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Active Calculus
Matthew Boelkins
Contents
Index
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Annotations
Contents
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Front Matter
Colophon
Acknowledgements
Contributors
Active Calculus: Our Goals
Features of the Text
How to Use this Text
1
Understanding the Derivative
How do we measure velocity?
The notion of limit
The derivative of a function at a point
The derivative function
Interpreting, estimating, and using the derivative
The second derivative
Limits, Continuity, and Differentiability
The Tangent Line Approximation
2
Computing Derivatives
Elementary derivative rules
The sine and cosine functions
The product and quotient rules
Derivatives of other trigonometric functions
The chain rule
Derivatives of Inverse Functions
Derivatives of Functions Given Implicitly
Using Derivatives to Evaluate Limits
3
Using Derivatives
Using derivatives to identify extreme values
Using derivatives to describe families of functions
Global Optimization
Applied Optimization
Related Rates
4
The Definite Integral
Determining distance traveled from velocity
Riemann Sums
The Definite Integral
The Fundamental Theorem of Calculus
5
Evaluating Integrals
Constructing Accurate Graphs of Antiderivatives
The Second Fundamental Theorem of Calculus
Integration by Substitution
Integration by Parts
Other Options for Finding Algebraic Antiderivatives
Numerical Integration
6
Using Definite Integrals
Using Definite Integrals to Find Area and Length
Using Definite Integrals to Find Volume
Density, Mass, and Center of Mass
Physics Applications: Work, Force, and Pressure
Improper Integrals
7
Differential Equations
An Introduction to Differential Equations
Qualitative behavior of solutions to DEs
Euler's method
Separable differential equations
Modeling with differential equations
Population Growth and the Logistic Equation
8
Sequences and Series
Sequences
Geometric Series
Series of Real Numbers
Alternating Series
Taylor Polynomials and Taylor Series
Power Series
Back Matter
A Short Table of Integrals
Answers to Activities
Answers to Selected Exercises
Index
Colophon
Feedback
Authored in PreTeXt
Chapter
4
The Definite Integral
ΒΆ
4.1
Determining distance traveled from velocity
4.2
Riemann Sums
4.3
The Definite Integral
4.4
The Fundamental Theorem of Calculus