Library · Mathematics

Active Calculus

Matthew Boelkins, David Austin, Steven Schlicker

First-year university · CC BY-SA 4.0 · read it here, in full

This page titled Active Calculus (Boelkins et al.) was last modified on Sat, 22 Aug 2026 21:09:35 GMT and is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Matthew Boelkins, David Austin, and Steven Schlicker ( ScholarWorks @Grand Valley State University ) via source content that was edited to the style and standards of the LibreTexts platform.

1. 1: Understanding the Derivative

   1. 1.1: How do we Measure Velocity?
   2. 1.2: The Notion of Limit
   3. 1.3: The Derivative of a Function at a Point
   4. 1.4: The Derivative Function
   5. 1.5: Interpretating, Estimating, and Using the Derivative
   6. 1.6: The Second Derivative
   7. 1.7: Limits, Continuity, and Differentiability
   8. 1.8: The Tangent Line Approximation
   9. 1.E: Understanding the Derivative (Exercises)

2. 2: Computing Derivatives

   1. 2.1: Elementary Derivative Rules
   2. 2.2: The Sine and Cosine Function
   3. 2.3: The Product and Quotient Rules
   4. 2.4: Derivatives of Other Trigonometric Functions
   5. 2.5: The Chain Rule
   6. 2.6: Derivatives of Inverse Functions
   7. 2.7: Derivatives of Functions Given Implicitly
   8. 2.8: Using Derivatives to Evaluate Limits
   9. 2.E: Computing Derivatives (Exercises)

3. 3: Using Derivatives

   1. 3.1: Using Derivatives to Identify Extreme Values
   2. 3.2: Using Derivatives to Describe Families of Functions
   3. 3.3: Global Optimization
   4. 3.4: Applied Optimization
   5. 3.5: Related Rates
   6. 3.E: Using Derivatives (Exercises)

4. 4: The Definite Integral

   1. 4.1: Determining Distance Traveled from Velocity
   2. 4.2: Riemann Sums
   3. 4.3: The Definite Integral
   4. 4.4: The Fundamental Theorem of Calculus
   5. 4.E: The Definite Integral (Exercises)

5. 5: Finding Antiderivatives and Evaluating Integrals

   1. 5.1: Construction Accurate Graphs of Antiderivatives
   2. 5.2: The Second Fundamental Theorem of Calculus
   3. 5.3: Integration by Substitution
   4. 5.4: Integration by Parts
   5. 5.5: Other Options for Finding Algebraic Derivatives
   6. 5.6: Numerical Integration
   7. 5.E: Finding Antiderivatives and Evaluating Integrals (Exercises)

6. 6: Using Definite Integrals

   1. 6.1: Using Definite Integrals to Find Area and Length
   2. 6.2: Using Definite Integrals to Find Volume
   3. 6.3: Density, Mass, and Center of Mass
   4. 6.4: Physics Applications - Work, Force, and Pressure
   5. 6.5: Improper Integrals
   6. 6.E: Using Definite Integrals (Exercises)

7. 7: Differential Equations

   1. 7.1: An Introduction to Differential Equations
   2. 7.2: Qualitative Behavior of Solutions to Differential Equations
   3. 7.3: Euler's Method
   4. 7.4: Separable Differential Equations
   5. 7.5: Modeling with Differential Equations
   6. 7.6: Population Growth and the Logistic Equation
   7. 7.E: Differential Equations (Exercises)

8. 8: Sequences and Series

   1. 8.1: Sequences
   2. 8.2: Geometric Series
   3. 8.3: Series of Real Numbers
   4. 8.4: Alternating Series
   5. 8.5: Taylor Polynomials and Taylor Series
   6. 8.6: Power Series
   7. 8.E: Sequences and Series (Exercises)

9. 9: Multivariable and Vector Functions

   1. 9.1: Functions of Several Variables and Three Dimensional Space
   2. 9.2: Vectors
   3. 9.3: Dot Product
   4. 9.4: The Cross Product
   5. 9.5: Lines and Planes in Space
   6. 9.6: Vector-Valued Functions
   7. 9.7: Derivatives and Integrals of Vector-Valued Functions
   8. 9.8: Arc Length and Curvature

10. 10: Derivatives of Multivariable Functions

    1. 10.1: Limits
    2. 10.2: First-Order Partial Derivatives
    3. 10.3: Second-Order Partial Derivatives
    4. 10.4: Linearization- Tangent Planes and Differentials
    5. 10.5: The Chain Rule
    6. 10.6: Directional Derivatives and the Gradient
    7. 10.7: Optimization
    8. 10.8: Constrained Optimization - Lagrange Multipliers

11. 11: Multiple Integrals

    1. 11.1: Double Riemann Sums and Double Integrals over Rectangles
    2. 11.2: Iterated Integrals
    3. 11.3: Double Integrals over General Regions
    4. 11.4: Applications of Double Integrals
    5. 11.5: Double Integrals in Polar Coordinates
    6. 11.6: Surfaces Defined Parametrically and Surface Area
    7. 11.7: Triple Integrals
    8. 11.8: Triple Integrals in Cylindrical and Spherical Coordinates
    9. 11.9: Change of Variables

12. 12: Appendices

    1. 12.1: A- Short Table of Integrals
    2. 12.2: B- Answers to Activities
    3. 12.3: C- Answers to Selected Exercises

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