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For freshman-level, two-semester or three-semester courses in Calculus for Life Sciences.
This package includes MyLab Math.
Shows students how calculus is used to analyze phenomena in nature – while providing flexibility for instructors to teach at their desired level of rigor
Calculus for Biology and Medicine motivates life and health science majors to learn calculus through relevant and strategically placed applications to their chosen fields. It presents the calculus in such a way that the level of rigor can be adjusted to meet the specific needs of the audience – from a purely applied course to one that matches the rigor of the standard calculus track.
In the 4th Edition, new co-author Marcus Roper (UCLA) partners with author Claudia Neuhauser to preserve these strengths while adding an unprecedented number of real applications and an infusion of modeling and technology.
Reach every student by pairing this text with MyLab Math
MyLab™ Math is the teaching and learning platform that empowers instructors to reach every student. By combining trusted author content with digital tools and a flexible platform, MyLab Math personalizes the learning experience and improves results for each student. For the first time, instructors teaching with Calculus for Biology and Medicine can assign text-specific online homework and other resources to students outside of the classroom.
0134845048 / 9780134845043 Calculus for Biology and Medicine plus MyLab Math with Pearson eText — Access Card Package, 4/e
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0134070046 / 9780134070049 Calculus for Biology and Medicine
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Claudia Neuhauser, PhD, is Associate Vice President for Research and Director of Research Computing in the Office of the Vice President for Research at University of Minnesota. In her role as Director of Research Computing she oversees the University of Minnesota Informatics Institute (UMII), the Minnesota Supercomputing Institute (MSI), and U-Spatial. UMII fosters and accelerates data-intensive research across all disciplines in the University and develops partnership with industry. Neuhauser’s research is at the interface of mathematics and biology, and focuses on the analysis of ecological and evolutionary models and the development of statistical methods in biomedical applications. She received her Diplom in mathematics from the Universität Heidelberg (Germany) in 1988, and a Ph.D. in mathematics from Cornell University in 1990. She is a fellow of the American Association for the Advancement of Science (AAAS) and a fellow of the American Mathematical Society (AMS).
Marcus Roper, PhD, is a Professor of Mathematics at UCLA. He specializes in developing mathematical models inspired by physics and biology. His particular research interests include biological transport networks, such as fungal mycelia and the microvascular system. Although many of the projects he works on are experimentally inspired, his goal is to develop new image analysis methods and to gain a better understanding of the world around us.
(NOTE: Each chapter concludes with Key Terms and Review Problems.)
1. Preview and Review
1.1 Precalculus Skills Diagnostic Test
1.2 Preliminaries
1.2.1 The Real Numbers 1.2.2 Lines in the Plane 1.2.3 Equation of the Circle 1.2.4 Trigonometry 1.2.5 Exponentials and Logarithms 1.2.6 Complex Numbers and Quadratic Equations
1.2.1 The Real Numbers
1.2.2 Lines in the Plane
1.2.3 Equation of the Circle
1.2.4 Trigonometry
1.2.5 Exponentials and Logarithms
1.2.6 Complex Numbers and Quadratic Equations
1.3 Elementary Functions
1.3.1 What Is a Function? 1.3.2 Polynomial Functions 1.3.3 Rational Functions 1.3.4 Power Functions 1.3.5 Exponential Functions 1.3.6 Inverse Functions 1.3.7 Logarithmic Functions 1.3.8 Trigonometric Functions
