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).
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Chapter
Applications of differentiation
Chapter
Applying our simulation approach beyond null hypothesis testing: parameter estimation, bayesian, and model-selection contexts
This chapter talks about the performance of a simulation analysis in the contexts of the precision of a parameter estimate, Bayesian statistics, and model selection. It describes null hypothesis testing as an absolute dominant form of statistical inquiry that remains important to a lot of researchers in many different areas of the life sciences. It also reviews toolkits of statistical approaches which are in common use alongside null hypothesis testing. The chapter reviews the simulation approach to power in the context of null hypothesis testing and other contexts that estimate the size of a parameter or select from a number of alternative models. It uses examples to explore simulation analysis in the context of precision of parameter estimation.
Chapter
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.
Chapter
Communication
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.
Book
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.
Book
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.
Chapter
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.
Chapter
Dealing with multiple hypotheses
This chapter explains how to evaluate the powers for study designs with multiple hypotheses, emphasizing that no single design will offer exactly the same power for all the hypotheses. It points out that ranking hypotheses in terms of their importance can help with the process of finding a design that offers sufficient power for all the hypotheses of prime importance. It also discusses how to evaluate whether the power available for testing ancillary hypotheses is sufficient to continue to include them in the study after one has settled on a design with sufficient power for the most important hypotheses. The chapter explores how to select a single experiment offering good power to address all hypotheses and involve further simulations after carrying out power analyses for each of the hypotheses in isolation. It examines power analyses that flag up situations where bonus investigations are of little value. This is because an experiment optimized to address questions of primary interest means there is low power to address bonus hypotheses.
Chapter
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.
Chapter
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.
Chapter
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.
Chapter
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.
Chapter
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.
Chapter
Extensions to other designs
This chapter covers how to simulate experimental studies where the predictor variable is continuous and there is a straight-line relationship between the response and the predictor. Moreover, it explores simulations that include other kinds of relationships, such as a quadratic one. It also demonstrates how to deal with response variables which are not continuous, as well as with variation that does not follow a normal distribution. The chapter looks at another common type of biological study in which both the response and the predictor variable are continuous. It talks about the possibility of having a different number of replicates at each level and explores situations where the relationship is either linear or non-linear.
Chapter
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.
Chapter
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.
Chapter
How to quantify power by simulation
This chapter focuses on the necessity of being able to make estimates concerning a potential study design to assess its power. It explains how to generate assumptions and convert them into a good estimate of statistical power by conducting a simulation of a potential study in R or in any other programming language. The chapter also shows how to modify one’s approach to compare multiple possible versions of a study to decide which one offers the power that is needed most efficiently. The key aspects involved in estimating power covered in the chapter can be applied to any study. The chapter also introduces basic computational skills which form part of the toolkit that is needed to perform power analyses.
Chapter
Improving the power of an experiment
This chapter discusses why inherent variation is a challenge to achieving good statistical power and how design decisions can reduce or control the effects of inherent variation. It demonstrates the increase of power by changing the experimental design to increase the strength of the effect that should be detected. It also addresses why increasing sample size increases power and why higher power will be more costly to achieve. The chapter talks about changes to the experiment that increase its power but may incur other costs, which could be felt financially, in terms of increased effort required, or could involve ethical issues. It explores the quantification of power benefits and the costs of different options for an experiment that have a rational basis for choosing the alternative with the most attractive trade-off between power and costs.
Chapter
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.
Chapter
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.
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