Description
Book SynopsisClaudia Neuhauser, PhD, is Associate Vice President for Research and Director of Research Computing in the Office of the Vice President for Research at University of Minnesota. In her role as Director of Research Computing she oversees the University of Minnesota Informatics Institute (UMII), the Minnesota Supercomputing Institute (MSI), and U-Spatial. UMII fosters and accelerates data-intensive research across all disciplines in the University and develops partnership with industry. Neuhauser's research is at the interface of mathematics and biology, and focuses on the analysis of ecological and evolutionary models and the development of statistical methods in biomedical applications. She received her Diplom in mathematics from the Universität Heidelberg (Germany) in 1988, and a Ph.D. in mathematics from Cornell University in 1990. She is a fellow of the American Association for the Advancement of Science (AAAS) and a fellow of the American Mathematic
Table of Contents
(NOTE: Each chapter concludes with Key Terms and Review Problems.) 1. Preview and Review
- 1.1 Precalculus Skills Diagnostic Test
- 1.2 Preliminaries
- 1.3 Elementary Functions
- 1.4 Graphing
2. Discrete-Time Models, Sequences, and Difference Equations
- 2.1 Exponential Growth and Decay
- 2.2 Sequences
- 2.3 Modeling with Recurrence Equations
3. Limits and Continuity
- 3.1 Limits
- 3.2 Continuity
- 3.3 Limits at Infinity
- 3.4 Trigonometric Limits and the Sandwich Theorem
- 3.5 Properties of Continuous Functions
- 3.6 A Formal Definition of Limits (Optional)
4. Differentiation
- 4.1 Formal Definition of the Derivative
- 4.2 Properties of the Derivative
- 4.3 Power Rules and Basic Rules
- 4.4 The Product and Quotient Rules, and the Derivatives of Rational and Power Functions
- 4.5 Chain Rule
- 4.6 Implicit Functions and Implicit Differentiation
- 4.7 Higher Derivatives
- 4.8 Derivatives of Trigonometric Functions
- 4.9 Derivatives of Exponential Functions
- 4.10 Inverse Functions and Logarithms
- 4.11 Linear Approximation and Error Propagation
5. Applications of Differentiation
- 5.1 Extrema and the Mean-Value Theorem
- 5.2 Monotonicity and Concavity
- 5.3 Extrema and Inflection Points
- 5.4 Optimization
- 5.5 L'Hôpital's Rule
- 5.6 Graphing and Asymptotes
- 5.7 Recurrence Equations: Stability (Optional)
- 5.8 Numerical Methods: The Newton - Raphson Method (Optional)
- 5.9 Modeling Biological Systems Using Differential Equations (Optional)
- 5.10 Antiderivatives
6. Integration
- 6.1 The Definite Integral
- 6.2 The Fundamental Theorem of Calculus
- 6.3 Applications of Integration
7. Integration Techniques and Computational Methods
- 7.1 The Substitution Rule
- 7.2 Integration by Parts and Practicing Integration
- 7.3 Rational Functions and Partial Fractions
- 7.4 Improper Integrals (Optional)
- 7.5 Numerical Integration
- 7.6 The Taylor Approximation (optional)
- 7.7 Tables of Integrals (Optional)
8. Differential Equations
- 8.1 Solving Separable Differential Equations
- 8.2 Equilibria and Their Stability
- 8.3 Differential Equation Models
- 8.4 Integrating Factors and Two-Compartment Models
9. Linear Algebra and Analytic Geometry
- 9.1 Linear Systems
- 9.2 Matrices
- 9.3 Linear Maps, Eigenvectors, and Eigenvalues
- 9.4 Demographic Modeling
- 9.5 Analytic Geometry
10. Multivariable Calculus
- 10.1 Two or More Independent Variables
- 10.2 Limits and Continuity (optional)
- 10.3 Partial Derivatives
- 10.4 Tangent Planes, Differentiability, and Linearization
- 10.5 The Chain Rule and Implicit Differentiation (Optional)
- 10.6 Directional Derivatives and Gradient Vectors (Optional)
- 10.7 Maximization and Minimization of Functions (Optional)
- 10.8 Diffusion (Optional)
- 10.9 Systems of Difference Equations (Optional)
11. Systems of Differential Equations
- 11.1 Linear Systems: Theory
- 11.2 Linear Systems: Applications
- 11.3 Nonlinear Autonomous Systems: Theory
- 11.4 Nonlinear Systems: Lotka - Volterra Model of Interspecific Interactions
- 11.5 More Mathematical Models (Optional)
12. Probability and Statistics
- 12.1 Counting
- 12.2 What Is Probability?
- 12.3 Conditional Probability and Independence
- 12.4 Discrete Random Variables and Discrete Distributions
- 12.5 Continuous Distributions
- 12.6 Limit Theorems
- 12.7 Statistical Tools
Appendices
- A: Frequently Used Symbols
- B: Table of the Standard Normal Distribution
Answers to Odd-Numbered Problems References Photo Credits Index
