Biocalculus: Calculus, Probability, and Statistics for the Life Sciences

Description

About the Cover Images......Page 3 About the Authors......Page 8 Contents......Page 9 Preface......Page 17 To the Student......Page 27 Calculators, Computers, and Other Graphing Devices......Page 28 Diagnostic Tests......Page 30 Prologue: Mathematics and Biology......Page 35 Case Studies in Mathematical Modeling......Page 43 Case Study 1: Kill Curves and Antibiotic Effectiveness......Page 44 Case Study 2: Hosts, Parasites, and Time-Travel......Page 48 Ch 1: Functions and Sequences......Page 51 1.1: Four Ways to Represent a Function......Page 52 1.2: A Catalog of Essential Functions......Page 67 1.3: New Functions from Old Functions......Page 81 1.4: Exponential Functions......Page 91 1.5: Logarithms; Semilog and Log-Log Plots......Page 102 1.6: Sequences and Difference Equations......Page 120 Chapter 1: Review......Page 130 Ch 2: Limits......Page 139 2.1: Limits of Sequences......Page 140 2.2: Limits of Functions at Infinity......Page 152 2.3: Limits of Functions at Finite Numbers......Page 161 2.4: Limits: Algebraic Methods......Page 175 2.5: Continuity......Page 187 Chapter 2: Review......Page 199 Ch 3: Derivatives......Page 205 3.1: Derivatives and Rates of Change......Page 206 3.2: The Derivative as a Function......Page 218 3.3: Basic Differentiation Formulas......Page 231 3.4: The Product and Quotient Rules......Page 244 3.5: The Chain Rule......Page 252 3.6: Exponential Growth and Decay......Page 265 3.7: Derivatives of the Logarithmic and Inverse Tangent Functions......Page 272 3.8: Linear Approximations and Taylor Polynomials......Page 280 Chapter 3: Review......Page 290 Ch 4: Applications of Derivatives......Page 299 4.1: Maximum and Minimum Values......Page 300 4.2: How Derivatives Affect the Shape of a Graph......Page 311 4.3: L'Hospital's Rule: Comparing Rates of Growth......Page 324 4.4: Optimization Problems......Page 335 4.5: Recursions: Equilibria and Stability......Page 349 4.6: Antiderivatives......Page 356 Chapter 4: Review......Page 362 Ch 5: Integrals......Page 365 5.1: Areas, Distances, and Pathogenesis......Page 366 5.2: The Definite Integral......Page 379 5.3: The Fundamental Theorem of Calculus......Page 392 5.4: The Substitution Rule......Page 404 5.5: Integration by Parts......Page 412 5.6: Partial Fractions......Page 418 5.7: Integration Using Tables and Computer Algebra Systems......Page 421 5.8: Improper Integrals......Page 426 Chapter 5: Review......Page 431 Ch 6: Applications of Integrals......Page 437 6.1: Areas between Curves......Page 438 6.2: Average Values......Page 447 6.3: Further Applications to Biology......Page 450 6.4: Volumes......Page 455 Chapter 6: Review......Page 462 Ch 7: Differential Equations......Page 469 7.1: Modeling with Differential Equations......Page 470 7.2: Phase Plots, Equilibria, and Stability......Page 481 7.3: Direction Fields and Euler's Method......Page 490 7.4: Separable Equations......Page 499 7.5: Systems of Differential Equations......Page 509 7.6: Phase Plane Analysis......Page 518 Chapter 7: Review......Page 530 Ch 8: Vectors and Matrix Models......Page 537 8.1: Coordinate Systems......Page 538 8.2: Vectors......Page 546 8.3: The Dot Product......Page 555 8.4: Matrix Algebra......Page 564 8.5: Matrices and the Dynamics of Vectors......Page 570 8.6: The Inverse and Determinant of a Matrix......Page 578 8.7: Eigenvectors and Eigenvalues......Page 587 8.8: Iterated Matrix Models......Page 597 Chapter 8: Review......Page 610 Ch 9: Multivariable Calculus......Page 615 9.1: Functions of Several Variables......Page 616 9.2: Partial Derivatives......Page 635 9.3: Tangent Planes and Linear Approximations......Page 646 9.4: The Chain Rule......Page 654 9.5: Directional Derivatives and the Gradient Vector......Page 660 9.6: Maximum and Minimum Values......Page 669 Chapter 9: Review......Page 678 Ch 10: Systems of Linear Differential Equations......Page 681 10.1: Qualitative Analysis of Linear Systems......Page 682 10.2: Solving Systems of Linear Differential Equations......Page 690 10.3: Applications......Page 702 10.4: Systems of Nonlinear Differential Equations......Page 715 Chapter 10: Review......Page 726 Ch 11: Descriptive Statistics......Page 733 11.1: Numerical Descriptions of Data......Page 734 11.2: Graphical Descriptions of Data......Page 743 11.3: Relationships between Variables......Page 753 11.4: Populations, Samples, and Inference......Page 763 Chapter 11: Review......Page 772 Ch 12: Probability......Page 777 12.1: Principles of Counting......Page 778 12.2: What is Probability?......Page 787 12.3: Conditional Probability......Page 801 12.4: Discrete Random Variables......Page 817 12.5: Continuous Random Variables......Page 836 Chapter 12: Review......Page 849 Ch 13: Inferential Statistics......Page 853 13.1: The Sampling Distribution......Page 854 13.2: Confidence Intervals......Page 862 13.3: Hypothesis Testing......Page 871 13.4: Contingency Table Analysis......Page 879 Chapter 13: Review......Page 885 Appendixes......Page 889 Appendix A: Intervals, Inequalities, and Absolute Values......Page 890 Appendix B: Coordinate Geometry......Page 895 Appendix C: Trigonometry......Page 905 Appendix D: Precise Definitions of Limits......Page 914 Appendix E: A Few Proofs......Page 920 Appendix F: Sigma Notation......Page 924 Appendix G: Complex Numbers......Page 930 Appendix H: Statistical Tables......Page 938 Glossary of Biological Terms......Page 941 Answers to Odd-Numbered Exercises......Page 943 Biological Index......Page 997 Index......Page 1007