Pharmaceutical Mathematics with Application to Pharmacy

Pharmaceutical Mathematics with Application to Pharmacy

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(iii) Pharmaceutical Mathe ematics with Application to Pharmacy D.H. Panchaksharappa Gowda Assistant Professor, J.S.S. College of Pharmacy, Sri Shivarathreeshwara Nagar, Mysore-570 015. PharmaMed Press An imprint of Pharma Book Syndicate A unit of BSP Books Pvt. Ltd. 4-4-309/316, Giriraj Lane, Sultan Bazar, Hyderabad - 500 095.

(iv) Pharmaceutical Mathematics with Application to Pharmacy- D.H. Panchaksharappa Gowda 2014, by Publisher All rights reserved. No part of this book or parts thereof may be reproduced, stored in a retrieval system or transmitted in any language or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publishers. Published by PharmaMed Press An imprint of Pharma Book Syndicate A unit of BSP Books Pvt. Ltd. 4-4-309/ /316, Giriraj Lane, Sultan Bazar, Hyderabadd - 500 095. Phone: 040-23445605, 23445688; Fax: 91+40-23445611 E-mail: info@pharmamedpress.com www.pharmamedpress.com/pharmamedpress.net India by Bhavish Graphics. Printed at Aditya Offset Process (I) Pvt. Ltd. Hyderabad. Price : Rs. 295.00 ISBN : 978-93-83635-12-2

(v) Foreword It gives me immense pleasure in introducing this book on Pharmaceutical Mathematics with Application to Pharmacy authored by Mr. Panchaksharappa Gowda D.H. This book describes the fundamental aspects of Pharmaceutical Mathematics a core subject, Industrial Pharmacy and Pharmacokinetics application in a very easy to read and understandable language with number of pharmaceutical examples. This book not only deals with Pharmaceutical Mathematics but also with related aspects such as Factorial design problems (formulations), Pharmacokinetics and Physical pharmaceutics, etc. The book is not just theoretical, but based on the practical application by the author over the years. As far as I know there has been no attempts earlier by an Indian author to write a textbook in this area specially related to Pharmacy course. The author of this book is known for his expertise in teaching for more than twenty seven years, a dedicated, student caring teacher was instrumental in introducing Pharmaceutical Mathematics at B.Pharm and Pharm D level. He worked very hard in restructuring the JSS University, Mysore syllabus of Pharmaceutical Mathematics and Biostatistics and Computer application in Pharmacy. He has conducted workshops on Design of experiments (DOE) and Biostatistics and its application for M.Pharm and Doctoral students. I am sure this book will be highly informative and interesting reading material for the students of B.Pharm, Pharm D and M.Pharm and other related course in the field of Pharmaceutical Sciences. It is my great pleasure to commend this textbook, as it will strengthen and support in learning the related Pharmacy subjects. I wish every success to Mr. Panchaksharappa Gowda D. H. for all such endeavors. Dr. B. Suresh President, Pharmacy Council of India Vice chancellor, JSS University, JSS Medical Institutions campus, Mysore-570 015. (v)

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(vii) Preface Mathematics has been given an important place in pharmacy course to solve various equations in pharmacokinetics. No students of these discipline can afford without the knowledge of those topics which are explained in this book. The present text is designed to introduce students about the methods and applications of Mathematics in Industrial Pharmacy and Pharmacokinetics. This book covers theoretical, Practical and applied aspects of mathematics in a clear and exhaustive way. The author has attempted to give as many illustrations as possible in order to make the students to understand various methods in solving or deriving the Pharmacokinetics Problems. I claim no originality for the matter presented in the text, however, the methods of presentation and arrangement of the subject-matter is my own, which reflects an approach for mathematical methods to solve kinetic equation and also understandability based on the long teaching experience of degree (B.Pharm, Pharm.D) and Post-graduate classes. I hope that the book will be found more useful by the students. I invite suggestions from students as well as my teacher-friends for any improvement (addition or deletion) of the book in future. I would like to thank my colleagues for their encourage and suggestions. The author accepts full responsibility for any typographical or subject-matter errors, which may be there in the book of this dimension and subject. - Author (vii)