1.3.1 What Is a Function?
1.3.2 Polynomial Functions
1.3.3 Rational Functions
1.3.4 Power Functions
1.3.5 Exponential Functions
1.3.6 Inverse Functions
1.3.7 Logarithmic Functions
1.3.8 Trigonometric Functions
1.4 Graphing
1.4.1 Graphing and Basic Transformations of Functions 1.4.2 The Logarithmic Scale 1.4.3 Transformations into Linear Functions 1.4.4 From a Verbal Description to a Graph (Optional)
1.4.1 Graphing and Basic Transformations of Functions
1.4.2 The Logarithmic Scale
1.4.3 Transformations into Linear Functions
1.4.4 From a Verbal Description to a Graph (Optional)
2. Discrete-Time Models, Sequences, and Difference Equations
2.1 Exponential Growth and Decay
2.1.1 Modeling Population Growth in Discrete Time 2.1.2 Recurrence Equations 2.1.3 Visualizing Recurrence Equations
2.1.1 Modeling Population Growth in Discrete Time
2.1.2 Recurrence Equations
2.1.3 Visualizing Recurrence Equations
2.2 Sequences
2.2.1 What Are Sequences? 2.2.2 (Optional) Using Spreadsheets to Calculate a Recursive Sequence 2.2.3 Limits 2.2.4 Recurrence Equations 2.2.5 Using Notation to Represent Sums of Sequences
2.2.1 What Are Sequences?
2.2.2 (Optional) Using Spreadsheets to Calculate a Recursive Sequence
2.2.3 Limits
2.2.4 Recurrence Equations
2.2.5 Using Notation to Represent Sums of Sequences
2.3 Modeling with Recurrence Equations
2.3.1 Density-Dependent Population Growth 2.3.2 Density-Dependent Population Growth: The Beverton - Holt Model 2.3.3 The Discrete Logistic Equation 2.3.4 Modeling Drug Absorption (optional)
2.3.1 Density-Dependent Population Growth
2.3.2 Density-Dependent Population Growth: The Beverton - Holt Model
2.3.3 The Discrete Logistic Equation
2.3.4 Modeling Drug Absorption (optional)
3. Limits and Continuity
3.1 Limits
3.1.1 A Non-Rigorous Discussion of Limits 3.1.2 Pitfalls of Finding Limits 3.1.3 Limit Laws
3.1.1 A Non-Rigorous Discussion of Limits
3.1.2 Pitfalls of Finding Limits
3.1.3 Limit Laws
3.2 Continuity
3.2.1 What Is Continuity? 3.2.2 Combinations of Continuous Functions
3.2.1 What Is Continuity?
3.2.2 Combinations of Continuous Functions
3.3 Limits at Infinity
3.4 Trigonometric Limits and the Sandwich Theorem
3.4.1 Geometric Argument for Trigonometric Limits 3.4.2 The Sandwich Theorem (Optional)
3.4.1 Geometric Argument for Trigonometric Limits
3.4.2 The Sandwich Theorem (Optional)
3.5 Properties of Continuous Functions
3.5.1 The Intermediate-Value Theorem and The Bisection Method 3.5.2 Using a Spreadsheet to Implement the Bisection Method (Optional) 3.5.3 A Final Remark on Continuous Functions
3.5.1 The Intermediate-Value Theorem and The Bisection Method
3.5.2 Using a Spreadsheet to Implement the Bisection Method (Optional)
3.5.3 A Final Remark on Continuous Functions
3.6 A Formal Definition of Limits (Optional)
4. Differentiation
4.1 Formal Definition of the Derivative
4.2 Properties of the Derivative
4.2.1 Interpreting the Derivative 4.2.2 Differentiability and Continuity
4.2.1 Interpreting the Derivative
4.2.2 Differentiability and Continuity
4.3 Power Rules and Basic Rules
4.4 The Product and Quotient Rules, and the Derivatives of Rational and Power Functions
4.4.1 The Product Rule 4.4.2 The Quotient Rule
4.4.1 The Product Rule
4.4.2 The Quotient Rule
