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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index
asymptotically semi-stable solution
y=ky=k if it is neither asymptotically stable nor asymptotically unstable
asymptotically stable solution
y=ky=k if there exists ε>0ε>0 such that for any value c(kε,k+ε)c(kε,k+ε) the solution to the initial-value problem y=f(x,y),y(x0)=cy=f(x,y),y(x0)=c approaches kk as xx approaches infinity
asymptotically unstable solution
y=ky=k if there exists ε>0ε>0 such that for any value c(kε,k+ε)c(kε,k+ε) the solution to the initial-value problem y=f(x,y),y(x0)=cy=f(x,y),y(x0)=c never approaches kk as xx approaches infinity
autonomous differential equation
an equation in which the right-hand side is a function of yy alone
carrying capacity
the maximum population of an organism that the environment can sustain indefinitely
differential equation
an equation involving a function y=y(x)y=y(x) and one or more of its derivatives
direction field (slope field)
a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point
equilibrium solution
any solution to the differential equation of the form y=c,y=c, where cc is a constant
Euler’s Method
a numerical technique used to approximate solutions to an initial-value problem
general solution (or family of solutions)
the entire set of solutions to a given differential equation
growth rate
the constant r>0r>0 in the exponential growth function P(t)=P0ertP(t)=P0ert
initial population
the population at time t=0t=0
initial value(s)
a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable
initial velocity
the velocity at time t=0t=0
initial-value problem
a differential equation together with an initial value or values
integrating factor
any function f(x)f(x) that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions
linear
description of a first-order differential equation that can be written in the form a(x)y+b(x)y=c(x)a(x)y+b(x)y=c(x)
logistic differential equation
a differential equation that incorporates the carrying capacity KK and growth rate rr into a population model
order of a differential equation
the highest order of any derivative of the unknown function that appears in the equation
particular solution
member of a family of solutions to a differential equation that satisfies a particular initial condition
phase line
a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions
separable differential equation
any equation that can be written in the form y=f(x)g(y)y=f(x)g(y)
separation of variables
a method used to solve a separable differential equation
solution curve
a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field
solution to a differential equation
a function y=f(x)y=f(x) that satisfies a given differential equation
standard form
the form of a first-order linear differential equation obtained by writing the differential equation in the form y+p(x)y=q(x)y+p(x)y=q(x)
step size
the increment hh that is added to the xx value at each step in Euler’s Method
threshold population
the minimum population that is necessary for a species to survive
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