Active Calculus 907G4C4EJ1

URI	`m2://res/907G4C4EJ1`
Kind	textbook
Creators	Matthew Boelkins, David Austin, Steven Schlicker
Scope	deferred (no test decides the record, and the classifier has no model)

Expressions

e1: edition

Contents

chapter 1: Understanding the Derivative
section 1.1: How do we measure velocity?
subsection 1.1.1: Introduction
subsection 1.1.2: Position and average velocity
subsection 1.1.3: Instantaneous Velocity
subsection 1.1.4: Summary
section 1.2: The notion of limit
subsection 1.2.1: Introduction
subsection 1.2.2: The Notion of Limit
subsection 1.2.3: Instantaneous Velocity
subsection 1.2.4: Summary
section 1.3: The derivative of a function at a point
subsection 1.3.1: Introduction
subsection 1.3.2: The Derivative of a Function at a Point
subsection 1.3.3: Summary
section 1.4: The derivative function
subsection 1.4.1: Introduction
subsection 1.4.2: How the derivative is itself a function
subsection 1.4.3: Summary
section 1.5: Interpreting, estimating, and using the derivative
subsection 1.5.1: Introduction
subsection 1.5.2: Units of the derivative function
subsection 1.5.3: Toward more accurate derivative estimates
subsection 1.5.4: Summary
section 1.6: The second derivative
subsection 1.6.1: Introduction
subsection 1.6.2: Increasing or decreasing
subsection 1.6.3: The Second Derivative
subsection 1.6.4: Concavity
subsection 1.6.5: Summary
section 1.7: Limits, continuity, and differentiability
subsection 1.7.1: Introduction
subsection 1.7.2: Having a limit at a point
subsection 1.7.3: Being continuous at a point
subsection 1.7.4: Being differentiable at a point
subsection 1.7.5: Summary
section 1.8: The tangent line approximation
subsection 1.8.1: Introduction
subsection 1.8.2: The tangent line
subsection 1.8.3: The local linearization
subsection 1.8.4: Summary
chapter 2: Computing Derivatives
section 2.1: Elementary derivative rules
subsection 2.1.1: Introduction
subsection 2.1.2: Some Key Notation
subsection 2.1.3: Constant, Power, and Exponential Functions
subsection 2.1.4: Constant Multiples and Sums of Functions
subsection 2.1.5: Summary
section 2.2: The sine and cosine functions
subsection 2.2.1: Introduction
subsection 2.2.2: The sine and cosine functions
subsection 2.2.3: Summary
section 2.3: The product and quotient rules
subsection 2.3.1: Introduction
subsection 2.3.2: The product rule
subsection 2.3.3: The quotient rule
subsection 2.3.4: Combining rules
subsection 2.3.5: Summary
section 2.4: Derivatives of other trigonometric functions
subsection 2.4.1: Introduction
subsection 2.4.2: Derivatives of the cotangent, secant, and cosecant functions
subsection 2.4.3: Summary
section 2.5: The chain rule
subsection 2.5.1: Introduction
subsection 2.5.2: The chain rule
subsection 2.5.3: Using multiple rules simultaneously
subsection 2.5.4: The composite version of basic function rules
subsection 2.5.5: Summary
section 2.6: Derivatives of inverse functions
subsection 2.6.1: Introduction
subsection 2.6.2: Basic facts about inverse functions
subsection 2.6.3: The derivative of the natural logarithm function
subsection 2.6.4: Inverse trigonometric functions and their derivatives
subsection 2.6.5: The link between the derivative of a function and the derivative of its inverse
subsection 2.6.6: Summary
section 2.7: Derivatives of functions given implicitly
subsection 2.7.1: Introduction
subsection 2.7.2: Implicit Differentiation
subsection 2.7.3: Summary
chapter 3: Using Derivatives
section 3.1: Related rates
subsection 3.1.1: Introduction
subsection 3.1.2: Related Rates Problems
subsection 3.1.3: Summary
section 3.2: Using derivatives to evaluate limits
subsection 3.2.1: Introduction
subsection 3.2.2: Using derivatives to evaluate indeterminate limits of the form $\frac{0}{0}$.
subsection 3.2.3: Limits involving $\infty$
subsection 3.2.4: Summary
section 3.3: Using derivatives to identify extreme values
subsection 3.3.1: Introduction
subsection 3.3.2: Critical numbers and the first derivative test
subsection 3.3.3: The second derivative test
subsection 3.3.4: Summary
section 3.4: Using derivatives to describe families of functions
subsection 3.4.1: Introduction
subsection 3.4.2: Describing families of functions in terms of parameters
subsection 3.4.3: Summary
section 3.5: Global optimization
subsection 3.5.1: Introduction
subsection 3.5.2: Global Optimization
subsection 3.5.3: Moving toward applications
subsection 3.5.4: Summary
section 3.6: Applied optimization
subsection 3.6.1: Introduction
subsection 3.6.2: More applied optimization problems
subsection 3.6.3: Summary
chapter 4: The Definite Integral
section 4.1: Determining distance traveled from velocity
subsection 4.1.1: Introduction
subsection 4.1.2: Area under the graph of the velocity function
subsection 4.1.3: Two approaches: area and antidifferentiation
subsection 4.1.4: When velocity is negative
subsection 4.1.5: Summary
section 4.2: Riemann sums
subsection 4.2.1: Introduction
subsection 4.2.2: Sigma Notation
subsection 4.2.3: Riemann Sums
subsection 4.2.4: When the function is sometimes negative
subsection 4.2.5: Summary
