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Core maths for the biosciences:

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Bibliographische Detailangaben
Beteilige Person:	Reed, Martin B. (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Oxford [u.a.] Oxford Univ. Press 2011
Schlagwörter:	Mathematik Biowissenschaften Einführung
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021182972&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XXVIII, 576 S. Ill., graph. Darst.
ISBN:	9780199216345

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adam_text	Titel: Core maths for the biosciences Autor: Reed, Martin B. Jahr: 2011 Contents in full The search for The Ultimate Answer xxi Introduction: why and how to use this book xxii Acknowledgements xxv Technical notes: using the electronic support that accompanies this book xxvi PART I ARITHMETIC, ALGEBRA, AND FUNCTIONS 1 1 Arithmetic and algebra 3 Case Study A1: Introduction to models of population growth 4 Case Study B1: Introduction to models of cancer 4 Case Study C1: Introduction to predator-prey relationships 5 1.1 Numerical and algebraic expressions 5 Case Study B2: Angiogenic cancer cells 6 1.2 The real numbers 7 1.2.1 Integers and reals 7 1.2.2 The real line 8 1.3 Arithmetic operations 9 1.3.1 Negation 9 1.3.2 Addition and subtraction 9 1.3.3 Multiplication and division 10 1.3.4 Absolute value 13 1.3.5 Percentages 14 1.3.6 Basic rules for manipulating equations 15 1.4 Brackets and the distributive law 16 1.4.1 How to use brackets 16 1.4.2 Rule of precedence 18 1.4.3 The distributive law 18 CONTENTS IN FULL 1.5 Exponents 19 1.5.1 Definition of exponents 19 1.5.2 Rules for exponents 20 Case study A2: Formula for geometric growth 22 1.5.3 Products and factors 22 1.6 Roots 23 1.6.1 Definition of roots 23 1.6.2 Roots and exponents 24 1.6.3 Irrational numbers 25 1.6.4 Surds 25 1.6.5 A third operation in manipulating equations 26 1.7 Evaluating expressions 26 1.7.1 Order of operations 27 1.7.2 Handling complex fractions 27 1.7.3 Numerical expressions in Excel 28 Case Study A3: Birth and death rates 30 Case Study B3: Evaluating the angiogenic cancer cell density 31 1.8 Extension: intervals and inequalities 32 1.8.1 Intervals on the real line 32 1.8.2 Inequalities 33 Case Study A4: Birth rate, death rate and extinction 35 Case Study B4: Conditions for angiogenic cell line extinction 36 Summary 37 Problems 38 Units; precision and accuracy 45 2.1 Scientific notation 46 2.1.1 Definition of scientific notation 46 2.1.2 Converting numbers between decimal and scientific notation 47 2.1.3 Performing addition and subtraction in scientific notation 48 2.1.4 Performing multiplication and division in scientific notation 49 2.1.5 An aside: floating point notation 49 2.2 SI units 50 2.2.1 Base, supplementary, and derived SI units 50 2.2.2 SI prefixes 53 Case Study C2: Velocity 54 2.2.3 More problems with SI units; units of volume 55 2.2.4 Non-SI units 56 2.3 Calculations using SI units 56 Case Study C3: Force and acceleration 57 2.4 Dimensional analysis 58 2.5 Rounding, precision, and accuracy 60 2.5.1 Rounding numbers 60 2.5.2 Significant figures 61 2.5.3 Uncertainty intervals 62 2.6 Extension: accuracy and errors 64 2.6.1 Errors in addition and subtraction 65 2.6.2 Errors in multiplication and division 66 2.6.3 Errors in exponentiation 68 2.6.4 Delta notation 71 Case Study A5: Error analysis for geometric growth 72 Summary 74 Problems 75 3 Data tables, graphs, interpolation 81 3.1 Constructing a data table and a data plot 82 3.1.1 Independent and dependent variables 