This chapter reviews differential calculus, which gave a tremendous impetus to the development of both pure and applied mathematics. It covers optimization and polynomial approximation of functions, together with a powerful numerical method for finding roots. It also shows how calculus can provide all the ‘pre-calculus’ results about roots and turning-points for the basic functions, including an analysis of more complicated differentiable functions. The chapter emphasizes the crucial step of interpreting the function or graph in order to extract significant information from which practical conclusions can be drawn. It highlights the use of the function f (x) and its first and second derivatives f '(x) and f ''(x) to identify significant features of the graph of a general smooth function f (x).
Applications of differentiation
Arithmetic and algebra
This chapter gives an abstract approach that builds up the rules from basic concepts of real numbers and arithmetic operations. It presents mathematics in a bioscience context and includes case studies that show how a range of different mathematical techniques are needed in the development of a biological topic. It also introduces geometric growth, which is used to model colonies of certain populations in situations where the birth rate exceeds the mortality rate, there are sufficient supplies of food, and there is an absence of predators or diseases. The chapter describes a mathematical model that expresses the rates of change of the densities of the three types of cell: healthy, stage one cancer, and angiogenic cells. It examines the evaluation of numerical expressions and algebraic expressions involving fractions, exponents, and roots.
This chapter describes communication in the context of biomedical science practice ranging from the everyday interactions between colleagues regarding clinical matters to the dissemination of the results of research. It outlines how to communicate effectively with various colleagues, such as presenting the findings of studies internally or externally as abstracts to learned societies, writing formal documents, and preparing data for publication in a journal. It also describes the key aspects of the layout of a PowerPoint presentation, the major sections of a written communication, the significant elements of an abstract, and the leading features of a research paper. The chapter promotes the correct passage of information, which is crucial for the safe and efficient working of a routine laboratory and is a key part of good laboratory practice. It emphasizes how failing to communicate in a clinical setting endangers life or can lead to poor experimental or process results.
Martin B Reed
Core Maths for the Biosciences consists of two parts. Part 1 looks atconsiders arithmetic, algebra, and functions. Here, chapters cover precision and accuracy, data tables, graphs, molarity and dilutions, variables, functions, equations, and linear functions. They also look at quadratic and polynomial functions, fitting curves, periodic functions, and exponential and logarithmic functions. Part 2 looks atfocusses on calculus and differential equations. It Chapters examines instantaneous rate-of-change, the rules of differentiation, applications of differentiation, techniques of integration, and the definite integral.
Andrew D. Blann
Data Handling starts off with an analysis of information in the biomedical sciences. It then considers handling quantities which encompasses mass, volume, and concentration. It moves on to obtaining and verifying data. Next, it looks at presenting data in graphic form. Another chapter considers quality, audit, and good laboratory practice. The next three chapters are about research, setting the scene, the analysis of modest data sets, and large data sets. Finally, the text ends with an examination of communication methods.
Data tables, graphs, interpolation
This chapter talks about finding patterns as the essence of mathematics, which can be done if the data involves measurement of two or more physical variables. It illustrates how to plot graphs of data-points on paper and in MS Excel, and it demonstrates the basic patterns to look for, such as direct and inverse proportion, and linear relations. It also elaborates how to construct a data table and a data plot, which begins by tabulating the data and then plotting the data values as points on a graph. The chapter highlights a typical experiment, wherein the experimenter allows the independent variable to increase or decrease and measures the value of the dependent variable. It clarifies how data-point plots show the relationship between the variables through a graph.
The definite integral
This chapter provides a geometric and physical interpretation of the integral, which leads to the concept of the definite integral as a numerical quantity instead of a function. It highlights practical applications of the definite integral, including a powerful method for evaluating definite integrals numerically. It also demonstrates how to calculate a definite integral by working out the indefinite integral F(x)| = (x) dx and omitting the constant of integration. The chapter points out that areas below the x-axis are counted as negative, noting the importance of checking that the function does not cross the x-axis anywhere within the interval. It talks about a concept of the definite integral that is independent of the idea of differentiation, which can be used to define the definite integral mathematically as a process that involves taking the limit called Riemann integration.
