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Calculus for The Life Sciences



Calculus for The Life Sciences

Sebastian J. Schreiber, Karl Smith, Wayne Getz

ISBN: 978-1-118-89336-4 January 2014 744 Pages

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Authored by two distinguished researchers/teachers and an experiences, successful textbook author, Calculus for Life Sciences is a valuable resource for Life Science courses. As life-science departments increase the math requirements for their majors, there is a need for greater mathematic knowledge among students. This text balances rigorous mathematical training with extensive modeling of biological problems. The biological examples from health science, ecology, microbiology, genetics, and other domains, many based on cited data, are key features of this text.

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Preview of Modeling and Calculus

1 Modeling with Functions

1.1 Real Numbers and Functions

1.2 Data Fitting with Linear and Periodic Functions

1.3 Power Functions and Scaling Laws

1.4 Exponential Growth

1.5 Function Building

1.6 Inverse Functions and Logarithms

1.7 Sequences and Difference Equations

2 Limits and Derivatives

2.1 Rates of Change and Tangent Lines

2.2 Limits

2.3 Limit Laws and Continuity

2.4 Asymptotes and Infinity

2.5 Sequential Limits

2.6 Derivative at a Point

2.7 Derivatives as Functions

Group Projects

3 Derivative Rules and Tools

3.1 Derivatives of Polynomials and Exponentials

3.2 Product and Quotient Rules

3.3 Chain Rule and Implicit Differentiation

3.4 Derivatives of Trigonometric Functions

3.5 Linear Approximation

3.6 Higher Derivatives and Approximations

3.7 l’Hoˆ pital’s Rule

Group Projects

4 Applications of Differentiation

4.1 Graphing Using Calculus

4.2 Getting Extreme

4.3 Optimization in Biology

4.4 Decisions and Optimization

4.5 Linearization and Difference Equations

Group Projects

5 Integration

5.1 Antiderivatives

5.2 Accumulated Change and Area under a Curve

5.3 The Definite Integral

5.4 The Fundamental Theorem of Calculus

5.5 Substitution

5.6 Integration by Parts and Partial Fractions

5.7 Numerical Integration

5.8 Applications of Integration

Group Projects

6 Differential Equations

6.1 A Modeling Introduction to Differential Equations

6.2 Solutions and Separable Equations

6.3 Linear Models in Biology

6.4 Slope Fields and Euler’s Method

6.5 Phase Lines and Classifying Equilibria

6.6 Bifurcations

Group Projects

7 Probabilistic Applications of Integration

7.1 Histograms, PDFs, and CDFs

7.2 Improper Integrals

7.3 Mean and Variance

7.4 Bell-Shaped Distributions

7.5 Life Tables

Group Projects

8 Multivariable Extensions

8.1 Multivariate Modeling

8.2 Matrices and Vectors

8.3 Eigenvalues and Eigenvectors

8.4 Systems of Linear Differential Equations

8.5 Nonlinear Systems

Group Projects

  • Topics are motivated where possible by significant biological applications, several of which appear in no other introductory calculus texts. Examples involve real world data and whenever possible, these examples motivate and develop formal definitions, procedures, and theorems.
  •  Each chapter includes one or more projects which can be used for individual or group work. These projects will be diverse in scope ranging from a study of enzyme kinetics to the heart rates in mammals to disease outbreaks.
  •  Difference equations and their applications are interwoven at the sectional level in the first four chapters.
  •  Introduces two topics, bifurcation diagrams and life history tables, which are often not covered in other calculus books. Bifurcation diagrams for univariate differential equations are a conceptually rich yet accessible topic. They provide an opportunity to illustrate that small parameter changes can have large dynamical effects.
  •  Throughout the text are problems described as Historical Quest. These problems are not just historical notes to help one see mathematics and biology as living and breathing disciplines, but are designed to involve the student in the quest of pursing some great ideas in the history of science.
  •  Throughout the book, concepts are presented visually, numerically, algebraically, and verbally. By presenting these different perspectives, it enhances as well as reinforces the students understanding of and appreciation for the main ideas.

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