82 3.1.2 Data plots 82 3.2 Drawing graphs 85 3.2.1 Three basic types of graph 85 3.2.2 Drawing graphs in Excel 86 3.3 Straight-line graphs: finding the slope 88 3.3.1 Direct proportion 89 3.3.2 Linear relationship 91 3.3.3 Calculating the slope 91 3.4 Inverse proportion 93 3.5 Application: allometry 94 3.6 Extension: interpolation 96 3.6.1 Performing interpolation by hand 96 3.6.2 Linear interpolation between two data values 97 3.6.3 Piecewise linear interpolation 99 3.6.4 Linear interpolation using Excel 101 Case Study A6: Cobwebbing 102 Problems 104 4 Molarity and dilutions 109 4.1 Basic concepts 109 4.1.1 Simple solutions 109 XJI CONTENTS IN FULL 4.1.2 Atomic mass 110 4.1.3 The mole 110 4.1.4 The molar mass of a substance 111 4.1.5 The molarity of a solution 112 4.1.6 Application: measurements of cholesterol level 113 4.2 Calculations involving moles and molarity 113 4.2.1 Calculating the number of moles in a sample 113 4.2.2 Calculating the molar mass of a compound 114 4.2.3 Calculating the molarity of a solution 114 4.2.4 Calculating the moles present in a sample of solution 115 4.2.5 Calculating the moles to add in making a solution 115 4.2.6 Calculating the mass to add in making a solution 116 4.3 Calculations for dilutions of solutions 116 4.3.1 Calculating the new concentration after diluting 116 4.3.2 Calculating how much to dilute to obtain a specific concentration 117 4.3.3 Serial dilutions 119 4.3.4 Application: serial dilution in homeopathy 120 4.4 Excel spreadsheets 121 Problems 122 5 Variables, functions, and equations 125 5.1 What is a function? 126 5.2 Some simple functions 127 5.2.1 Functions on your calculator, and in Excel 127 5.2.2 Direct and inverse proportionality functions 128 5.2.3 Quadratic functions, parameters 129 5.3 Creating tables of values in Excel 131 Case Study B5: Graphing the relationship in Excel 132 5.4 Manipulating equations 133 5.4.1 What is an equation? 133 5.4.2 Linear equations and their graphs 134 5.4.3 Rearranging linear equations 135 5.4.4 Equation of a circle 135 5.4.5 Rearranging more complicated equations 138 5.5 Graphs of the direct and inverse proportion functions 141 5.6 Algebra of functions 142 5.6.1 Changing the argument 142 5.6.2 Arithmetic operations with functions 143 5.6.3 Composition of functions (or function of a function ) 145 5.7 Extension: solving simultaneous linear equations 145 Problems 149 CONTENTS IN FULL XIII Linear functions and curve sketching 6.1 Graph sketching with FNGraph 6.2 Constant functions y= c 6.3 Linear functions 6.3.1 The identity function y = x 6.3.2 Proportionality functions y=mx Case Study C4: Equations of motion 1 6.3.3 General linear functions y= mx+c Case Study C5: A constant-velocity chase 6.3.4 Trend lines revisited: goodness of fit 6.4 Limits and asymptotes 6.4.1 Limits of functions 6.4.2 Horizontal asymptotes 6.4.3 Vertical asymptotes 6.4.4 Limits of sequences 6.5 Extension: linear transformations 6.5.1 Vertical stretch/squash/flip: y=a.f(x) 6.5.2 Vertical shift: y= f x) + d 6.5.3 Horizontal stretch/squash/flip: y= f(bx) 6.5.4 Horizontal shift: y= f(x-c) Problems 151 152 155 156 156 157 159 159 164 165 166 166 167 168 168 169 171 171 171 171 173 7 Quadratic and polynomial functions 176 7.1 Simple quadratic functions 176 7.1.1 The squaring function: y=x2 176 7.1.2 The proportional to the square function: y= ax2 178 7.1.3 Adding a constant term: y= ax2 + d X79 7A.4 The shifted squaring function: y= a{x-a)2 180 7.2 General quadratic functions 181 7.2.1 The completed square form: y= a(x- a)2 + d 181 7.2.2 The polynomial form: y= ax2 + bx + c 184 Case Study C6: Equations of motion 2 189 Case Study A7: Fibonacci s Rabbits: geometric growth in a population structured by age 191 7.2.3 Calculating the polynomial coefficients 192 7.2.4 Application: Reduction of cholesterol level 193 Case Study B6: Derivation of the equilibrium density 195 7.2.5 The factorized form: y= a x- a)(x- /3) 197 XIV CONTENTS IN FULL 7.3 Cubic and higher-degree polynomials 198 7.3.1 Power functions and geometric series 198 7.3.2 General properties of polynomial functions 201 7.3.3 Algebraic long division 201 7.4 Logistic growth 204 7.4.1 The logistic function 204 Case Study C7: Fisheries management 205 7.4.2 Logistic growth of populations 206 Case Study B7: Logistic growth of cancer cells 208 Case Study A8: Cobwebbing of logistic growth model 208 7.5 Extension: quadratic interpolation 210 Problems 213 8 Fitting curves; rational and inverse functions 220 8.1 Reciprocal functions 220 8.1.1 Definition of the reciprocal of f(x) 221 11 x 8.1.2 Rational functions y = -L, y =------r,V =------r 221 1 * ax + b ax + b Case Study C8: A hyperbolic model of animal speed 223 8.2 General rational functions p(} 224 8.2.1 Finding the x-intercepts 225 8.2.2 Finding the y-intercept 225 8.2.3 Finding the horizontal (and sloping) asymptotes 225 8.2.4 Finding the vertical asymptotes 226 8.2.5 Example of graph sketching 226 8.3 Fitting curves to data 227 8.3.1 Inverse proportion 228 8.3.2 Rational function y = ?^- 230 ax + b 8.3.3 Quadratic functions 230 8.3.4 Rational function v = ~ + b 231 x 8.4 Application: enzyme kinetics 231 8.4.1 The Michaelis-Menten equation 232 8.4.2 The Lineweaver-Burk transformation 233 8.4.3 Error analysis 234 8.4.4 Allosteric regulation 235 8.5 Inverse functions 237 8.5.1 Definition of the inverse of f(x) 237 8.5.2 The inverse of rational functions 238 8.6 Bracketing methods 239 8.6.1 Root-finding algorithms 241 8.6.2 Minimization algorithms 246 CONTENTS IN FULL J\|y, Case Study C9: Fisheries management: finding the Maximum Economic Yield 250 8.7 Extension: finding the equation of a trend line 251 Problems 255 9 Periodic functions 259 9.1 Sawtooth functions 259 9.1.1 Basic sawtooth function 259 9.1.2 Specifying the period and amplitude 261 9.1.3 Specifying the vertical shift and phase 262 9.2 Revision of school trigonometry 262 9.3 Measurement of angles in radians 265 9.4 The sine and cosine functions 268 9.5 Periodic functions of time 273 9.5.1 General sine and cosine functions 273 Case Study C10: A simple model of predator-prey population dynamics 274 9.5.2 Application: modelling tidal data 276 9.5.3 Application: modelling temperature variations 279 9.6 Reciprocal and inverse trigonometric functions 280 9.6.1 Reciprocal trigonometric functions 280 9.6.2 Inverse trigonometric functions 282 9.7 More trigonometric identities 283 9.8 The tangent function and the gradient of a curve 284 9.8.1 Definition of the tangent function 284 9.8.2 The tangent function and the slope of a line 284 9.8.3 The geometric tangent 285 9.8.4 An approximation to the gradient 287 Problems 289 10 Exponential and logarithmic functions 290 10.1 Exponential functions to the base a 290 10.1.1 Discrete and continuous models 290 10.1.2 Exponential function to the base a:y=a 291 10.2 Exponential growth function y = Aekx 294 Case study A9: Exponential growth of populations 296 10.3 Logarithms 296 10.3.1 Definition of logarithms to base a 296 10.3.2 Laws of logarithms 298 10.3.3 Logarithms to base 2 299 10.3.4 Logarithms to base 10 (common logarithms) 300 10.3.5 Logarithms to base e (natural logarithms) 301 XVI