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This volume teaches calculus in the biology context without compromising the level of regular calculus. Each new concept is typically introduced with a biological example; the concept is then developed without the biological context; and then the concept is tied into additional biological examples. This allows readers to first see why a certain concept is important; then lets them focus on how to use the concepts without getting distracted by applications; and then, once they feel more comfortable with the concepts, it revisits the biological applications to make sure that they can apply the concepts. The book features exceptionally detailed step-by-step worked examples and a variety of problems, including an unusually large number of word problems in a biological context. Limits and Continuity. Derivatives. Integration. Integration Techniques. Differential Equations. Applications. Linear Algebra and Analytic Geometry. Multivariable Calculus. Systems of Differential Equations. For anyone in the biological sciences who needs a working knowledge of calculus.
1. Preview and Review.
Preliminaries. Elementary Functions. Graphing. Review Problems.
2. Limits and Continuity.
Limits. Continuity. The Sandwich Theorem and Some Trigonometric Limits. Limits at Infinity. Continuity.
3. Differentiation.
Formal Definition of the Derivative. The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials. The Product and Quotient Rules, and the Derivatives of Rational and Power Functions. The Chain Rule and Higher Derivatives. Derivatives of Trigonometric Functions. Derivatives of Exponential Functions. Derivatives of Inverse and Logarithmic Functions. Approximation and Local Linearity.
4. Applications of Differentiation.
Local Extrema and the Mean Value Theorem. Monotonicity and Concavity. Extrema, Inflection Points and Graphing. Optimization. L'Hospital's Rule. Numerical Methods. Antiderivatives. Review Problems.
5. Integration.
The Definite Integral. The Fundamental Theorem of Calculus. Applications of Integration.
6. Integration Techniques and Computational Methods.
The Substitution Rule. Integration by Parts. Improper Integrals. Numerical Integration. Tables of Integration. The Taylor Approximation.
7. Differential Equations.
Solving Differential Equations. Equilibria and Their Stability. Systems of Autonomous Equations.
8. Applications of Integration.
Rectification of Curves. Densities and Histograms. Average Values. The Normal Distribution. Age-Structured Populations.
9. Linear Algebra and Analytic Geometry.
Linear Systems. Matrices. Linear Maps, Eigenvectors and Eignvalues. Analytic Geometry.
10. Multivariable Calculus.
Functions of Two or More Independent Variables. Limits and Continuity. Partial Derivatives. Tangent Planes, Differentiability, and Linearization. More About Derivatives.
11. Systems of Differential Equations.
Linear Systems -- Theory. Linear Systems -- Applications. Nonlinear Autonomous Systems -- Theory. Nonlinear Systems -- Applications.
12. Probability and Statistics.
Counting. What Is Probability? Conditional Probability and Independence. Discrete Random Variables and Discrete Distributions. Continuous Distributions. Statistical Tools.
This volume teaches calculus in the biology context without compromising the level of regular calculus. Each new concept is typically introduced with a biological example; the concept is then developed without the biological context; and then the concept is tied into additional biological examples. This allows readers to first see why a certain concept is important; then lets them focus on how to use the concepts without getting distracted by applications; and then, once they feel more comfortable with the concepts, it revisits the biological applications to make sure that they can apply the concepts. The book features exceptionally detailed step-by-step worked examples and a variety of problems, including an unusually large number of word problems in a biological context. Limits and Continuity. Derivatives. Integration. Integration Techniques. Differential Equations. Applications. Linear Algebra and Analytic Geometry. Multivariable Calculus. Systems of Differential Equations. For anyone in the biological sciences who needs a working knowledge of calculus.
Table of Contents
1. Preview and Review.
Preliminaries. Elementary Functions. Graphing. Review Problems.
2. Limits and Continuity.
Limits. Continuity. The Sandwich Theorem and Some Trigonometric Limits. Limits at Infinity. Continuity.
3. Differentiation.
Formal Definition of the Derivative. The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials. The Product and Quotient Rules, and the Derivatives of Rational and Power Functions. The Chain Rule and Higher Derivatives. Derivatives of Trigonometric Functions. Derivatives of Exponential Functions. Derivatives of Inverse and Logarithmic Functions. Approximation and Local Linearity.
4. Applications of Differentiation.
Local Extrema and the Mean Value Theorem. Monotonicity and Concavity. Extrema, Inflection Points and Graphing. Optimization. L'Hospital's Rule. Numerical Methods. Antiderivatives. Review Problems.
5. Integration.
The Definite Integral. The Fundamental Theorem of Calculus. Applications of Integration.
6. Integration Techniques and Computational Methods.
The Substitution Rule. Integration by Parts. Improper Integrals. Numerical Integration. Tables of Integration. The Taylor Approximation.
7. Differential Equations.
Solving Differential Equations. Equilibria and Their Stability. Systems of Autonomous Equations.
8. Applications of Integration.
Rectification of Curves. Densities and Histograms. Average Values. The Normal Distribution. Age-Structured Populations.
9. Linear Algebra and Analytic Geometry.
Linear Systems. Matrices. Linear Maps, Eigenvectors and Eignvalues. Analytic Geometry.
10. Multivariable Calculus.
Functions of Two or More Independent Variables. Limits and Continuity. Partial Derivatives. Tangent Planes, Differentiability, and Linearization. More About Derivatives.
11. Systems of Differential Equations.
Linear Systems -- Theory. Linear Systems -- Applications. Nonlinear Autonomous Systems -- Theory. Nonlinear Systems -- Applications.
12. Probability and Statistics.
Counting. What Is Probability? Conditional Probability and Independence. Discrete Random Variables and Discrete Distributions. Continuous Distributions. Statistical Tools.