Claudia 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 Mathematical Society (AMS).

Marcus Roper, PhD, is a Professor of Mathematics at UCLA. He specializes in developing mathematical models inspired by physics and biology. His particular research interests include biological transport networks, such as fungal mycelia and the microvascular system. Although many of the projects he works on are experimentally inspired, his goal is to develop new image analysis methods and to gain a better understanding of the world around us.

1.1     Precalculus Skills Diagnostic Test    

1.2     Preliminaries    

1.2.1     The Real Numbers    

1.2.2     Lines in the Plane    

1.2.3     Equation of the Circle   

1.2.4     Trigonometry    

1.2.5     Exponentials and Logarithms    

1.2.6     Complex Numbers and Quadratic Equations  

1.3        Elementary Functions    

1.3.1     What Is a Function?    

1.3.2     Polynomial Functions    

1.3.3     Rational Functions    

1.3.4     Power Functions    

1.3.5     Exponential Functions    

1.3.6     Inverse Functions    

1.3.7     Logarithmic Functions    

1.3.8     Trigonometric Functions    

1.4        Graphing    

1.4.1     Graphing and Basic Transformations of Functions    

1.4.2     The Logarithmic Scale    

1.4.3     Transformations into Linear Functions    

1.4.4     From a Verbal Description to a Graph (Optional)    

2.1     Exponential Growth and Decay    

2.1.1     Modeling Population Growth in Discrete Time    

2.1.2     Recurrence Equations    

2.1.3     Visualizing Recurrence Equations    

2.2     Sequences    

2.2.1     What Are Sequences?    

2.2.2     (Optional) Using Spreadsheets to Calculate a Recursive Sequence    

2.2.3     Limits    

2.2.4     Recurrence Equations    

2.2.5     Using Notation to Represent Sums of Sequences    

2.3     Modeling with Recurrence Equations    

2.3.1     Density-Dependent Population Growth    

2.3.2     Density-Dependent Population Growth: The Beverton - Holt Model    

2.3.3     The Discrete Logistic Equation    

2.3.4     Modeling Drug Absorption (optional)  

3.1     Limits    

3.1.1     A Non-Rigorous Discussion of Limits    

3.1.2     Pitfalls of Finding Limits    

3.1.3     Limit Laws    

3.2     Continuity    

3.2.1     What Is Continuity?    

3.2.2     Combinations of Continuous Functions    

3.3     Limits at Infinity    

3.4     Trigonometric Limits and the Sandwich Theorem    

3.4.1     Geometric Argument for Trigonometric Limits    

3.4.2     The Sandwich Theorem (Optional)    

3.5     Properties of Continuous Functions    

3.5.1     The Intermediate-Value Theorem and The Bisection Method    

3.5.2     Using a Spreadsheet to Implement  the Bisection Method (Optional)    

3.5.3     A Final Remark on Continuous Functions    

3.6     A Formal Definition of Limits (Optional)    

4.1     Formal Definition of the Derivative    

4.2     Properties of the Derivative    

4.2.1     Interpreting the Derivative    

4.2.2     Differentiability and Continuity    

4.3     Power Rules and Basic Rules    

4.4     The Product and Quotient Rules, and the Derivatives of Rational and Power Functions    

4.4.1     The Product Rule    

4.4.2     The Quotient Rule    

4.5     Chain Rule    

4.5.1     The Chain Rule    

4.5.2     Proof of the Chain Rule    

4.6     Implicit Functions and Implicit Differentiation    

4.6.1     Implicit Differentiation    

4.6.2     Related Rates    

4.7     Higher Derivatives    

4.8     Derivatives of Trigonometric Functions    

4.9     Derivatives of Exponential Functions    

4.10     Inverse Functions and Logarithms    

4.10.1     Derivatives of Inverse Functions    

4.10.2     The Derivative of the Logarithmic Function    

4.10.3     Logarithmic Differentiation    

4.11     Linear Approximation and Error Propagation    

5.1     Extrema and the Mean-Value Theorem    

5.1.1     The Extreme-Value Theorem    

5.1.2     Local Extrema    

5.1.3     The Mean-Value Theorem  

5.2     Monotonicity and Concavity    

5.2.1     Monotonicity    

5.2.2     Concavity    

