In Example 4e of Section 1. In Section 1.

## Introductory mathematics for the life sciences

Again we have separated, in advance, a simple idea from a more advanced one. Of course Problem 12 of Problems 1. Early treatment of summation notation: This topic is necessary for study of the definite integral in Chapter 14 but it is useful long before that. Since it is a notation that is new to most students at this level, but no more than a notation, we get it out of the way in Chapter 1. By using it when convenient, before coverage of the definite integral , it is not a distraction from that challenging concept.

Section 1. The very definition is stated in a fashion that paves the way for the more important and more basic definition of function in Chapter 2. In summing the terms of a sequence we are able to practice the use of summation notation introduced in the preceding section. Both the present and the future values of an annuity are obtained by summing finite geometric sequences. Sum of an infinite sequence: In the course of summing the terms of a finite sequence, it is natural to raise the possibility of summing the terms of an infinite sequence.

This is a nonthreatening environment in which to provide a first foray into the world of limits. We simply explain how certain infinite geometric sequences have well-defined sums and phrase the results in a way that creates a toehold for the introduction of limits in Chapter These particular infinite sums enable us to introduce the idea of a perpetuity, first informally in the sequence section, and then again in more detail in a separate section in Chapter 5.

Section 2. Once we have done some calculus there are particular ways to use calculus in the study of functions of several variables, but these aspects should not be confused with the basics that we use throughout the book. We begin by describing what we call the Leontief matrix A as an encoding of the input and output relationships between sectors of an economy. Since this matrix can often be assumed to be constant, for a substantial period of time, we begin by assuming that A is a given.

The simpler problem is then to determine the production X which is required to meet an external demand D for an economy whose Leontief matrix is A. We provide a careful account of this as the solution of. Since A can be assumed to be fixed while various demands D are investigated, there is some justification to compute. However, use of a matrix inverse should not be considered an essential part of the solution.

Finally, we explain how the Leontief matrix can be found from a table of data that might be available to a planner. Birthday probability in Section 8. While this problem is given as an example in many texts, the recursive formula that we give for calculating the probability as a function of n is not a common feature. It is reasonable to include it in this book because recursively defined sequences appear explicitly in Section 1.

Markov Chains: We noticed that considerable simplification to the problem of finding steady state vectors is obtained by writing state vectors as columns rather than rows.

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This does necessitate that a transition matrix T D. Sign Charts for a function in Chapter The sign charts that we introduced in the 12th edition now make their appearance in Chapter Our point is that these charts can be made for any real-valued function of a real variable and their help in graphing a function begins prior to the introduction of derivatives. When this is possible, the graph of the function becomes almost self-evident. MCN focuses on several seminal mathematical models used in neuronal coding, neuronal and network dynamics, and learning in neuronal networks.

The main objective of the course is to develop in students the ability to interpret, analyze, and develop these models. A secondary objective is to stimulate a deeper understanding and active use of the mathematical concepts used in the formulating and analyses of models. Concerning the introductory sequence, it is now the main mathematics requirement for the biology major. Unfortunately, student math background and aptitude vary widely. We attempt to balance continuity with high school math training, as varied as it is, with emphases on modeling and problem solving, which are usually new to our students.

To manage quite demanding syllabi, there is a mandatory additional contact hour that allows review of material, question-and-answer, and more frequent testing; so, students have 3 lecture hours and 1 lab hour each week. And Emory College offers a supplemental instruction program based on peer tutoring, an important resource.

Substantial integration of life science topics is key both to the educational purpose and to student interest. This has been underscored with presentations by biology faculty on topics such as population biology and modeling physiological systems such as the neuron. In teaching inferential statistics, it is important to emphasize experimental design and to use real experimental data. Coordination of several text and Web sources is required.

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Unlike the situation in physics-oriented calculus education, there are not many texts that address the range of topics required. The choice of proper content and textbooks for MCN is also a challenge. There are several excellent textbooks on the application of mathematical methods in the neurosiences, including Foundations of Cellular Neurophysiology Johnston and Wu, , Computational Cell Biology Fall et al. From our perspective, the most balanced representation of mathematical concepts used in the modern neurosciences is presented in Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems Dayan and Abbott, For example, Dayan and Abbott consider not only excitability of individual neurons and dynamics of activity in neuronal networks but also information-theoretic approaches to neuronal code and learning in neuronal networks—important themes that are often underrepresented in textbooks.

Dayan and Abbott clearly demonstrate how a mathematical approach helps to solve many fundamental problems in neuroscience.

Unfortunately, mathematics is just used and not discussed in the book. Our challenge is to retain the research spirit of the book but adapt the content to undergraduates who have just begun to understand the role of mathematics in neuroscience. However, during the course we referred to other textbooks on specific topics.

