Calculus for Biology and Medicine 4th Edition

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 11
Acknowledgments......Page 18
1.1 Precalculus Skills Diagnostic Test......Page 20
1.2.1 The Real Numbers......Page 23
1.2.2 Lines in the Plane......Page 26
1.2.4 Trigonometry......Page 28
1.2.5 Exponentials and Logarithms......Page 30
1.2.6 Complex Numbers and Quadratic Equations......Page 32
1.3.1 What Is a Function?......Page 37
1.3.2 Polynomial Functions......Page 40
1.3.3 Rational Functions......Page 42
1.3.4 Power Functions......Page 43
1.3.5 Exponential Functions......Page 44
1.3.6 Inverse Functions......Page 47
1.3.7 Logarithmic Functions......Page 49
1.3.8 Trigonometric Functions......Page 52
1.4.1 Graphing and Basic Transformations of Functions......Page 59
1.4.2 The Logarithmic Scale......Page 61
1.4.3 Transformations into Linear Functions......Page 63
1.4.4 * From a Verbal Description to a Graph......Page 68
Review Problems......Page 77
2.1.1 Modeling Population Growth in Discrete Time......Page 81
2.1.2 Recurrence Equations......Page 83
2.1.3 Visualizing Recurrence Equations......Page 84
2.2.1 What Are Sequences?......Page 87
2.2.3 Limits......Page 90
2.2.4 Recurrence Equations......Page 94
2.2.5 Using ∑ Notation to Represent Sums of Sequences......Page 97
2.3.1 Density-Dependent Population Growth......Page 100
2.3.2 Density-Dependent Population Growth: The Beverton–Holt Model......Page 103
2.3.3 The Discrete Logistic Equation......Page 104
2.3.4 * Modeling Drug Absorption......Page 107
Review Problems......Page 117
3.1 Limits......Page 120
3.1.1 A Non-Rigorous Discussion of Limits......Page 121
3.1.2 Pitfalls of Finding Limits......Page 125
3.1.3 Limit Laws......Page 127
3.2.1 What Is Continuity?......Page 131
3.2.2 Combinations of Continuous Functions......Page 134
3.3 Limits at Infinity......Page 139
3.4.1 Geometric Argument for Trigonometric Limits......Page 143
3.4.2 * The Sandwich Theorem......Page 145
3.5.1 The Intermediate-Value Theorem and The Bisection Method......Page 148
3.5.2 * Using a Spreadsheet to Implement the Bisection Method......Page 151
3.6 * A Formal Definition of Limits......Page 153
Review Problems......Page 158
4 Differentiation......Page 161
4.1 Formal Definition of the Derivative......Page 162
4.2.1 Interpreting the Derivative......Page 167
4.2.2 Differentiability and Continuity......Page 169
4.3 The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials......Page 173
4.4.1 The Product Rule......Page 179
4.4.2 The Quotient Rule......Page 181
4.5.1 The Chain Rule......Page 187
4.5.2 Proof of the Chain Rule......Page 191
4.6.1 Implicit Differentiation......Page 193
4.6.2 Related Rates......Page 196
4.7 Higher Derivatives......Page 199
4.8 Derivatives of Trigonometric Functions......Page 203
4.9 Derivatives of Exponential Functions......Page 207
4.10.1 Derivatives of Inverse Functions......Page 213
4.10.2 The Derivative of the Logarithmic Function......Page 218
4.10.3 * Logarithmic Differentiation......Page 220
4.11 Linear Approximation and Error Propagation......Page 223
Review Problems......Page 230
5.1.1 The Extreme-Value Theorem......Page 232
5.1.2 Local Extrema......Page 234
5.1.3 The Mean-Value Theorem......Page 238
5.2 Monotonicity and Concavity......Page 244
5.2.1 Monotonicity......Page 245
5.2.2 Concavity......Page 247
5.3.1 Extrema......Page 253
5.3.2 Inflection Points......Page 259
5.4 Optimization......Page 261
5.5 L’Hôpital’s Rule......Page 272
5.6 Graphing and Asymptotes......Page 279
5.7.1 Exponential Growth......Page 290
5.7.2 Stability: General Case......Page 291
5.7.3 Population Growth Models......Page 294
5.8 * Numerical Methods: The Newton–Raphson Method......Page 298
5.9.1 Modeling Population Growth......Page 304