Differential equations I
This chapter deals with the differential equation, which is an equation involving a derivative and its solution requires finding y as a function of x. It points out that finding solutions for differential equations require basic algebra, the theory of functions, and differential and integral calculus. It also classifies the different types of differential equation, including methods for solving first-order differential equations. The chapter covers solution techniques applied to mathematical models of biological processes and numerical techniques implemented in MS Excel, which produce the solution as a set of data-points that can be graphed. It mentions the main classification of differential equations in terms of their order and the concept of boundary conditions.
Differential equations II
This chapter introduces numerical methods that can be used on problems that are too hard to solve by standard algebra, such as finding roots of complicated equations. It demonstrates numerical methods for solving a first-order ordinary differential equations (ODE) with an initial condition. It also looks at problems involving two or more simultaneous ODEs, particularly the equations relating the growth of populations of different species in a predator–prey situation. The chapter explains how a numerical method produces a set of data-points that lie on or close to the true solution curve. It analyses the accuracy of Euler's method, which can be improved by reducing the steplength or taking shorter steps between data-points.
Exponential and logarithmic functions
This chapter deals with exponential and logarithmic functions, which constitute arguably the most important area of mathematics for bioscientists. This is because many nonlinear natural processes involve some form of exponential growth or exponential decay. The chapter talks about the natural process of exponential growth and relates it to the geometric growth model. It also cites an example wherein a cell culture of 100 cells doubles in size each hour, which can be modelled by the function N(t) = 100.2t . The chapter introduces allometry, which is the study of power-law relationships between physical characteristics of an organism; these usually take the mass of the organism as the basic characteristic.
Extension: dynamical systems
This chapter focuses on dynamical systems, wherein the simplest nonlinear mathematical models could give rise to incredibly complex behaviour. It mentions scientists studying biological and physical processes of nature that were in the forefront of the revolution in mathematics for discovering dynamical systems. It also introduces some basic concepts of complexity theory: equilibria and stability, bifurcations, and chaos. The chapter begins with a brief snapshot of the birth of chaos theory, particularly theories of pattern formation and ecological implications. It cites the truncation error, which is another source of error in numerical methods that happens by approximating the true function by a Taylor series that is truncated after the first two or three terms.
Fitting curves; rational and inverse functions
This chapter looks at two concepts that can provide functions: the reciprocal of a function and the inverse of a function. It discusses a class of functions called rational functions, which have important applications in the biosciences. It also explores how the functions produced can be used with real data, such as how to use data-points obtained from experiment or observation and how to determine the values of the parameters in the function. The chapter demonstrates how to find the values of the parameters m and c from two data-points, including how to put a best-fit straight line through to more than two data-points. It reviews a technique that can fit general curves to data and reveals how the equation of the best-fit straight line is calculated in MS Excel.
Handling quantities: mass, volume, and concentration
This chapter highlights important aspects of chemistry central to biomedical science practice, such as the Système International (SI) convention. It describes the structure of the atom and key features of the periodic table. It also reviews the concepts of atomic number, atomic mass, acids, bases, and buffers, including the determination of the mole and molarity. The chapter introduces tools in the field of chemistry that are used for measuring and characterizing different substances. These have central importance to laboratory-based diagnosis. Finally, it discusses electrophoresis, ion-specific electrodes, colorimetry, and high-performance liquid chromatography.
Information in biomedical science
This chapter looks at the variety of different types of information in the biomedical sciences, the bases of different types of data, and key features of qualitative information. It covers the fundamentals of quantitative information, the mean, standard deviation, the median and interquartile range, the numerical basis of the reference range, and scaling methods that permit the quantitative analysis of qualitative information. The chapter provides a broad overview of the issues surrounding information handling and statistical analysis as applied to biomedical science. It explores the different ways in which information can be described and analysed, including the issues that relate to best practice when it is being handled.