4.5 Chain Rule
4.5.1 The Chain Rule 4.5.2 Proof of the Chain Rule
4.5.1 The Chain Rule
4.5.2 Proof of the Chain Rule
4.6 Implicit Functions and Implicit Differentiation
4.6.1 Implicit Differentiation 4.6.2 Related Rates
4.6.1 Implicit Differentiation
4.6.2 Related Rates
4.7 Higher Derivatives
4.8 Derivatives of Trigonometric Functions
4.9 Derivatives of Exponential Functions
4.10 Inverse Functions and Logarithms
4.10.1 Derivatives of Inverse Functions 4.10.2 The Derivative of the Logarithmic Function 4.10.3 Logarithmic Differentiation
4.10.1 Derivatives of Inverse Functions
4.10.2 The Derivative of the Logarithmic Function
4.10.3 Logarithmic Differentiation
4.11 Linear Approximation and Error Propagation
5. Applications of Differentiation
5.1 Extrema and the Mean-Value Theorem
5.1.1 The Extreme-Value Theorem 5.1.2 Local Extrema 5.1.3 The Mean-Value Theorem
5.1.1 The Extreme-Value Theorem
5.1.2 Local Extrema
5.1.3 The Mean-Value Theorem
5.2 Monotonicity and Concavity
5.2.1 Monotonicity 5.2.2 Concavity
5.2.1 Monotonicity
5.2.2 Concavity
5.3 Extrema and Inflection Points
5.3.1 Extrema 5.3.2 Inflection Points
5.3.1 Extrema
5.3.2 Inflection Points
5.4 Optimization
5.5 L'Hôpital's Rule
5.6 Graphing and Asymptotes
5.7 Recurrence Equations: Stability (Optional)
5.7.1 Exponential Growth 5.7.2 Stability: General Case 5.7.3 Population Growth Models
5.7.1 Exponential Growth
5.7.2 Stability: General Case
5.7.3 Population Growth Models
5.8 Numerical Methods: The Newton - Raphson Method (Optional)
5.9 Modeling Biological Systems Using Differential Equations (Optional)
5.9.1 Modeling Population Growth 5.9.2 Interpreting the Mathematical Model 5.9.3 Passage of Drugs Through the Human Body
5.9.1 Modeling Population Growth
5.9.2 Interpreting the Mathematical Model
5.9.3 Passage of Drugs Through the Human Body
5.10 Antiderivatives
6. Integration
6.1 The Definite Integral
6.1.1 The Area Problem 6.1.2 The General Theory of Riemann Integrals 6.1.3 Properties of the Riemann Integral 6.1.4 Order Properties of the Riemann Integral (Optional)
6.1.1 The Area Problem
6.1.2 The General Theory of Riemann Integrals
6.1.3 Properties of the Riemann Integral
6.1.4 Order Properties of the Riemann Integral (Optional)
6.2 The Fundamental Theorem of Calculus
6.2.1 The Fundamental Theorem of Calculus (Part I) 6.2.2 Leibniz's Rule and a Rigorous Proof of the Fundamental Theorem of Calculus (Optional) 6.2.3 Antiderivatives and Indefinite Integrals 6.2.4 The Fundamental Theorem of Calculus (Part II)
6.2.1 The Fundamental Theorem of Calculus (Part I)
6.2.2 Leibniz's Rule and a Rigorous Proof of the Fundamental Theorem of Calculus (Optional)
6.2.3 Antiderivatives and Indefinite Integrals
6.2.4 The Fundamental Theorem of Calculus (Part II)
6.3 Applications of Integration
6.3.1 Cumulative Change 6.3.2 Average Values 6.3.3 The Mean Value Theorem (Optional) 6.3.4 Areas (Optional) 6.3.5 The Volume of a Solid (Optional) 6.3.6 Rectification of Curves (Optional)
6.3.1 Cumulative Change
6.3.2 Average Values
6.3.3 The Mean Value Theorem (Optional)
6.3.4 Areas (Optional)
6.3.5 The Volume of a Solid (Optional)
6.3.6 Rectification of Curves (Optional)
7. Integration Techniques and Computational Methods
7.1 The Substitution Rule