5.3     Extrema and Inflection Points    

5.3.1     Extrema    

5.3.2     Inflection Points    

5.4     Optimization    

5.5     L'Hôpital's Rule    

5.6     Graphing and Asymptotes    

5.7     Recurrence Equations: Stability (Optional)    

5.7.1     Exponential Growth    

5.7.2     Stability: General Case    

5.7.3     Population Growth Models    

5.8     Numerical Methods: The Newton - Raphson Method (Optional)    

5.9     Modeling Biological Systems Using Differential Equations (Optional)    

5.9.1     Modeling Population Growth    

5.9.2     Interpreting the Mathematical Model    

5.9.3     Passage of Drugs Through the Human Body    

5.10     Antiderivatives    

6.1     The Definite Integral    

6.1.1     The Area Problem    

6.1.2     The General Theory of Riemann Integrals    

6.1.3     Properties of the Riemann Integral    

6.1.4     Order Properties of the Riemann Integral (Optional)   

6.2     The Fundamental Theorem of Calculus    

6.2.1     The Fundamental Theorem of Calculus (Part I)    

6.2.2     Leibniz's Rule and a Rigorous Proof of the Fundamental Theorem of Calculus (Optional)    

6.2.3     Antiderivatives and Indefinite Integrals    

6.2.4     The Fundamental Theorem of Calculus (Part II)    

6.3     Applications of Integration    

6.3.1     Cumulative Change    

6.3.2     Average Values    

6.3.3     The Mean Value Theorem (Optional)    

6.3.4     Areas (Optional)    

6.3.5     The Volume of a Solid (Optional)    

6.3.6     Rectification of Curves (Optional)       

7.1     The Substitution Rule    

7.1.1     Indefinite Integrals    

7.1.2     Definite Integrals    

7.2     Integration by Parts and Practicing Integration    

7.2.1     Integration by Parts    

7.2.2     Practicing Integration    

7.3     Rational Functions and Partial Fractions    

7.3.1     Proper Rational Functions    

7.3.2     Partial-Fraction Decomposition    

7.3.3     Repeated Linear Factors    

7.3.4     Irreducible Quadratic Factors (optional)    

7.3.5     Summary    

7.4     Improper Integrals (Optional)    

7.4.1     Type 1: Unbounded Intervals    

7.4.2     Type 2: Unbounded Integrand    

7.4.3     A Comparison Result for Improper Integrals    

7.5     Numerical Integration    

7.5.1     The Midpoint Rule    

7.5.2     The Trapezoidal Rule    

7.5.3     Using a Spreadsheet for Numerical Integration    

7.5.4     Estimating Error in a Numerical Integration (Optional)  

7.6     The Taylor  Approximation (optional)    

7.6.1     Taylor Polynomials    

7.6.2     The Taylor Polynomial about   x   =  a    

7.6.3     How Accurate Is the Approximation? (Optional)  

7.7     Tables of Integrals (Optional)    

8.1     Solving Separable Differential Equations    

8.1.1     Pure-Time Differential Equations    

8.1.2     Autonomous Differential Equations    

8.1.3     General Separable Equations    

8.2     Equilibria and Their Stability    

8.2.1     Equilibrium Points    

8.2.2     Graphical Approach to Finding Equilibria    

8.2.3     Stability of Equilibrium Points    

8.2.4     Sketching Solutions Using the Vector Field Plot    

8.2.5     Behavior Near an Equilibrium    

8.3     Differential Equation Models    

8.3.1     Compartment Models    

8.3.2     An Ecological Model    

8.3.3     Modeling a Chemical Reaction    

8.3.4     The Evolution of Cooperation    

8.3.5     Epidemic Model    

8.4     Integrating Factors and Two-Compartment Models    

8.4.1     Integrating Factors    

8.4.2     Two-Compartment Models    

9.1     Linear Systems    

9.1.1     Graphical Solution    

9.1.2     Solving Equations Using Elimination    

9.1.3     Solving Systems of Linear Equations    

9.1.4     Representing Systems of Equations Using Matrices    

9.2     Matrices    

9.2.1     Matrix Operations    

9.2.2     Matrix Multiplication    

9.2.3     Inverse Matrices    

9.2.4     Computing Inverse Matrices (Optional)    

9.3     Linear Maps, Eigenvectors, and Eigenvalues    

9.3.1     Graphical Representation    

9.3.2     Eigenvalues and Eigenvectors    

