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For a two-semester course in Calculus for Life Sciences.
This text addresses the needs of students in the biological sciences by teaching calculus in a biological context without reducing the course level. It is a calculus text, written so that a math professor without a biology background can teach from it successfully. New concepts are introduced in a three step manner. First, a biological example motivates the topic; second, the topic is then developed via a simple mathematical example; and third the concept is tied to deeper biological examples. This allows students: to see why a concept is important; to understand how to use the concept computationally; to make sure that they can apply the concept.
Features
1. Preview and Review.
Preliminaries. Elementary Functions. Graphing. Key Terms. Review Problems.
2. Discrete Time Models, Sequences, and Difference Equations.
Exponential Growth and Decay. Sequences. More Population Models. Key Terms. Review Problems.
3. Limits and Continuity.
Limits. Continuity. Limits at Infinity. The Sandwich Theorem and Some Trigonometric Limits. Properties of Continuous Functions. Formal Definition of Limits. Key Terms. Review Problems.
4. Differentiation.
Formal Definition of the Derivative. The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials. Product Rule and Quotient Rule. The Chain Rule and Higher Derivatives. Derivatives of Trigonometric Functions. Derivatives of Exponential Functions. Derivatives of Inverse and Logarithmic Functions. Approximation and Local Linearity. Key Terms. Review Problems.
5. Applications of Differentiation.
Extrema and the Mean Value Theorem. Monotonicity and Concavity. Extrema, Inflection Points and Graphing. Optimization. L'Hopital's Rule. Difference Equations - Stability. Numerical Methods: The Newton-Raphson Method. Antiderivatives. Key Terms. Review Problems.
6. Integration.
The Definite Integral. The Fundamental Theorem of Calculus. Applications of Integration. Key Terms. Review Problems.
7. Integration Techniques and Computational Methods.
The Substitution Rule. Integration by Parts. Practicing Integration and Partial Fractions. Improper Integrals. Numerical Integration. Tables of Integration. The Taylor Approximation. Key Terms. Review Problems.
8. Differential Equations.
Solving Differential Equations. Equilibria and Their Stability. Systems of Autonomous Equations. Key Terms. Review Problems.
9. Linear Algebra and Analytic Geometry.
Linear Systems. Matrices. Linear Maps, Eigenvectors and Eignvalues. Analytic Geometry. Key Terms. Review Problems.
10. Multivariable Calculus.
Functions of Two or More Independent Variables. Limits and Continuity. Partial Derivatives. Tangent Planes, Differentiability, and Linearization. More About Derivatives. Applications. Systems of Difference Equations. Key Terms. Review Problems.
11. Systems of Differential Equations.
Linear Systems: Theory. Linear Systems: Applications. Nonlinear Autonomous Systems: Theory. Nonlinear Systems: Applications. Key Terms. Review Problems.
12. Probability and Statistics.
Counting. What Is Probability? Conditional Probability and Independence. Discrete Random Variables and Discrete Distributions. Continuous Distributions. Limit Theorems. Statistical Tools. Key Terms. Review Problems.
For a two-semester course in Calculus for Life Sciences.
This text addresses the needs of students in the biological sciences by teaching calculus in a biological context without reducing the course level. It is a calculus text, written so that a math professor without a biology background can teach from it successfully. New concepts are introduced in a three step manner. First, a biological example motivates the topic; second, the topic is then developed via a simple mathematical example; and third the concept is tied to deeper biological examples. This allows students: to see why a concept is important; to understand how to use the concept computationally; to make sure that they can apply the concept.
Features
Table of Contents
1. Preview and Review.
Preliminaries. Elementary Functions. Graphing. Key Terms. Review Problems.
2. Discrete Time Models, Sequences, and Difference Equations.
Exponential Growth and Decay. Sequences. More Population Models. Key Terms. Review Problems.
3. Limits and Continuity.
Limits. Continuity. Limits at Infinity. The Sandwich Theorem and Some Trigonometric Limits. Properties of Continuous Functions. Formal Definition of Limits. Key Terms. Review Problems.
4. Differentiation.
Formal Definition of the Derivative. The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials. Product Rule and Quotient Rule. The Chain Rule and Higher Derivatives. Derivatives of Trigonometric Functions. Derivatives of Exponential Functions. Derivatives of Inverse and Logarithmic Functions. Approximation and Local Linearity. Key Terms. Review Problems.
5. Applications of Differentiation.
Extrema and the Mean Value Theorem. Monotonicity and Concavity. Extrema, Inflection Points and Graphing. Optimization. L'Hopital's Rule. Difference Equations - Stability. Numerical Methods: The Newton-Raphson Method. Antiderivatives. Key Terms. Review Problems.
6. Integration.
The Definite Integral. The Fundamental Theorem of Calculus. Applications of Integration. Key Terms. Review Problems.
7. Integration Techniques and Computational Methods.
The Substitution Rule. Integration by Parts. Practicing Integration and Partial Fractions. Improper Integrals. Numerical Integration. Tables of Integration. The Taylor Approximation. Key Terms. Review Problems.
8. Differential Equations.
Solving Differential Equations. Equilibria and Their Stability. Systems of Autonomous Equations. Key Terms. Review Problems.
9. Linear Algebra and Analytic Geometry.
Linear Systems. Matrices. Linear Maps, Eigenvectors and Eignvalues. Analytic Geometry. Key Terms. Review Problems.
10. Multivariable Calculus.
Functions of Two or More Independent Variables. Limits and Continuity. Partial Derivatives. Tangent Planes, Differentiability, and Linearization. More About Derivatives. Applications. Systems of Difference Equations. Key Terms. Review Problems.
11. Systems of Differential Equations.
Linear Systems: Theory. Linear Systems: Applications. Nonlinear Autonomous Systems: Theory. Nonlinear Systems: Applications. Key Terms. Review Problems.
12. Probability and Statistics.
Counting. What Is Probability? Conditional Probability and Independence. Discrete Random Variables and Discrete Distributions. Continuous Distributions. Limit Theorems. Statistical Tools. Key Terms. Review Problems.