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Discover Calculus
Single-Variable Calculus Topics with Motivating Activities
Peter Keep
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Front Matter
Colophon
Acknowledgements
Disclosure about the Use of AI
Notes for Students
Notes for Instructors
1
Limits
1.1
The Definition of the Limit
Defining a Limit
Approximating Limits Using Our New Definition
Practice Problems
1.2
Evaluating Limits
Adding Precision to Our Estimations
Limit Properties
Practice Problems
1.3
First Indeterminate Forms
A First Introduction to Indeterminate Forms
What if There Is No Algebra Trick?
Practice Problems
1.4
Limits Involving Infinity
Infinite Limits
End Behavior Limits
Practice Problems
1.5
The Squeeze Theorem
Weird Functions, Weird Behavior
Squeeze Theorem
Practice Problems
1.6
Continuity and the Intermediate Value Theorem
Continuity as Connectedness
Defining Continuity
Discontinuities
Intermediate Value Theorem
Practice Problems
1.7
Exploration: Limits
Precise Definition of Limits
Practice Problems
2
Derivatives
2.1
Introduction to Derivatives
Defining the Derivative
Calculating a Bunch of Slopes at Once
Practice Problems
2.2
Interpreting Derivatives
The Derivative is a Slope
The Derivative is a Rate of Change
The Derivative is a Limit
The Derivative is a Function
Notation for Derivatives
Practice Problems
2.3
Some Early Derivative Rules
Derivatives of Power Functions
Derivatives of Trigonometric and Exponential Functions
Some Properties of Derivatives in General
Practice Problems
2.4
The Product and Quotient Rules
The Product Rule
What about Dividing?
Derivatives of (the Rest of the) Trigonometric Functions
Practice Problems
2.5
The Chain Rule
Composition and Decomposition
The Chain Rule, Intuitively
Doing is Different than Knowing
Generalizing the Derivative of the Exponential
Practice Problems
3
Implicit Differentiation
3.1
Implicit Differentiation
Explicit vs. Implicit Definitions
Using a Derivative as an Operator
Some Summary and Strategy
Practice Problems
3.2
Derivatives of Inverse Functions
Wielding Implicit Differentiation
Derivatives of the Inverse Trigonometric Functions
Practice Problems
3.3
Logarithmic Differentiation
Logs Are Friends!
Wow, So Friendly!
Practice Problems
4
Applications of Derivatives
4.1
Mean Value Theorem
Slopes
The Mean Value Theorem
More Results due to the Mean Value Theorem
4.2
Increasing and Decreasing Functions
Critical Points, Local Maximums, and Local Minimums
Direction of a Function (and Where it Changes)
Using the Graph of the First Derivative
Strange Domains
Practice Problems
4.3
Concavity
Defining the Curvature of a Curve
Interpreting the Concavity at Critical Points
Practice Problems
4.4
Global Maximums and Minimums
When Do We Guarantee Both a Global Maximum and a Global Minimum?
What about Domains of Functions that Aren’t Closed?
Practice Problems
4.5
Optimization
Optimization Framework
Balancing Volume and Surface Area
What Other Examples Can We Do?
Practice Problems
4.6
Linear Approximations
Linearly Approximating a Function
Approximating Zeros of a Function
Practice Problems
4.7
L’Hôpital’s Rule
Indeterminate Forms
L’Hôpital’s Rule
Forcing Division
Products!
Differences!
Exponentials!
Practice Problems
4.8
Exploration: Applications of Derivatives
5
Antiderivatives and Integrals
5.1
Antiderivatives and Indefinite Integrals
Antiderivatives
Initial Value Problems
Indefinite Integrals
Practice Problems
5.2
Riemann Sums and Area Approximations
Rectangular Approximations
Selection Strategies
Practice Problems
5.3
The Definite Integral
Evaluating Areas (Instead of Approximating Them)
Signed Area
Properties of Definite Integrals
Practice Problems
5.4
The Fundamental Theorem of Calculus
Areas and Antiderivatives
Evaluating Definite Integrals
Practice Problems
5.5
More Results about Definite Integrals
Symmetry
Average Value of a Function
Practice Problems
5.6
Introduction to
\(u\)
-Substitution
Undoing the Chain Rule
Substitution for Definite Integrals
Antidifferentiate, then Evaluate
A More Wholistic Substitution
More to Translate
Practice Problems
6
Applications of Integrals
6.1
Integrals as Net Change
Estimating Movement
Position, Velocity, and Acceleration
Displacement, Distance, and Speed
Finding the Future Value of a Function
Practice Problems
6.2
Area Between Curves
Remembering Riemann Sums
Building an Integral Formula for the Area Between Curves
Changing Perspective
Practice Problems
6.3
Volumes of Solids of Revolution
From Area To Volume
Solids of Revolution
Reorienting our Rectangles
Practice Problems
6.4
More Volumes: Shifting the Axis of Revolution
What Changes?
