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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 |
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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 |
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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 |
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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 -- 4.1.3. 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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 -- 15.2.1. 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 |
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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 |
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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. |
| General note | "With interactive Excel® workbooks online to help you master the essentials."--P.1 of cover. |
| Bibliography note | Includes bibliographical references (p. [571]-572) and index. |
| ISBN | 9780199216345 (pbk.) |
| ISBN | 0199216347 (pbk.) |