5.9.2 Interpreting the Mathematical Model......Page 306
5.9.3 Passage of Drugs Through the Human Body......Page 308
5.10 Antiderivatives......Page 313
Key Terms......Page 320
Review Problems......Page 321
6.1.1 The Area Problem......Page 325
6.1.2 The General Theory of Riemann Integrals......Page 327
6.1.3 Properties of the Riemann Integral......Page 333
6.1.4 * Order Properties of the Riemann Integral......Page 335
6.2.1 The Fundamental Theorem of Calculus (Part I)......Page 341
6.2.2 * Leibniz’s Rule and a Rigorous Proof of the Fundamental Theorem of Calculus......Page 342
6.2.3 Antiderivatives and Indefinite Integrals......Page 345
6.2.4 The Fundamental Theorem of Calculus (Part II)......Page 348
6.3.1 Cumulative Change......Page 353
6.3.2 Average Values......Page 355
6.3.3 * The Mean Value Theorem......Page 357
6.3.4 * Areas......Page 359
6.3.5 * The Volume of a Solid......Page 362
6.3.6 * Rectification of Curves......Page 365
Review Problems......Page 371
7.1.1 Indefinite Integrals......Page 374
7.1.2 Definite Integrals......Page 379
7.2.1 Integration by Parts......Page 384
7.2.2 Practicing Integration......Page 389
7.3.1 Proper Rational Functions......Page 393
7.3.2 Partial-Fraction Decomposition......Page 394
7.3.3 Repeated Linear Factors......Page 398
7.3.4 * Irreducible Quadratic Factors......Page 399
7.3.5 Summary......Page 404
7.4.1 Type 1: Unbounded Intervals......Page 407
7.4.2 Type 2: Unbounded Integrand......Page 411
7.4.3 A Comparison Result for Improper Integrals......Page 414
7.5.1 The Midpoint Rule......Page 417
7.5.2 The Trapezoidal Rule......Page 420
7.5.3 Using a Spreadsheet for Numerical Integration......Page 421
7.5.4 * Estimating Error in a Numerical Integration......Page 425
7.6.1 Taylor Polynomials......Page 428
7.6.2 The Taylor Polynomial about x = a......Page 433
7.6.3 How Accurate Is the Approximation?......Page 434
7.7 * Tables of Integrals......Page 439
Review Problems......Page 443
8 Differential Equations......Page 446
8.1 Solving Separable Differential Equations......Page 447
8.1.1 Pure-Time Differential Equations......Page 448
8.1.2 Autonomous Differential Equations......Page 449
8.1.3 General Separable Equations......Page 455
8.2 Equilibria and Their Stability......Page 460
8.2.2 Graphical Approach to Finding Equilibria......Page 461
8.2.3 Stability of Equilibrium Points......Page 462
8.2.4 Sketching Solutions Using the Vector Field Plot......Page 467
8.2.5 Behavior Near an Equilibrium......Page 469
8.3.1 Compartment Models......Page 474
8.3.2 An Ecological Model......Page 475
8.3.3 Modeling a Chemical Reaction......Page 476
8.3.4 The Evolution of Cooperation......Page 478
8.3.5 Epidemic Model......Page 482
8.4.1 Integrating Factors......Page 490
8.4.2 Two-Compartment Models......Page 494
Review Problems......Page 503
9.1 Linear Systems......Page 506
9.1.1 Graphical Solution......Page 507
9.1.2 Solving Equations Using Elimination......Page 510
9.1.3 Solving Systems of Linear Equations......Page 511
9.1.4 Representing Systems of Equations Using Matrices......Page 515
9.2.1 Matrix Operations......Page 520
9.2.2 Matrix Multiplication......Page 522
9.2.3 Inverse Matrices......Page 525
9.2.4 * Computing Inverse Matrices......Page 532
9.3 Linear Maps, Eigenvectors, and Eigenvalues......Page 537
9.3.1 Graphical Representation......Page 538
9.3.2 Eigenvalues and Eigenvectors......Page 542
9.3.3 * Iterated Maps......Page 550
9.4.1 Modeling with Leslie Matrices......Page 554
9.4.2 Stable Age Distributions in Demographic Models......Page 559
9.5.1 Points and Vectors in Higher Dimensions......Page 566
9.5.2 The Dot Product......Page 570
9.5.3 Parametric Equations of Lines......Page 574
Key Terms......Page 577
Review Problems......Page 578