Formalizing These Changes in the Washers and Shells
Practice Problems
6.5
Arc Length and Surface Area
Integrals for Evaluating the Length of a Curve
Integrals for Evaluating the Surface Area of a Solid
Practice Problems
6.6
Other Applications of Integrals
Physics Application: Mass
Physics Application: Work
Work: Springs
Work: Pumping Problems
Practice Problems
7
Techniques for Antidifferentiation
7.1
Improper Integrals
Improper Integrals
Strategies for Evaluating Improper Integrals
Convergence and Divergence of an Improper Integral
Practice Problems
7.2
More on
\(u\)
-Substitution
Variable Substitution in Integrals
7.3
Manipulating Integrands
Rewriting the Integrand
Antidifferentiating Rational Functions
Practice Problems
7.4
Integration By Parts
Discovering the Integration by Parts Formula
Intuition for Selecting the Parts
Some Flexible Choices for Parts
Solving for the Integral
Practice Problems
7.5
Integrating Powers of Trigonometric Functions
Building a Strategy for Powers of Sines and Cosines
Building a Strategy for Powers of Secants and Tangents
Practice Problems
7.6
Trigonometric Substitution
Another Type of Variable Substitution
Practice Problems
7.7
Partial Fractions
When?
How?
More Specific Strategies
Partial Fraction Type: Simple Linear Factors
Partial Fraction Type: Irreducible Quadratic Factors
Partial Fraction Type: Repeated Linear Factors
Practice Problems
8
Infinite Series
8.1
Introduction to Infinite Sequences
Sequences as Functions
Graphing Sequences
Sequence Terminology
Some Cool Recursive Examples
Practice Problems
8.2
Introduction to Infinite Series
Partial Sums
Visualizing the Sequence of Partial Sums
Special Series
Practice Problems
8.3
The Divergence Test and the Harmonic Series
The Relationship Between a Sequence and Series
The Divergence Test
Squeeze Theorem and Growth Rates
Practice Problems
8.4
The Integral Test
Infinite Series As a Kind of Integral
The Integral Test
Why Do We Need These Conditions?
Practice Problems
8.5
Alternating Series and Conditional Convergence
Defining Alternating Series, and the Main Result
Convergence, More Carefully
Practice Problems
8.6
Common Series Types
Geometric Series
\(p\)
-Series
Recapping Our Mathematical Objects
Practice Problems
8.7
Comparison Tests
Comparing Partial Sums
(Un)Helpful Comparisons
Limit Comparison
Practice Problems
8.8
The Ratio and Root Tests
Eventually Geometric-ish
Inconclusive Results
Practice Problems
8.9
Exploration: Infinite Series
Interesting Infinite Series
9
Power Series
9.1
Polynomial Approximations of Functions
What Do We Want From a Polynomial Approximation?
How Do We Build a Polynomial Approximation?
Are These Partial Sums?
Practice Problems
9.2
Power Series Convergence
Interval of Convergence
Operations on Power Series
Practice Problems
9.3
How to Build Taylor Series
Constructing Directly
Connections To Other Taylor Series
Practice Problems
9.4
How to Use Taylor Series
Approximations
Integrals
Euler’s Formula
Complex Analysis
Trigonometric Identities
Practice Problems
9.5
Exploration: Power Series
The Basel Problem
The Normal Distribution
Backmatter
A
Carnation Letter
Colophon
Discover Calculus
Single-Variable Calculus Topics with Motivating Activities
Peter Keep
Mathematics
Moraine Valley Community College
Last revised: January 3, 2026
Colophon
Acknowledgements
Disclosure about the Use of AI
Notes for Students
Notes for Instructors
🔗
