| Description |
xxvii, 576 pages : illustrations ; 27 cm |
| Contents |
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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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 |
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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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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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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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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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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 |
| Summary |
This text introduces the range of mathematical concepts that bioscience students need to master during their studies. Starting from fundamental concepts, it blends clear explanations and biological examples throughout as it equips the reader with the full range of mathematical tools required by biologists today |
| Notes |
"With interactive Excel® workbooks online to help you master the essentials."--P.1 of cover |
| Bibliography |
Includes bibliographical references (pages [571]-572) and index |
| Subject |
Biomathematics -- Textbooks.
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Biomathematics.
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Life sciences -- Mathematics -- Textbooks.
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Mathematics -- Textbooks.
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Mathematics.
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| Genre/Form |
Textbooks.
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| LC no. |
2011377442 |
| ISBN |
0199216347 (paperback) |
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9780199216345 (paperback) |