7.1.1 Indefinite Integrals 7.1.2 Definite Integrals
7.1.1 Indefinite Integrals
7.1.2 Definite Integrals
7.2 Integration by Parts and Practicing Integration
7.2.1 Integration by Parts 7.2.2 Practicing Integration
7.2.1 Integration by Parts
7.2.2 Practicing Integration
7.3 Rational Functions and Partial Fractions
7.3.1 Proper Rational Functions 7.3.2 Partial-Fraction Decomposition 7.3.3 Repeated Linear Factors 7.3.4 Irreducible Quadratic Factors (optional) 7.3.5 Summary
7.3.1 Proper Rational Functions
7.3.2 Partial-Fraction Decomposition
7.3.3 Repeated Linear Factors
7.3.4 Irreducible Quadratic Factors (optional)
7.3.5 Summary
7.4 Improper Integrals (Optional)
7.4.1 Type 1: Unbounded Intervals 7.4.2 Type 2: Unbounded Integrand 7.4.3 A Comparison Result for Improper Integrals
7.4.1 Type 1: Unbounded Intervals
7.4.2 Type 2: Unbounded Integrand
7.4.3 A Comparison Result for Improper Integrals
7.5 Numerical Integration
7.5.1 The Midpoint Rule 7.5.2 The Trapezoidal Rule 7.5.3 Using a Spreadsheet for Numerical Integration 7.5.4 Estimating Error in a Numerical Integration (Optional)
7.5.1 The Midpoint Rule
7.5.2 The Trapezoidal Rule
7.5.3 Using a Spreadsheet for Numerical Integration
7.5.4 Estimating Error in a Numerical Integration (Optional)
7.6 The Taylor Approximation (optional)
7.6.1 Taylor Polynomials 7.6.2 The Taylor Polynomial about x = a 7.6.3 How Accurate Is the Approximation? (Optional)
7.6.1 Taylor Polynomials
7.6.2 The Taylor Polynomial about x = a
7.6.3 How Accurate Is the Approximation? (Optional)
7.7 Tables of Integrals (Optional)
8. Differential Equations
8.1 Solving Separable Differential Equations
8.1.1 Pure-Time Differential Equations 8.1.2 Autonomous Differential Equations 8.1.3 General Separable Equations
8.1.1 Pure-Time Differential Equations
8.1.2 Autonomous Differential Equations
8.1.3 General Separable Equations
8.2 Equilibria and Their Stability
8.2.1 Equilibrium Points 8.2.2 Graphical Approach to Finding Equilibria 8.2.3 Stability of Equilibrium Points 8.2.4 Sketching Solutions Using the Vector Field Plot 8.2.5 Behavior Near an Equilibrium
8.2.1 Equilibrium Points
8.2.2 Graphical Approach to Finding Equilibria
8.2.3 Stability of Equilibrium Points
8.2.4 Sketching Solutions Using the Vector Field Plot
8.2.5 Behavior Near an Equilibrium
8.3 Differential Equation Models
8.3.1 Compartment Models 8.3.2 An Ecological Model 8.3.3 Modeling a Chemical Reaction 8.3.4 The Evolution of Cooperation 8.3.5 Epidemic Model
8.3.1 Compartment Models
8.3.2 An Ecological Model
8.3.3 Modeling a Chemical Reaction
8.3.4 The Evolution of Cooperation
8.3.5 Epidemic Model
8.4 Integrating Factors and Two-Compartment Models
8.4.1 Integrating Factors 8.4.2 Two-Compartment Models
8.4.1 Integrating Factors
8.4.2 Two-Compartment Models
9. Linear Algebra and Analytic Geometry
9.1 Linear Systems
9.1.1 Graphical Solution 9.1.2 Solving Equations Using Elimination 9.1.3 Solving Systems of Linear Equations 9.1.4 Representing Systems of Equations Using Matrices
9.1.1 Graphical Solution
9.1.2 Solving Equations Using Elimination
9.1.3 Solving Systems of Linear Equations
9.1.4 Representing Systems of Equations Using Matrices
9.2 Matrices
