TY  - BOOK
AU  - Reed,Martin B.
TI  - Core maths for the biosciences
SN  - 9780199216345 (pbk.)
AV  - QH323.5 .R42 2011
U1  - 570.151 22
PY  - 2011///
CY  - Oxford, New York
PB  - Oxford University Press
KW  - Biomathematics
KW  - Textbooks
KW  - Life sciences
KW  - Mathematics
N1  - "With interactive Excel  workbooks online to help you master the essentials."--P.1 of cover; Includes bibliographical references (p. [571]-572) and index; Machine generated contents note; pt. I; ARITHMETIC, ALGEBRA, AND FUNCTIONS --; 1; Arithmetic and algebra --; Case Study A1; Introduction to models of population growth --; Case Study B1; Introduction to models of cancer --; Case Study C1; Introduction to predator-prey relationships --; 1.1; Numerical and algebraic expressions --; Case Study B2; Angiogenic cancer cells --; 1.2; The real numbers --; 1.2.1; Integers and reals --; 1.2.2; The real line --; 1.3; Arithmetic operations --; 1.3.1; Negation --; 1.3.2; Addition and subtraction --; 1.3.3; Multiplication and division --; 1.3.4; Absolute value --; 1.3.5; Percentages --; 1.3.6; Basic rules for manipulating equations --; 1.4; Brackets and the distributive law --; 1.4.1; How to use brackets --; 1.4.2; Rule of precedence --; 1.4.3; The distributive law --; 1.5; Exponents --; 1.5.1; Definition of exponents --; 1.5.2; Rules for exponents --; Case study A2; Formula for geometric growth --; 1.5.3; Products and factors --; 1.6; Roots --; 1.6.1; Definition of roots; 1.6.2; Roots and exponents --; 1.6.3; Irrational numbers --; 1.6.4; Surds --; 1.6.5; A third operation in manipulating equations --; 1.7; Evaluating expressions --; 1.7.1; Order of operations --; 1.7.2; Handling complex fractions --; 1.7.3; Numerical expressions in Excel --; Case Study A3; Birth and death rates --; Case Study B3; Evaluating the angiogenic cancer cell density --; 1.8; Extension: intervals and inequalities --; 1.8.1; Intervals on the real line --; 1.8.2; Inequalities --; Case Study A4; Birth rate, death rate and extinction --; Case Study B4; Conditions for angiogenic cell line extinction --; Summary --; Problems --; 2; Units; precision and accuracy --; 2.1; Scientific notation --; 2.1.1; Definition of scientific notation --; 2.1.2; Converting numbers between decimal and scientific notation --; 2.1.3; Performing addition and subtraction in scientific notation --; 2.1.4; Performing multiplication and division in scientific notation --; 2.1.5; An aside: floating point notation --; 2.2; SI units --; 2.2.1; Base, supplementary, and derived SI units --; 2.2.2; SI prefixes; Case Study C2; Velocity --; 2.2.3; More problems with SI units; units of volume --; 2.2.4; Non-SI units --; 2.3; Calculations using SI units --; Case Study C3; Force and acceleration --; 2.4; Dimensional analysis --; 2.5; Rounding, precision, and accuracy --; 2.5.1; Rounding numbers --; 2.5.2; Significant figures --; 2.5.3; Uncertainty intervals --; 2.6; Extension: accuracy and errors --; 2.6.1; Errors in addition and subtraction --; 2.6.2; Errors in multiplication and division --; 2.6.3; Errors in exponentiation --; 2.6.4; Delta notation --; Case Study A5; Error analysis for geometric growth --; Summary --; Problems --; 3; Data tables, graphs, interpolation --; 3.1; Constructing a data table and a data plot --; 3.1.1; Independent and dependent variables --; 3.1.2; Data plots --; 3.2; Drawing graphs --; 3.2.1; Three basic types of graph --; 3.2.2; Drawing graphs in Excel --; 3.3; Straight-line graphs: finding the slope --; 3.3.1; Direct proportion --; 3.3.2; Linear relationship --; 3.3.3; Calculating the slope --; 3.4; Inverse proportion --; 3.5; Application: allometry; 3.6; Extension: interpolation --; 3.6.1; Performing interpolation by hand --; 3.6.2; Linear interpolation between two data values --; 3.6.3; Piecewise linear interpolation --; 3.6.4; Linear interpolation using Excel --; Case Study A6; Cobwebbing --; Problems --; 4; Molarity and dilutions --; 4.1; Basic concepts --; 4.1.1; Simple solutions --; 4.1.2; Atomic mass --; g; The mole --; 4.1.4; The molar mass of a substance --; 4.1.5; The molarity of a solution --; 4.1.6; Application: measurements of cholesterol level --; 4.2; Calculations involving moles and molarity --; 4.2.1; Calculating the number of moles in a sample --; 4.2.2; Calculating the molar mass of a compound --; 4.2.3; Calculating the molarity of a solution --; 4.2.4; Calculating the moles present in a sample of solution --; 4.2.5; Calculating the moles to add in making a solution --; 4.2.6; Calculating the mass to add in making a solution --; 4.3; Calculations for dilutions of solutions --; 4.3.1; Calculating the new concentration after diluting --; 4.3.2; Calculating how much to dilute to obtain a specific concentration; 8.2; General rational functions p(x)/q(x) --; 