Instantaneous rate of change: the derivative
This chapter talks about the mathematical rate-of-change theory, in other words, calculus, which has two branches: differential calculus and integral calculus. It considers differential equation as the most powerful use of calculus in the biosciences as it is an equation involving the derivative of a function and the function itself. It also examines how epidemiologists come up with predictions for the likely course of a new strain of influenza by modelling the instantaneous rate at which the disease is spreading. The chapter clarifies that rate-of-change involves an independent variable and a dependent variable, such as the velocity of a moving object is the rate-of-change of its position. It illustrates this topic using the Michaelis–Menten equation as this describes an enzyme-catalysed biochemical reaction wherein the rate of reaction v depends on the substrate concentration s.
Linear functions and curve sketching
This chapter discusses the general formula for the function that involves x and various parameters and the shape of the graph of y = f(x). It shows useful points on the graph, such as where it crosses the x- or y-axes and demonstrates how to use information about a particular function to work out the equation of the curve. It also cites some of the physical processes that can be modelled by the function. The chapter mentions the graph-drawing package called FNGraph, which can show the direct and inverse proportion functions, with the constant of proportionality k as a parameter. It considers a function where the output value is the same, that is y stays constant whatever the value of x.
Molarity and dilutions
This chapter sets out the calculations required to work effectively with materials in solutions. It explains that a solution consists of a certain amount of solid or liquid chemical, called solutes, dissolved in a volume of liquid, called the solvent, which for biological purposes is usually water. It also defines the molar mass of a substance as the mass (in grams) of one mole of the substance, noting that the mass of a substance can help determine the number of moles and molecules present. The chapter points out that, as each molecule is made up of atoms, the mass required for one mole can easily be determined by addition of the masses of the individual atoms that make up the molecule. It discusses the concept of moles, which determines the concentration of solutions in terms of how many moles of a substance are contained in a solution; this is called molarity of the solution.
Obtaining and verifying data
This chapter clarifies how to obtain data from machines and people, how to ensure the results are correct, and how the results should be interpreted. It analyses sources of potential error and looks at crucial factors in laboratory practice, such as optimizing assay performance, ensuring reliability, and monitoring reliability. It also considers operator performance, the clinical value of data generated in the laboratory, and the formal comparison of different techniques. The chapter explores pre-analytical error, the importance of accuracy and precision, and the differences between intra- and inter-assay coefficients of variation. It outlines the principles of sensitivity and specificity and the methods for comparing a new method with an existing method.
This chapter deals with periodic functions, which are functions that have a certain basic pattern of behaviour or cycle that repeats. It refers to the basic trigonometric functions: the sine and cosine, which are the most commonly used and relevant periodic functions. It also discusses a simple MS Excel function that can be used to produce periodic graphs with sudden changes in value. The chapter explains that sawtooth functions are used in sound waves created in music synthesizers and are used to describe the horizontal scanning function in cathode-ray-tube televisions and monitors. It considers functions whose graphs are smooth oscillations, wherein the curves are called sinusoidal.
Presenting data in graphical form
This chapter provides a background on the most common formats in which data can be presented visually, and how such data can be interpreted. It looks at methods of presenting both categorical and continuously variable data and reviews software packages that can produce excellent figures suitable for presentation and publication, limiting the requirement for hand-drawn artistic skills. It also explains how to interpret numerical data when they are presented graphically in the form of a normal and non-normal distribution, scatterplot, histogram, pie chart, dot plot and box plot and whisker plot, line plot, and graph. The chapter clarifies that continuously variable data with a normal distribution is represented by a bell-shaped curve, while data with a non-normal distribution is skewed to the left or to the right. It mentions data with a categorical distribution that is presented as a histogram.