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.

See the preface for a comprehensive Table of Contents.

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    

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