8.2.1; Finding the x-intercepts --; 8.2.2; Finding the y-intercept --; 8.2.3; Finding the horizontal (and sloping) asymptotes --; 8.2.4; Finding the vertical asymptotes --; 8.2.5; Example of graph sketching --; 8.3; Fitting curves to data --; 8.3.1; Inverse proportion --; 8.3.2; Rational function y = 1/ax + b --; 8.3.3; Quadratic functions --; 8.3.4; Rational function y = a/x + b --; 8.4; Application: enzyme kinetics --; 8.4.1; The Michaelis -- Menten equation --; 8.4.2; The Lineweaver -- Burk transformation --; 8.4.3; Error analysis --; 8.4.4; Allosteric regulation --; 8.5; Inverse functions --; 8.5.1; Definition of the inverse of f(x) --; 8.5.2; The inverse of rational functions --; 8.6; Bracketing methods --; 8.6.1; Root-finding algorithms --; 8.6.2; Minimization algorithms --; Case Study C9; Fisheries management: finding the Maximum Economic Yield --; 8.7; Extension: finding the equation of a trend line --; Problems --; 9; Periodic functions --; 9.1; Sawtooth functions --; 9.1.1; Basic sawtooth function --; 9.1.2; Specifying the period and amplitude; 9.1.3; Specifying the vertical shift and phase --; 9.2; Revision of school trigonometry --; 9.3; Measurement of angles in radians --; 9.4; The sine and cosine functions --; 9.5; Periodic functions of time --; 9.5.1; General sine and cosine functions --; Case Study C10; A simple model of predator-prey population dynamics --; 9.5.2; Application: modelling tidal data --; 9.5.3; Application: modelling temperature variations --; 9.6; Reciprocal and inverse trigonometric functions --; 9.6.1; Reciprocal trigonometric functions --; 9.6.2; Inverse trigonometric functions --; 9.7; More trigonometric identities --; 9.8; The tangent function and the gradient of a curve --; 9.8.1; Definition of the tangent function; Note continued; 9.8.2; The tangent function and the slope of a line --; 9.8.3; The geometric tangent --; 9.8.4; An approximation to the gradient --; Problems --; 10; Exponential and logarithmic functions --; 10.1; Exponential functions to the base a --; 10.1.1; Discrete and continuous models --; 10.1.2; Exponential function to the base a: y = ax --; 10.2; Exponential growth function y = Aekx --; Case study A9; Exponential growth of populations --; 10.3; Logarithms --; 10.3.1; Definition of logarithms to base a --; 10.3.2; Laws of logarithms --; 10.3.3; Logarithms to base 2 --; 10.3.4; Logarithms to base 10 (common logarithms) --; 10.3.5; Logarithms to base e (natural logarithms) --; 10.4; Fitting exponential curves to data --; 10.4.1; Fitting an exponential growth model --; 10.4.2; Application: allometry --; 10.4.3; Application: allosteric regulation --; 10.5; Exponential decay --; 10.5.1; Exponential decay function: y=Ae -- kx; Case Study C11; An exponential model of animal speed --; 10.5.2; Application: sensitization and habituation --; 10.5.3; Application: drug administration --; 10.5.4; Example: radiocarbon dating --; Case study A10; An equation for logistic growth --; 10.6; Example: reduction of cholesterol level --; 10.7; Extension: a stochastic model of exponential decay --; Case study A11; Gompertz curve for population mortality --; Problems --; Revision Problems --; Historical interlude: finding the roots of polynomials --; pt. II; CALCULUS AND DIFFERENTIAL EQUATIONS --; 11; Instantaneous rate of change: the derivative --; 11.1; Introduction to the calculus --; 11.1.1; Differential calculus --; 11.1.2; Integral calculus --; 11.1.3; Differential equations --; Case Study B8; Constructing the angiogenic tumour model --; 11.2; Definition of the derivative --; 11.3; Differentiating polynomial functions --; 11.3.1; The derivative of power functions y=xn --; 11.3.2; Notation --; 11.3.3; The derivative of linear functions; 11.3.4; The derivative of polynomial functions --; Case Study C12; Differentiating the animal motion model --; 11.4; Differentiating roots and reciprocals --; 11.5; Differentiating functions of linear functions --; 11.6; Differentiating exponential functions --; 11.7; Extension: small changes and errors --; Case Study C13; Deriving the exponential model of animal speed --; Case Study A12; Differential equation for exponential growth --; Problems --; 12; Rules of differentiation --; 12.1; Differentiable functions --; 12.2; The chain rule --; 12.3; The product and quotient rules --; 12.3.1; The product rule --; 12.3.2; The quotient rule --; Case Study C14; Deriving the hyperbolic model of animal speed --; 12.4; Differentiating trigonometric functions --; 12.5; Implicit differentiation --; 12.6; Differentiating logarithmic functions --; 12.7; Differentiating inverse trigonometric functions --; 12.8; Higher-order derivatives --; 12.9; Summary of standard derivatives, and rules of differentiation --; Problems; 13; Applications of