section 4.3: The definite integral
subsection 4.3.1: Introduction
subsection 4.3.2: The definition of the definite integral
subsection 4.3.3: Some properties of the definite integral
subsection 4.3.4: How the definite integral is connected to a function's average value
subsection 4.3.5: Summary
section 4.4: The Fundamental Theorem of Calculus
subsection 4.4.1: Introduction
subsection 4.4.2: The Fundamental Theorem of Calculus
subsection 4.4.3: Basic antiderivatives
subsection 4.4.4: The total change theorem
subsection 4.4.5: Summary
chapter 5: Evaluating Integrals
section 5.1: Constructing accurate graphs of antiderivatives
subsection 5.1.1: Introduction
subsection 5.1.2: Constructing the graph of an antiderivative
subsection 5.1.3: Multiple antiderivatives of a single function
subsection 5.1.4: Functions defined by integrals
subsection 5.1.5: Summary
section 5.2: The Second Fundamental Theorem of Calculus
subsection 5.2.1: Introduction
subsection 5.2.2: The Second Fundamental Theorem of Calculus
subsection 5.2.3: Understanding Integral Functions
subsection 5.2.4: Differentiating an Integral Function
subsection 5.2.5: Summary
section 5.3: Integration by substitution
subsection 5.3.1: Introduction
subsection 5.3.2: Reversing the Chain Rule: First Steps
subsection 5.3.3: Reversing the Chain Rule: $u$-substitution
subsection 5.3.4: Evaluating Definite Integrals via $u$-substitution
subsection 5.3.5: Summary
section 5.4: Integration by parts
subsection 5.4.1: Introduction
subsection 5.4.2: Reversing the Product Rule: Integration by Parts
subsection 5.4.3: Some Subtleties with Integration by Parts
subsection 5.4.4: Using Integration by Parts Multiple Times
subsection 5.4.5: Evaluating Definite Integrals Using Integration by Parts
subsection 5.4.6: When $u$-substitution and Integration by Parts Fail to Help
subsection 5.4.7: Summary
section 5.5: Other options for finding algebraic antiderivatives
subsection 5.5.1: Introduction
subsection 5.5.2: The Method of Partial Fractions
subsection 5.5.3: Using an Integral Table
subsection 5.5.4: Using Computer Algebra Systems
subsection 5.5.5: Summary
section 5.6: Numerical integration
subsection 5.6.1: Introduction
subsection 5.6.2: The Trapezoid Rule
subsection 5.6.3: Comparing the Midpoint and Trapezoid Rules
subsection 5.6.4: Simpson's Rule
subsection 5.6.5: Overall observations regarding $L_n$, $R_n$, $T_n$, $M_n$, and $S_{2n}$.
subsection 5.6.6: Summary
chapter 6: Using Definite Integrals
section 6.1: Using definite integrals to find area and length
subsection 6.1.1: Introduction
subsection 6.1.2: The Area Between Two Curves
subsection 6.1.3: Finding Area with Horizontal Slices
subsection 6.1.4: Finding the length of a curve
subsection 6.1.5: Summary
section 6.2: Using definite integrals to find volume
subsection 6.2.1: Introduction
subsection 6.2.2: The Volume of a Solid of Revolution
subsection 6.2.3: Revolving about the $y$-axis
subsection 6.2.4: Revolving about horizontal and vertical lines other than the coordinate axes
subsection 6.2.5: Summary
section 6.3: Density, mass, and center of mass
subsection 6.3.1: Introduction
subsection 6.3.2: Density
subsection 6.3.3: Weighted Averages
subsection 6.3.4: Center of Mass
subsection 6.3.5: Summary
section 6.4: Physics applications: work, force, and pressure
subsection 6.4.1: Introduction
subsection 6.4.2: Work
subsection 6.4.3: Work: Pumping Liquid from a Tank
subsection 6.4.4: Force due to Hydrostatic Pressure
subsection 6.4.5: Summary
section 6.5: Improper integrals
subsection 6.5.1: Introduction
subsection 6.5.2: Improper Integrals Involving Unbounded Intervals
subsection 6.5.3: Convergence and Divergence
subsection 6.5.4: Improper Integrals Involving Unbounded Integrands
subsection 6.5.5: Summary
chapter 7: Differential Equations
section 7.1: An introduction to differential equations
subsection 7.1.1: Introduction
subsection 7.1.2: What is a differential equation?
subsection 7.1.3: Differential equations in the world around us
subsection 7.1.4: Solving a differential equation
subsection 7.1.5: Summary
section 7.2: Qualitative behavior of solutions to differential equations
subsection 7.2.1: Introduction
subsection 7.2.2: Slope fields
subsection 7.2.3: Equilibrium solutions and stability
subsection 7.2.4: Summary
section 7.3: Euler's method
subsection 7.3.1: Introduction
subsection 7.3.2: Euler's Method
subsection 7.3.3: The error in Euler's method
subsection 7.3.4: Summary
section 7.4: Separable differential equations
subsection 7.4.1: Introduction
subsection 7.4.2: Solving separable differential equations
subsection 7.4.3: Summary
section 7.5: Modeling with differential equations
subsection 7.5.1: Introduction
subsection 7.5.2: Developing a differential equation
subsection 7.5.3: Summary
section 7.6: Population growth and the logistic equation
subsection 7.6.1: Introduction
subsection 7.6.2: The earth's population
subsection 7.6.3: Solving the logistic differential equation
subsection 7.6.4: Summary
appendix A: A Short Table of Integrals

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