For example, when considering modeling neurons with simple nondifferential equations, we used some models and diagrams from Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting Izhikevich, Another challenge is building adequate enrollment. MCN is an elective course. The health career focus of the majority of majors presents a real challenge. It is hoped that courses such as MCN, supported by the introductory sequence and early independent research opportunities, will be useful in attracting more students intent on science graduate programs, as well as changing the ambitions of current students.

In the early s, the Science and Mathematics Departments at Emory College were meeting regularly to discuss Science , an ambitious faculty, program, and infrastructure development plan. We were eager to expand our majors, individually and in collaborative interdisciplinary programs. We also wanted to modify the premed orientation of the strong majority of lab science majors and to attract more students to our majors intent on graduate education in what have come to be called the science, technology, engineering, and mathematics STEM disciplines.

Even 20 yr ago, it was obvious that the life sciences were being transformed by the essential use of mathematics, statistics, and computing and that the standard physics-based calculus sequence did not provide the right quantitative training for life science majors. We began in — after extended discussions with faculty in biology and biostatistics. Version 1: — Main reference: Mathematics for the Biosciences Cullen, The second course was substantially different from standard Calculus 2, substituting DEs, discrete probability, and inferential statistics for integration techniques and series and sequences.

The course was designed with input from the Department of Biology and was recommended by it for its majors, but it was not required. The course was taught in two small sections each fall, with one section in the spring. During the same years, enrollments in standard, business, and Advanced Placement AP -based calculus were approximately — The department was pleased with the initial offerings but not with the available texts.

We found it difficult to maintain enough similarity with Math to allow students to switch between sequences and to introduce interesting life science applications.

Enrollments were disappointingly low, because the courses were not well enough publicized and although recommended by the Department of Biology, they were not required by any majors. Version 2: — The text and syllabus marked a substantial departure from the original objective of keeping the first term close to Calculus 1. Adler begins with a precalculus review and an introduction to discrete dynamics. This is followed by differential calculus and then by applications featuring the relationship between the derivative and stability of equilibria.

DEs and definite and indefinite integrals complete the first course. Substantial modeling topics are included throughout the syllabus. The second term begins with more DEs, including equilibria and stability for autonomous DEs. Systems of DEs and the phase plane are used to model competition, predator—prey interactions, and epidemics. An ambitious model of the neuron concluded the DE modeling segment of the second term, approximately one-third of the course.

This was followed by an introduction to probability and descriptive statistics, discrete and continuous random variables, the binomial and normal distributions, and the central limit theorem CLT. Inferential statistics, featuring estimates of the mean, confidence intervals, and hypothesis testing, are the last topics. This was used for both modeling and statistics. The department has an instructional computer lab with 32 networked work stations that allowed MATLAB training and assignments. Instructors were pleased with the evolving syllabus and choice of topics.

The course became difficult to offer logistically and just a bit overwhelming. Although the courses were still not required by any of the life science majors, Department of Biology advisors strongly suggested the sequence to freshmen and continuing students who had delayed taking mathematics.

Version 3: — This was the most significant departure from standard calculus that we attempted. We used Mathematical Models in Biology Allman and Rhodes, to introduce topics such as population dynamics and initially study them with discrete difference equations. Calculus was introduced, as needed, to understand continuous change.

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## Introductory Mathematics for the Life Sciences - David Phoenix - Google книги

In term 1, we developed discrete and continuous models of populations, infectious disease, and predator—prey systems. These were analyzed with cobwebbing and phase lines; stability of equilibria was determined with the derivative. After introducing the integral and DEs, we saw how autonomous DEs provide continuous versions of the previous population models. Term 2 began with systems of DEs, including predator—prey and the susceptible-infected-recovered SIR model of epidemics.

Phase planes and slope fields were used to provide qualitative information about solutions. Probability and statistics were then motivated by genetics and molecular evolution. Probability topics included Bayes theorem, rare disease examples, discrete and continuous random variables, binomial and geometric distributions, the normal distribution, and the CLT.

Statistical methods included descriptive and inferential statistics, with emphasis on hypothesis testing, sampling distributions, and analysis of variance. MATLAB was an important demonstration tool in the course, in part because it is used extensively in the Allman and Rhodes textbook. We had more opportunity for this because an extra lab hour was added to the course in —, a feature that we have retained. Students in each lecture section approximately 40 in each of two are divided into two labs that taught by the same lab instructor. Both course instructors were pleased with the syllabus.

Presenting students with both discrete and continuous versions of the same models seemed to work well.