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Calculus for Biological Scientists
Jeff Shriner
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Front Matter
Colophon
Acknowledgements
Our Goals
Students: Read this!
Instructors: Read this!
1
Functions As Models
1.1
Biology and Calculus
1.1.1
Mathematical Models
1.1.2
Mathematical Processes
1.1.3
Summary
1.1.4
Exercises
1.2
Functions
1.2.1
Functions and their Representations
1.2.2
Explicit and Recursive Functions
1.2.3
Variables and Parameters
1.2.4
Combinations of Functions
1.2.5
Inverse Functions
1.2.6
Average Rate of Change
1.2.7
Summary
1.2.8
Exercises
1.3
Units and Dimensions of Functions
1.3.1
Units and Dimensions
1.3.2
Conversion Factors and Fundamental Relations
1.3.3
Function Transformations
1.3.4
Summary
1.3.5
Exercises
1.4
Linear Functions
1.4.1
Linear and Proportional Relationships
1.4.2
Graphs of Linear Relationships
1.4.3
Solutions of Linear Equations
1.4.4
Summary
1.4.5
Exercises
1.5
Exponential and Logarithmic Functions
1.5.1
Exponential Functions
1.5.2
Logarithmic Functions and Solving Exponential Equations
1.5.3
Scaling and Fitting Data with Logarithms
1.5.4
Summary
1.5.5
Exercises
1.6
Trigonometric Functions
1.6.1
The Unit Circle and Trigonometric Functions
1.6.2
Graphs and Transformations of Trigonometric Functions
1.6.3
Summary
1.6.4
Exercises
1.7
Discrete-Time Dynamical Systems
1.7.1
Representing Discrete-Time Dynamical Systems
1.7.2
Solutions to Discrete-Time Dynamical Systems
1.7.3
Summary
1.7.4
Exercises
1.8
Analyzing Discrete-Time Dynamical Systems
1.8.1
Cob-Webbing
1.8.2
Equilibrium Points and Stability
1.8.3
Summary
1.8.4
Exercises
1.9
Applications: The Lung Model and Competing Species
1.9.1
The Lung Model
1.9.2
A Model for Competing Species
1.9.3
Summary
1.9.4
Exercises
2
The Derivative
2.1
Limits of Functions
2.1.1
The Notion of Limit
2.1.2
Instantaneous Velocity
2.1.3
Summary
2.1.4
Exercises
2.2
The Derivative of a Function at a Point
2.2.1
The Derivative of a Function at a Point
2.2.2
Summary
2.2.3
Exercises
2.3
The Derivative Function
2.3.1
How the derivative is itself a function
2.3.2
Summary
2.3.3
Exercises
2.4
The Second Derivative
2.4.1
Increasing or decreasing
2.4.2
The Second Derivative
2.4.3
Concavity
2.4.4
Summary
2.4.5
Exercises
2.5
Elementary Derivative Rules
2.5.1
Some Key Notation
2.5.2
Constant, Power, and Exponential Functions
2.5.3
Constant Multiples and Sums of Functions
2.5.4
Summary
2.5.5
Exercises
2.6
Derivatives of the Sine and Cosine Functions
2.6.1
The sine and cosine functions
2.6.2
Summary
2.6.3
Exercises
2.7
Derivatives of Products and Quotients
2.7.1
The product rule
2.7.2
The quotient rule
2.7.3
Combining rules
2.7.4
Summary
2.7.5
Exercises
2.8
Derivatives of Compositions
2.8.1
The chain rule
2.8.2
Using multiple rules simultaneously
2.8.3
The composite version of basic function rules
2.8.4
Summary
2.8.5
Exercises
2.9
Derivatives of Inverse Functions
2.9.1
Basic facts about inverse functions
2.9.2
The derivative of the natural logarithm function
2.9.3
The link between the derivative of a function and the derivative of its inverse
2.9.4
Summary
2.9.5
Exercises
3
Using the Derivative
3.1
Linear and Quadratic Approximation
3.1.1
The tangent line
3.1.2
Linear Approximation
3.1.3
Quadratic Approximation
3.1.4
Summary
3.1.5
Exercises
3.2
The Stability Theorem
3.2.1
Testing stability of equilibria using the derivative
3.2.2
An inconclusive case using the derivative
3.2.3
Summary
3.2.4
Exercises
3.3
The Logistic Discrete-Time Dynamical System
3.3.1
The logistic map
3.3.2
Equilibrium points and stability of the logistic map
3.3.3
Summary
3.3.4
Exercises
3.4
Identifying Extreme Values of Functions
3.4.1
Critical numbers and the first derivative test
3.4.2
The second derivative test
3.4.3
Summary
3.4.4
Exercises
3.5
Global Optimization and Applications
3.5.1
Global Optimization
3.5.2
Applications
3.5.3
Summary
3.5.4
Exercises
3.6
Limits: L’Hôpital’s Rule
3.6.1
Using derivatives to evaluate indeterminate limits of the form
\(\frac{0}{0}\text{.}\)
3.6.2
Limits involving
\(\infty\)
3.6.3
Summary
3.6.4
Exercises
3.7
Limits: Leading Behaviors
3.7.1
Leading Behavior at
\(\infty\)
3.7.2
Leading Behavior at
\(-\infty\)
3.7.3
Summary
3.7.4
Exercises
4
Continuous-Time Dynamical Systems
4.1
Introduction to Differential Equations and Antiderivatives
4.1.1
Differential Equations
4.1.2
Solutions to Differential Equations and Antiderivatives
4.1.3
Summary
4.1.4
Exercises
4.2
Solving Pure-Time Differential Equations
4.2.1
Basic Antiderivatives
4.2.2
\(u\)
-Substitution
4.2.3
Summary
4.2.4
Exercises
4.3
Riemann Sums
4.3.1
Areas, Distance, and Displacement
4.3.2
Sigma Notation
4.3.3
Riemann Sums
4.3.4
When the function is sometimes negative
4.3.5
Summary
4.3.6
Exercises
4.4
The Definite Integral
4.4.1
The definition of the definite integral
4.4.2
Some properties of the definite integral
4.4.3
How the definite integral is connected to a function’s average value
4.4.4
Summary
4.4.5
Exercises
4.5
The Fundamental Theorem of Calculus
4.5.1
The Fundamental Theorem of Calculus
4.5.2
Evaluating Definite Integrals via
\(u\)
-substitution
4.5.3
The total change theorem
4.5.4
Summary
4.5.5
Exercises
4.6
Approximations of Solutions
4.6.1
Euler’s Method
4.6.2
The error in Euler’s method
4.6.3
Summary
4.6.4
Exercises
Backmatter
A
Answers to Selected Exercises
B
List of Symbols
Index
Colophon
Colophon
Colophon
This book was authored in PreTeXt.