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(ix) Foreword... (v) Preface... (vii) Chapter 1 Partial Fractions 1.1 Introduction... 1 1.1.1 Polynomial... 1 1.1.2 Rational Fraction... 1 1.1.3 Proper and Improper Fractions... 2 1.2 Partial Fraction... 3 1.2.1 Rules to Resolve into Partial Fraction... 3 1.3 Application of Partial Fraction in Chemical Kinetics and Pharmacokinetics... 7 Chapter 2 Logarithms 2.1 Introduction... 13 2.1.1 Definition of Logarithm... 13 2.2 Theorems on Logarithms... 15 2.3 Common Logarithms... 17 2.4 Characteristic and Mantissa... 17 2.4.1 Natural Logarithm... 18 2.5 Applications of Logarithm to Solve Pharmaceutical Problems... 22 Chapter 3 Matrices and Determinants 3.1 Introduction... 23 3.2 Recapitulation of types of Matrices... 23 3.2.1 Types of Matrices... 24 3.3 Operation on Matrices... 27 (ix)

(x) 3.4 Determinants... 45 3.4.1 Properties of Determinants... 50 3.4.2 Product of Determinants... 63 3.5 Minors and Co-factors... 70 3.6 Inverse of a Matrix... 80 3.7 Solution of System of Linear of Equations using Matrix Method... 90 3.7.1 Cramer s Rule... 99 3.8 Characteristic Equation and Roots of a Square Matrix... 104 3.9 Cayley Hamilton Theorem... 107 3.10 Application of Matrices in Solving Pharmacokinetics Equations... 119 Chapter 4 Analytical Geometry 4.1 Introduction... 134 4.2 Cartesian Rectangular Co-ordinate System... 134 4.2.1 Signs of the Coordinates... 135 4.2.2 Distance Formula... 137 4.2.3 Section Formula... 143 4.3 Straight Line... 153 4.3.1 Slope or Gradient of a Straight Line... 153 4.3.2 Conditions for Parallelism and Perpendicularity of Two Lines... 153 4.3.3 Slope of a Line Joining Two Points... 155 4.3.4 Slope-Intercept form of a Straight Line... 156 4.3.5 Point Slope Form... 157 4.4 Two Point Form... 158 4.4.1 Equation of a Straight Line passing through Two Points... 158 4.4.2 Intercept Form... 159 4.4.3 Normal Form... 160 4.5 Application in Pharmacy... 164

(xi) Chapter 5 Introduction to Functions 5.1 Introduction... 166 5.2 Classification Real Valued Functions... 167 5.3 Types of Real Valued Functions... 168 5.4 Limits and Continuity... 169 5.4.1 Limit of a Function... 169 5.4.2 Interval... 171 Chapter 6 Differentiation 6.1 Introduction... 178 6.2 Derivative of a Function... 178 6.2.1 Right and Left Hand Derivatives... 179 6.3 Derivative of a Constant... 183 6.3.1 Derivative of a Product of a Constant and a Function... 183 6.3.2 Derivative of the Sum or Difference of Two Functions... 184 6.3.3 Derivative of the Product of Two Functions (Product Formula)... 184 6.3.4 Derivative of the Quotient of Two Functions (Quotient Formula)... 186 6.3.5 Derivative of x n w.r.t. where n is any Rational Number... 187 6.3.6 Derivative of e x... 188 6.3.7 Derivative of log e x... 189 6.3.8 Derivative of a x... 189 6.4 Derivative of Trigonometric Functions from First Principles... 190 6.5 Differentiation of Inverse Trigonometric Functions... 195 6.6 Function of a Function or a Composite Function... 200 6.7 Logarithmic Differentiation... 204 6.8 Differentiation of Implicit Functions... 207 6.9 Maxima and Minima... 208 6.9.1 Conditions for a Function to be a Maximum or a Minimum at a Point... 209

(xii) Chapter 7 Integration 7.1 Introduction... 216 7.2 Rules of Integration... 219 7.3 Method of Substitution... 222 7.4 Method of Partial Fractions... 228 7.5 Integration by Parts... 234 7.6 Definite Integrals and their Applications... 238 7.6.1 Application... 241 Chapter 8 Differential Equations 8.1 Some Basic Definitions... 243 8.1.1 Order and Degree... 244 8.2 Homogeneous Equations... 248 8.3 Linear Equations... 252 8.4 Exact Equations... 256 Chapter 9 Laplace Transforms 9.1 Laplace Transform... 273 9.1.1 Properties of Laplace Transform... 274 9.1.2 Laplace Transforms of Elementary Functions... 275 9.1.3 Existence of Laplace Transform... 277 9.2 Inverse Laplace Transforms... 281 9.3 Laplace Transform of Derivatives... 285 9.4 Application to solve Linear Differential Equations... 289 9.4.1 Transformation of Ordinary Differential Equations... 289 9.4.2 Working Rule to solve Differential Equations... 290 9.5 Application to Derive Pharmacokinetics Equations... 292 Index... 297