9.3.3     Iterated Maps (Needed for Section 10.7)    

9.4     Demographic Modeling    

9.4.1     Modeling with Leslie Matrices    

9.4.2     Stable Age Distributions in Demographic Models    

9.5     Analytic Geometry    

9.5.1     Points and Vectors in Higher Dimensions    

9.5.2     The Dot Product    

9.5.3     Parametric Equations of Lines    

10.1    Two or More Independent Variables

10.1.1    Defining a Function of Two or More Variables

10.1.2    The Graph of a Function of Two Independent Variables - Surface plot

10.1.3    Heat Maps

10.1.4    Contour plots

10.2    Limits and Continuity (optional)

10.2.1    Informal Definition of Limits

10.2.2    Continuity

10.2.3    Formal Definition of Limits

10.3    Partial Derivatives

10.3.1    Functions of Two Variables

10.3.2    Functions of More Than Two Variables

10.3.3    Higher-Order Partial Derivatives

10.4    Tangent Planes, Differentiability, and Linearization

10.4.1    Functions of Two Variables

10.4.2    Vector-Valued Functions

10.5    The Chain Rule and Implicit Differentiation (Optional)

10.5.1    The Chain Rule for Functions of Two Variables

10.5.2    Implicit Differentiation

10.6    Directional Derivatives and Gradient Vectors (Optional)

10.6.1    Deriving the Directional Derivative

10.6.2    Properties of the Gradient Vector

10.7    Maximization and Minimization of Functions (Optional)

10.7.1    Local Maxima and Minima

10.7.2    Global Extrema

10.7.3    Extrema with Constraints

10.7.4    Least-Squares Data Fitting

10.8    Diffusion (Optional)

10.9    Systems of Difference Equations (Optional)

10.9.1    A Biological Example

10.9.2    Equilibria and Stability in Systems of Linear Recurrence Equations

10.9.3    Equilibria and Stability of Nonlinear Systems of Difference Equations

11.1    Linear Systems: Theory

11.1.1    The Direction Field

11.1.2    Solving Linear Systems

11.1.3    Equilibria and Stability

11.1.4    Systems with Complex Conjugate Eigenvalues

11.1.5    Summary of the Theory of Linear Systems

11.2    Linear Systems: Applications

11.2.1    Two-Compartment Models

11.2.2    A Mathematical Model for Love

11.2.3    The Harmonic Oscillator (Optional)

11.3    Nonlinear Autonomous Systems: Theory

11.3.1    Analytical Approach

11.3.2    Graphical Approach for 2 x 2 Systems (Optional)

11.4    Nonlinear Systems: Lotka - Volterra Model of Interspecific Interactions

11.4.1    Competition

11.4.2    A Predator--Prey Model

11.5    More Mathematical Models (Optional)

11.5.1    The Community Matrix

11.5.2    Neuron Activity

11.5.3    Enzymatic Reactions

12.1     Counting    

12.1.1     The Multiplication Principle    

12.1.2     Permutations    

12.1.3     Combinations    

12.1.4     Combining the Counting Principles    

12.2     What Is Probability?    

12.2.1     Basic Definitions    

12.2.2     Equally Likely Outcomes    

12.3     Conditional Probability and Independence    

12.3.1     Conditional Probability    

12.3.2     The Law of Total Probability    

12.3.3     Independence    

12.3.4     The Bayes Formula    

12.4     Discrete Random Variables and Discrete Distributions    

12.4.1     Discrete Distributions    

12.4.2     Mean and Variance    

12.4.3     The Binomial Distribution    

12.4.4     The Multinomial Distribution    

12.4.5     Geometric Distribution    

12.4.6     The Poisson Distribution    

12.5     Continuous Distributions    

12.5.1     Density Functions    

12.5.2     The Normal Distribution    

12.5.3     The Uniform Distribution    

12.5.4     The Exponential Distribution    

12.5.5     The Poisson Process    

12.5.6    Aging    

12.6     Limit Theorems    

12.6.1     The Law of Large Numbers    

12.6.2     The Central Limit Theorem    

12.7     Statistical Tools    

12.7.1     Describing Univariate Data    

12.7.2     Estimating Parameters    

12.7.3     Linear Regression    

Appendix A Frequently Used Symbols

Appendix B Table of the Standard Normal Distribution

Answers to Odd-Numbered Problems

References

Photo Credits

Index

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