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

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