Summary of Convergence and Divergence Tests for Series TEST nth-term Geometric series p-series Integral SERIES ∑ an CONVERGENCE OR DIVERGENCE Diverges if lim n→∞ an ≠ 0 (i) Converges with sum S = (ii) Diverges if r ≥ 1 1 p COMMENTS Inconclusive if lim n →∞ an = 0 Useful for the comparison tests if the nth term an of a series is similar to arn-1 Useful for the comparison tests if the nth term an of a series is similar to 1/np The function f obtained from an = f ( n ) must be continuous‚ ∑
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About the Author Tony Crilly is a freelance writer‚ having previously taught at the University of Michigan‚ the City University in Hong Kong‚ Middlesex University and the Open University. His principal research interest is the history of mathematics‚ and he has written and edited many works on fractals‚ chaos and computing. He is the author of the acclaimed biography of the English mathematician Arthur Cayley and popular maths book How Big is Infinity? Reflection 50 Ideas You Really Need To Know
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The main application of plasmids is as cloning vectors in gene cloning. In gene cloning‚ a fragment of DNA‚ containing the gene to be cloned is inserted into a circular molecule called the “vector” to produce recombinant DNA molecule. Plasmids are one of the most commonly used “vectors” for this purpose. They transport the gene into a host cell‚ such as a bacterium‚ which is said to be transformed with the recombinant molecule. Here‚ these plasmid vectors multiply‚ producing numerous identical copies
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The Vector equation of a plane To find the vector equation of a plane a point on the plane and two different direction vectors are required. The equation is defined as: where a is the point on the plane and b and c are the vectors. This equation can then be written as: The Cartesian equation of a plane The cartesian equation of the plane is easier to use. The equation is defined as: One of the advantages to writing the equation in cartesian form is that we can easily find the normal
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Contents Pages OBJECTIVES 3 ACKNOWLEDGEMENTS 4 INTRODUCTION 5 PART 1 6 PART 2 7-12 PART 3 13-17 FURTHER EXPLORATION 18-19 REFLECTION 20-21 OBJECTIVES We students taking Additional Mathematics are required to carry out a project while we are in Form Five. This project can be done in groups or individually‚ but each of us is expected to submit an individually report. Upon completion of the Additional Mathematics Project Work‚ we are to gain valueable experiences and able to:
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This page intentionally left blank A Course in Financial Calculus A Course in Financial Calculus Alison Etheridge University of Oxford CAMBRIDGE UNIVERSITY PRESS Cambridge‚ New York‚ Melbourne‚ Madrid‚ Cape Town‚ Singapore‚ São Paulo Cambridge University Press The Edinburgh Building‚ Cambridge CB2 8RU‚ UK Published in the United States of America by Cambridge University Press‚ New York www.cambridge.org Information on this title: www.cambridge.org/9780521890779 © Cambridge University
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HIPPARCHUS (c 190 – c120 BC) Hipparchus was a Greek astronomer‚ mathematician and geographer. Surprisingly little is known about his early life. He was most certainly born in Nicaea‚ Bithynia‚ (which is now known as Iznik in Turkey)‚ around 190 BC. Most of what we do about him comes from the books of other scholars who came after him‚ such as Ptolemy and Strabo. It seems likely that Hipparchus studied in Alexandria but spent his later life in Rhodes. Strabo‚ another Greek geographer writing about
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| | (0‚ ∞)Question 8 Find the largest open interval(s) where the function is Increasing y = x4 - 18x2 + 81Answer | | (-∞‚ 0) | | | (-3‚ 0) | | | (-3‚ 3) | | | (3‚ ∞) | Question 9 S(x) = -x3 + 6x2 + 288x + 4000‚ 4 ≤ x ≤ 20 is an approximation to the number of salmon swimming upstream to spawn‚ where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon.Answer | | 8°C | | | 20°C | | | 4°C | | | 12°C |
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ABSTRACT Philippines is experiencing an erratic change in weather which is causing a lot of problems‚ including the increase of population growth of mosquitoes which could cause various kinds of diseases like Encephalitis‚ West Nile Fever‚ Dengue Fever and Malaria. People should be aware of the things that might happen because of the said diseases. The researcher conducted a study that could be a possible solution to this kind of problem. The main purpose of the study is to make
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needs no “cause.” Slide 4-8 Seat Belts: An Application of Newton’s First Law Slide 4-9 What Is a Force? A force... force ... is a push or pull. ... is a vector. ... acts on an object. object ... requires an agent. agent ... is a contact force or a long-range force. Slide 4-10 Force Vectors Slide 4-11 → A Short Catalog of Forces: Weight w → Slide 4-12 → Spring Force Fsp Slide 4-13 Tension Force T Slide 4-14 Normal Force n
Free Force Classical mechanics