline we know today. From the notched bones of early man
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therefore‚ the object is said to have either gained or lost a certain amount of energy of a particular type. The total work done on a particle by all forces that act on it is equal to the change in its kinetic energy‚ also known as the work-energy theorem. This can derived from: W=Fdx equation 1 W=maxdx where ax=vdvdx W=mvdvdxdx=mvdv W=v1v2mvdv =12mv22-12mv12 =K2-K1 W=ΔK equation 2 For a body moving along s ‚ displacement with a constant force F‚ work can be
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Distributions 1. Uniform 2. Binomial 3. Hypergeometric 4. Negative Binomial 5. Geometric 6. Poisson SKIPPING: Multinomial (p/149-150) Discrete Uniform Distribution Bernoulli Process Binomial Distribution f(x;n‚p)= =average number of successes in n trials Binomial Tables (in text) Problem • The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive? Negative Binomial Distribution k
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