differentiation --; 13.1; Interpretation of graphs --; 13.1.1; Gradients --; 13.1.2; Roots --; 13.1.3; Critical points --; 13.1.4; Curvature --; Case study A13; Analysing the Ricker update equation --; 13.1.5; Summary --; Case study A14; The point of inflection in the logistic growth curve --; 13.2; Optimization --; 13.2.1; Optimization in the biosciences --; 13.2.2; One-dimensional unconstrained optimization --; Case study C15; Fisheries management: using calculus to find the Maximum Economic Yield --; 13.2.3; Application: tubular bones --; 13.3; Related rates --; 13.4; Polynomial approximation of functions --; 13.4.1; Linear approximation of f(x) around x=0 --; 13.4.2; Quadratic approximation of f(x) around x=0 --; 13.4.3; Maclaurin series expansions of functions --; 13.4.4; Taylor series expansions of functions --; 13.5; Extension: numerical methods for finding roots and critical points --; 13.5.1; Newton -- Raphson method for finding roots --; 13.5.2; Newton's method for optimization --; Problems; 14; Techniques of integration --; 14.1; The integral as anti-derivative --; 14.1.1; Definition and notation --; 14.1.2; The integrals of power functions, and the coefficient rule --; 14.1.3; The sum rule, and the integrals of polynomial functions --; 14.1.4; Integrals of some standard functions --; Case study C16; Integrating the hyperbolic and exponential models of animal speed --; 14.2; Integration by substitution --; 14.3; Integration by parts --; Case study A15; Solving the differential equation for exponential growth --; 14.4; Integration by partial fractions --; 14.5; Integrating trigonometric functions --; 14.5.1; The general sine and cosine functions --; 14.5.2; The tangent function --; 14.5.3; Powers of sines and cosines --; 14.5.4; Integrating excos x --; 14.5.5; Integrating inverse trigonometric functions --; 14.6; Extension: integration using power series approximations --; 14.7; Summary of standard integrals --; Problems --; 15; The definite integral --; 15.1; The integral as area under the curve; 15.1.1; The link between the integral and area --; 15.1.2; Speed-time graphs --; 15.1.3; Definition of the definite integral --; 15.2; The integral as limit of a sum --; The Riemann integral --; 15.2.2; Application: chemotherapy drug delivery --; 15.2.3; Application: laminar blood flow --; 15.3; Using techniques of integration with definite integrals --; 15.3.1; Integration by substitution --; 15.3.2; Integration by parts --; 15.3.3; Integration by partial fractions --; 15.4; Improper integrals --; 15.5; Extension: numerical integration --; 15.5.1; The trapezium rule --; 15.5.2; Simpson's rule --; 15.5.3; Using Simpson's rule with data-sets --; Problems --; 16; Differential equations I --; 16.1; Overview of differential equations --; 16.1.1; Order of a differential equation --; 16.1.2; Boundary conditions --; 16.1.3; ODEs and PDEs --; 16.2; Solution by separation of variables --; 16.2.1; Right-hand side a function of x only --; 16.2.2; Right-hand side a function of y only; 16.2.3; Variables separable --; Case Study B9; The Gompertz model of tumour growth --; Case Study A16; Solving the ODE for logistic growth --; Case Study C17; A harvesting model for fish stocks --; 16.2.4; Change of variable --; Case Study B10; The Gompertz model revisited --; 16.3; Linear first-order ODEs --; 16.4; Extension: partial differentiation --; 16.4.1; Reducing a PDE to an ODE --; 16.4.2; Error analysis in several variables --; 16.4.3; Minimization in two variables --; Problems --; 17; Differential equations II --; 17.1; Numerical methods for first-order ODEs --; 17.1.1; Euler's method --; 17.1.2; Heun's method --; Case Study C18; Numerical solution offish harvesting model --; 17.1.3; Runge -- Kutta method RK4 --; 17.2; Systems of first-order ODEs --; 17.2.1; Lotka -- Volterra models of predator -- prey dynamics --; 17.2.2; Kermack -- McKendrick model of epidemics --; Case Study A17; The peak of an epidemic --; 17.3; Extension: analytic solutions --; 17.3.1; Solving second-order ODEs; 17.3.2; Solving first-order systems --; 17.3.3; Solving partial differential equations --; 17.3.4; Further reading --; Problems --; 18; Extension: dynamical systems --; 18.1; The butterfly effect --; 18.1.1; The birth of a new science --; 18.1.2; Numerical experiments --; 18.2; Equilibria and stability --; 18.2.1; Points of equilibrium for differential equations --; 18.2.2; Stability of equilibria for differential equations --; Case Study C19; Analysing the equilibria of the harvesting model --; 18.2.3; Stability of equilibria for update equations --; 18.2.4; Numerical experiments with the update equation --; 18.3; Bifurcations... --; 18.4; ... and Chaos --; 18.5; Postscript --; Problems
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