CONTENTS IN FULL 10.4 Fitting exponential curves to data 302 10.4.1 Fitting an exponential growth model 303 10.4.2 Application: allometry 303 10.4.3 Application: allosteric regulation 305 10.5 Exponential decay 306 10.5.1 Exponential decay function: y=Ae-kx 307 Case Study C11: An exponential model of animal speed 309 10.5.2 Application: sensitization and habituation 309 10.5.3 Application: drug administration 310 10.5.4 Example: radiocarbon dating 311 Case study A10: An equation for logistic growth 312 10.6 Example: reduction of cholesterol level 314 10.7 Extension: a stochastic model of exponential decay 317 Case study All: Gompertz curve for population mortality 319 Problems 320 Revision Problems 328 Historical interlude: finding the roots of polynomials 331 PART II CALCULUS AND DIFFERENTIAL EQUATIONS 335 11 Instantaneous rate of change: the derivative 337 11.1 Introduction to the calculus 337 11.1.1 Differential calculus 338 11.1.2 Integral calculus 340 11.1.3 Differential equations 341 Case Study B8: Constructing the angiogenic tumour model 342 11.2 Definition of the derivative 344 11.3 Differentiating polynomial functions 347 11.3.1 The derivative of power functions y=x 348 11.3.2 Notation 351 11.3.3 The derivative of linear functions 351 11.3.4 The derivative of polynomial functions 352 Case Study C12: Differentiating the animal motion model 354 11.4 Differentiating roots and reciprocals 354 11.5 Differentiating functions of linear functions 355 11.6 Differentiating exponential functions 357 CONTENTS IN FULL XVH 11.7 Extension: small changes and errors 360 Case Study CI3: Deriving the exponential model of animal speed 360 Case Study A12: Differential equation for exponential growth 362 Problems 363 12 Rules of differentiation 366 12.1 Differentiable functions 366 12.2 The chain rule 367 12.3 The product and quotient rules 370 12.3.1 The product rule 370 12.3.2 The quotient rule 372 Case Study C14: Deriving the hyperbolic model of animal speed 376 12.4 Differentiating trigonometric functions 376 12.5 Implicit differentiation 379 12.6 Differentiating logarithmic functions 381 12.7 Differentiating inverse trigonometric functions 382 12.8 Higher-order derivatives 383 12.9 Summary of standard derivatives, and rules of differentiation 386 Problems 387 13 Applications of differentiation 389 13.1 Interpretation of graphs 389 13.1.1 Gradients 391 13.1.2 Roots 392 13.1.3 Critical points 393 13.1.4 Curvature 395 Case study A13: Analysing the Ricker update equation 397 13.1.5 Summary 398 Case study A14: The point of inflection in the logistic growth curve 402 13.2 Optimization 403 13.2.1 Optimization in the biosciences 403 13.2.2 One-dimensional unconstrained optimization 404 Case study C15: Fisheries management: using calculus to find the Maximum Economic Yield 406 13.2.3 Application: tubular bones 407 13.3 Related rates 411 13.4 Polynomial approximation of functions 413 XViH CONTENTS IN FULL 13.4.1 Linear approximation of f x) around x = 0 414 13.4.2 Quadratic approximation of f x) around x = 0 416 13.4.3 Maclaurin series expansions of functions 417 13.4.4 Taylor series expansions of functions 418 13.5 Extension: numerical methods for finding roots and critical points 419 13.5.1 Newton-Raphson method for finding roots 420 13.5.2 Newton s method for optimization 423 Problems 424 14 Techniques of integration 426 14.1 The integral as anti-derivative 426 14.1.1 Definition and notation 427 14.1.2 The integrals of power functions, and the coefficient rule 428 14.1.3 The sum rule, and the integrals of polynomial functions 430 14.1.4 