10 Multivariable Calculus......Page 580
10.1.1 Defining a Function of Two or More Variables......Page 582
10.1.2 The Graph of a Function of Two Independent Variables–Surface Plot......Page 584
10.1.3 Heat Maps......Page 585
10.1.4 Contour Plots......Page 587
10.2.1 Informal Definition of Limits......Page 594
10.2.2 Continuity......Page 597
10.2.3 Formal Definition of Limits......Page 598
10.3.1 Functions of Two Variables......Page 601
10.3.3 Higher-Order Partial Derivatives......Page 605
10.4.1 Functions of Two Variables......Page 608
10.4.2 Vector-Valued Functions......Page 613
10.5.1 The Chain Rule for Functions of Two Variables......Page 618
10.5.2 Implicit Differentiation......Page 620
10.6.1 Deriving the Directional Derivative......Page 623
10.6.2 Properties of the Gradient Vector......Page 627
10.7.1 Local Maxima and Minima......Page 629
10.7.2 Global Extrema......Page 636
10.7.3 Extrema with Constraints......Page 640
10.7.4 Least-Squares Data Fitting......Page 645
10.8 * Diffusion......Page 654
10.9.1 A Biological Example......Page 659
10.9.2 Equilibria and Stability in Systems of Linear Recurrence Equations......Page 660
10.9.3 Equilibria and Stability of Nonlinear Systems of Recurrence Equations......Page 662
Review Problems......Page 669
11 Systems of Differential Equations......Page 672
11.1.1 The Vector Field......Page 674
11.1.2 Solving Linear Systems......Page 676
11.1.3 Equilibria and Stability......Page 683
11.1.4 Systems with Complex Conjugate Eigenvalues......Page 685
11.1.5 Summary of the Theory of Linear Systems......Page 690
11.2.1 Two-Compartment Models......Page 696
11.2.2 A Mathematical Model for Love......Page 701
11.2.3 * The Harmonic Oscillator......Page 703
11.3.1 Analytical Approach......Page 707
11.3.2 Graphical Approach for 2×2 Systems......Page 713
11.4.1 Competition......Page 717
11.4.2 A Predator–Prey Model......Page 723
11.5 * More Mathematical Models......Page 727
11.5.1 The Community Matrix......Page 728
11.5.2 Neuron Activity......Page 730
11.5.3 Enzymatic Reactions......Page 732
11.5.4 Microbial Growth in a Chemostat......Page 735
11.5.5 A Model for Epidemics......Page 737
Review Problems......Page 749
12.1.1 The Multiplication Principle......Page 753
12.1.2 Permutations......Page 754
12.1.3 Combinations......Page 756
12.1.4 Combining the Counting Principles......Page 757
12.2.1 Basic Definitions......Page 761
12.2.2 Equally Likely Outcomes......Page 765
12.3 Conditional Probability and Independence......Page 771
12.3.1 Conditional Probability......Page 772
12.3.2 The Law of Total Probability......Page 773
12.3.3 Independence......Page 774
12.3.4 The Bayes Formula......Page 777
12.4.1 Discrete Distributions......Page 782
12.4.2 Mean and Variance......Page 785
12.4.3 The Binomial Distribution......Page 793
12.4.4 The Multinomial Distribution......Page 797
12.4.5 Geometric Distribution......Page 798
12.4.6 The Poisson Distribution......Page 802
12.5.1 Density Functions......Page 812
12.5.2 The Normal Distribution......Page 818
12.5.3 The Uniform Distribution......Page 824
12.5.4 The Exponential Distribution......Page 826
12.5.5 The Poisson Process......Page 830
12.5.6 Aging......Page 831
12.6.1 The Law of Large Numbers......Page 838
12.6.2 The Central Limit Theorem......Page 842
12.7.1 Describing Univariate Data......Page 847
12.7.2 Estimating Parameters......Page 852
12.7.3 Linear Regression......Page 861
Key Terms......Page 867
Review Problems......Page 868
Appendix A Frequently Used Symbols......Page 870
Appendix B Table of the Standard Normal Distribution......Page 871
Answers to Odd-Numbered Problems......Page 872
References......Page 902
D......Page 908
F......Page 909
I......Page 910
M......Page 911
R......Page 912
T......Page 913
Z......Page 914