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Matthew Boelkins

Active Calculus

Single-variable calculus built around guided activities: understanding and computing derivatives, their applications, definite integrals, differential equations, and Taylor series. The activity-style problems have you construct each idea before using it — understanding first, procedure second.

Active Calculus

169 exercises45 sections
Status
Sections
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Understanding the Derivative

20
How do we measure velocity?3
The notion of limit3
The derivative of a function at a point2
The derivative function1
Interpreting, estimating, and using the derivative6
The second derivative2
The tangent line approximation3

Computing Derivatives

36
Elementary derivative rules6
The sine and cosine functions3
The product and quotient rules7
Derivatives of other trigonometric functions6
The chain rule6
Derivatives of inverse functions6
Derivatives of functions given implicitly2

Using Derivatives

16
Related rates1
Using derivatives to evaluate limits4
Using derivatives to identify extreme values2
Global optimization5
Applied optimization4

The Definite Integral

12
Determining distance traveled from velocity1
Riemann sums3
The definite integral3
The Fundamental Theorem of Calculus5

Evaluating Integrals

24
The Second Fundamental Theorem of Calculus3
Integration by substitution6
Integration by parts6
Other options for finding algebraic antiderivatives5
Numerical integration4

Using Definite Integrals

17
Using definite integrals to find area and length3
Using definite integrals to find volume3
Density, mass, and center of mass5
Physics applications: work, force, and pressure1
Improper integrals5

Differential Equations

22
An introduction to differential equations3
Qualitative behavior of solutions to differential equations1
Euler's method2
Separable differential equations6
Modeling with differential equations6
Population growth and the logistic equation4

Taylor Polynomials and Taylor Series

22
Extending local linearization3
Taylor polynomials4
Geometric sums4
Taylor series4
Finding and using Taylor series4
Quantifying the accuracy of approximations3

Adapted from Active Calculus © 2012–2025 Matthew Boelkins. Licensed under CC BY-SA 4.0. Source: Active Calculus. Imported at revision dc86ba3.