Integrals of some standard functions 432 Case study C16: Integrating the hyperbolic and exponential models 433 of animal speed 14.2 Integration by substitution 435 14.3 Integration by parts 438 Case study A15: Solving the differential equation for exponential growth 439 14.4 Integration by partial fractions 441 14.5 Integrating trigonometric functions 444 14.5.1 The general sine and cosine functions 444 14.5.2 The tangent function 444 14.5.3 Powers of sines and cosines 445 14.5.4 Integrating fcos x 446 14.5.5 Integrating inverse trigonometric functions 447 14.6 Extension: integration using power series approximations 448 14.7 Summary of standard integrals 449 Problems 450 15 The definite integral 451 15.1 The integral as area under the curve 451 15.1.1 The link between the integral and area 451 15.1.2 Speed-time graphs 452 15.1.3 Definition of the definite integral 453 15.2 The integral as limit of a sum 458 15.2.1 The Riemann integral 458 15.2.2 Application: chemotherapy drug delivery 460 15.2.3 Application: laminar blood flow 462 CONTENTS IN FULL xJX 15.3 Using techniques of integration with definite integrals 465 15.3.1 Integration by substitution 465 15.3.2 Integration by parts 466 15.3.3 Integration by partial fractions 467 15.4 Improper integrals 468 15.5 Extension: numerical integration 470 15.5.1 The trapezium rule 470 15.5.2 Simpson s rule 473 15.5.3 Using Simpson s rule with data-sets 476 Problems 479 16 Differential equations I 481 16.1 Overview of differential equations 481 16.1.1 Order of a differential equation 482 16.1.2 Boundary conditions 482 16.1.3 ODEsandPDEs 484 16.2 Solution by separation of variables 484 16.2.1 Right-hand side a function of x only 484 16.2.2 Right-hand side a function of y only 486 16.2.3 Variables separable 489 Case Study B9: The Gompertz model of tumour growth 490 Case Study A16: Solving the ODE for logistic growth 492 Case Study CI 7: A harvesting model for fish stocks 494 16.2.4 Change of variable 496 Case Study B10: The Gompertz model revisited 497 16.3 Linear first-order ODEs 499 16.4 Extension: partial differentiation 500 16.4.1 Reducing a PDE to an ODE 501 16.4.2 Error analysis in several variables 503 16.4.3 Minimization in two variables 504 Problems 505 17 Differential equations II 509 17.1 Numerical methods for first-order ODEs 509 17.1.1 Euler s method 510 17.1.2 Heun s method 515 Case Study C18: Numerical solution of fish harvesting model 518 17.1.3 Runge-Kutta method RK4 519 XX CONTENTS IN FULL 17.2 Systems of first-order ODEs 519 17.2.1 Lotka-Volterra models of predator-prey dynamics 520 17.2.2 Kermack-McKendrick model of epidemics 526 Case Study A17: The peak of an epidemic 527 17.3 Extension: analytic solutions 529 17.3.1 Solving second-order ODEs 529 17.3.2 Solving first-order systems 531 17.3.3 Solving partial differential equations 532 17.3.4 Further reading 534 Problems 534 18 Extension: dynamical systems 537 18.1 The butterfly effect 537 18.1.1 The birth of a new science 538 18.1.2 Numerical experiments 538 18.2 Equilibria and stability 542 18.2.1 Points of equilibrium for differential equations 542 18.2.2 Stability of equilibria for differential equations 543 Case Study C19: Analysing the equilibria of the harvesting model 547 18.2.3 Stability of equilibria for update equations 549 18.2.4 Numerical experiments with the update equation 551 18.3 Bifurcations... 553 18.4 ... and Chaos 557 18.5 Postscript 559 Problems 561 Answers to odd-numbered problems 563 Appendix: The Greek alphabet 570 References 571 Index 573
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