9.2.1 Matrix Operations 9.2.2 Matrix Multiplication 9.2.3 Inverse Matrices 9.2.4 Computing Inverse Matrices (Optional)
9.2.1 Matrix Operations
9.2.2 Matrix Multiplication
9.2.3 Inverse Matrices
9.2.4 Computing Inverse Matrices (Optional)
9.3 Linear Maps, Eigenvectors, and Eigenvalues
9.3.1 Graphical Representation 9.3.2 Eigenvalues and Eigenvectors 9.3.3 Iterated Maps (Needed for Section 10.7)
9.3.1 Graphical Representation
9.3.2 Eigenvalues and Eigenvectors
9.3.3 Iterated Maps (Needed for Section 10.7)
9.4 Demographic Modeling
9.4.1 Modeling with Leslie Matrices 9.4.2 Stable Age Distributions in Demographic Models
9.4.1 Modeling with Leslie Matrices
9.4.2 Stable Age Distributions in Demographic Models
9.5 Analytic Geometry
9.5.1 Points and Vectors in Higher Dimensions 9.5.2 The Dot Product 9.5.3 Parametric Equations of Lines
9.5.1 Points and Vectors in Higher Dimensions
9.5.2 The Dot Product
9.5.3 Parametric Equations of Lines
10. Multivariable Calculus
10.1 Two or More Independent Variables
10.1.1 Defining a Function of Two or More Variables 10.1.2 The Graph of a Function of Two Independent Variables - Surface plot 10.1.3 Heat Maps 10.1.4 Contour plots
10.1.1 Defining a Function of Two or More Variables
10.1.2 The Graph of a Function of Two Independent Variables - Surface plot
10.1.3 Heat Maps
10.1.4 Contour plots
10.2 Limits and Continuity (optional)
10.2.1 Informal Definition of Limits 10.2.2 Continuity 10.2.3 Formal Definition of Limits
10.2.1 Informal Definition of Limits
10.2.2 Continuity
10.2.3 Formal Definition of Limits
10.3 Partial Derivatives
10.3.1 Functions of Two Variables 10.3.2 Functions of More Than Two Variables 10.3.3 Higher-Order Partial Derivatives
10.3.1 Functions of Two Variables
10.3.2 Functions of More Than Two Variables
10.3.3 Higher-Order Partial Derivatives
10.4 Tangent Planes, Differentiability, and Linearization
10.4.1 Functions of Two Variables 10.4.2 Vector-Valued Functions
10.4.1 Functions of Two Variables
10.4.2 Vector-Valued Functions
10.5 The Chain Rule and Implicit Differentiation (Optional)
10.5.1 The Chain Rule for Functions of Two Variables 10.5.2 Implicit Differentiation
10.5.1 The Chain Rule for Functions of Two Variables
10.5.2 Implicit Differentiation
10.6 Directional Derivatives and Gradient Vectors (Optional)
10.6.1 Deriving the Directional Derivative 10.6.2 Properties of the Gradient Vector
10.6.1 Deriving the Directional Derivative
10.6.2 Properties of the Gradient Vector
10.7 Maximization and Minimization of Functions (Optional)
10.7.1 Local Maxima and Minima 10.7.2 Global Extrema 10.7.3 Extrema with Constraints 10.7.4 Least-Squares Data Fitting
10.7.1 Local Maxima and Minima
10.7.2 Global Extrema
10.7.3 Extrema with Constraints
10.7.4 Least-Squares Data Fitting
10.8 Diffusion (Optional)
10.9 Systems of Difference Equations (Optional)
10.9.1 A Biological Example 10.9.2 Equilibria and Stability in Systems of Linear Recurrence Equations 10.9.3 Equilibria and Stability of Nonlinear Systems of Difference Equations
10.9.1 A Biological Example
10.9.2 Equilibria and Stability in Systems of Linear Recurrence Equations
10.9.3 Equilibria and Stability of Nonlinear Systems of Difference Equations
11. Systems of Differential Equations
11.1 Linear Systems: Theory
11.1.1 The Direction Field 11.1.2 Solving Linear Systems 11.1.3 Equilibria and Stability 11.1.4 Systems with Complex Conjugate Eigenvalues 11.1.5 Summary of the Theory of Linear Systems
11.1.1 The Direction Field
11.1.2 Solving Linear Systems
11.1.3 Equilibria and Stability
11.1.4 Systems with Complex Conjugate Eigenvalues
11.1.5 Summary of the Theory of Linear Systems
11.2 Linear Systems: Applications
11.2.1 Two-Compartment Models 11.2.2 A Mathematical Model for Love 11.2.3 The Harmonic Oscillator (Optional)
11.2.1 Two-Compartment Models
11.2.2 A Mathematical Model for Love
11.2.3 The Harmonic Oscillator (Optional)
11.3 Nonlinear Autonomous Systems: Theory
11.3.1 Analytical Approach 11.3.2 Graphical Approach for 2 x 2 Systems (Optional)
11.3.1 Analytical Approach
11.3.2 Graphical Approach for 2 x 2 Systems (Optional)
11.4 Nonlinear Systems: Lotka - Volterra Model of Interspecific Interactions
11.4.1 Competition 11.4.2 A Predator--Prey Model
11.4.1 Competition
11.4.2 A Predator--Prey Model
11.5 More Mathematical Models (Optional)
11.5.1 The Community Matrix 11.5.2 Neuron Activity 11.5.3 Enzymatic Reactions
11.5.1 The Community Matrix
11.5.2 Neuron Activity
11.5.3 Enzymatic Reactions
12. Probability and Statistics
12.1 Counting
12.1.1 The Multiplication Principle 12.1.2 Permutations 12.1.3 Combinations 12.1.4 Combining the Counting Principles
12.1.1 The Multiplication Principle
12.1.2 Permutations
12.1.3 Combinations
12.1.4 Combining the Counting Principles
12.2 What Is Probability?
12.2.1 Basic Definitions 12.2.2 Equally Likely Outcomes
12.2.1 Basic Definitions
12.2.2 Equally Likely Outcomes
12.3 Conditional Probability and Independence
12.3.1 Conditional Probability 12.3.2 The Law of Total Probability 12.3.3 Independence 12.3.4 The Bayes Formula
12.3.1 Conditional Probability
12.3.2 The Law of Total Probability
12.3.3 Independence
12.3.4 The Bayes Formula
12.4 Discrete Random Variables and Discrete Distributions
12.4.1 Discrete Distributions 12.4.2 Mean and Variance 12.4.3 The Binomial Distribution 12.4.4 The Multinomial Distribution 12.4.5 Geometric Distribution 12.4.6 The Poisson Distribution
12.4.1 Discrete Distributions
12.4.2 Mean and Variance
12.4.3 The Binomial Distribution
12.4.4 The Multinomial Distribution
12.4.5 Geometric Distribution
12.4.6 The Poisson Distribution
12.5 Continuous Distributions
12.5.1 Density Functions 12.5.2 The Normal Distribution 12.5.3 The Uniform Distribution 12.5.4 The Exponential Distribution 12.5.5 The Poisson Process
12.5.1 Density Functions
12.5.2 The Normal Distribution
12.5.3 The Uniform Distribution
12.5.4 The Exponential Distribution
12.5.5 The Poisson Process
12.5.6 Aging
12.6 Limit Theorems 12.6.1 The Law of Large Numbers 12.6.2 The Central Limit Theorem
12.6 Limit Theorems
12.6.1 The Law of Large Numbers
12.6.2 The Central Limit Theorem
12.7 Statistical Tools
12.7.1 Describing Univariate Data 12.7.2 Estimating Parameters 12.7.3 Linear Regression
12.7.1 Describing Univariate Data
12.7.2 Estimating Parameters
12.7.3 Linear Regression
Appendix A Frequently Used Symbols
Appendix B Table of the Standard Normal Distribution
Answers to Odd-Numbered Problems
References
Photo Credits
Index
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