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Calculus I: Activities
Mark A. Fitch
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Front Matter
Colophon
Preface
1
Limits
1.1
Methods for Infinity
1.1.1
Finding the Slope of One Point
1.2
Discovering Limits
1.3
Discovering More Limits
1.4
Understanding the Limit Definitions
1.4.1
Epsilon-delta geometric interpretation
1.4.2
Epsilon-Delta Calculations
1.4.3
Epsilon-delta proofs of a limit
1.5
Basic Limit Forms
1.5.1
Basic Limits
1.5.2
Properties of Limits
1.5.3
More Limit Forms
1.6
Using Limit Properties
1.6.1
Using Basic Limit Properties
1.7
Techniques for Evaluating Limits
1.7.1
Oscillations (and more): Squeeze Theorem
1.7.2
Infinity minus infinity: Factoring
1.7.3
Infinity divided by infinity: Dividing
1.7.4
Infinity minus infinity: Conjugate
1.8
Discovering Continuity
1.9
Known Continuous Functions
1.9.1
Known Continuous Functions
1.9.2
Practice
2
Derivatives
2.1
Discovering Derivatives
2.2
Discovering Differentiability
2.2.1
Terminology
2.2.2
Differentiable Appearance
2.3
Derivative as a Function
2.3.1
Derivatives and Tangent Slopes
2.3.2
Known Derivatives
2.4
Developing Derivative Properties
2.4.1
Calculating Derivatives Using the Definition
2.4.2
Arithmetic Derivative Properties
2.4.2.1
Derivative of a Sum
2.4.2.2
Derivative of a Scalar Product
2.4.2.3
Derivative of a Product
2.4.2.4
Derivative of a Quotient
2.4.3
Using Properties to Calculate Derivatives
2.4.3.1
Derivative of Integer Powers
2.4.3.2
Derivatives of trigonometric functions
2.4.3.3
Calculating Using Multiple Properties
2.5
Chain Rule of Derivatives
2.5.1
Illustrating the Chain Rule
2.5.2
Property and Example
2.5.3
Implicit Differentiation
2.6
Related Rates
2.6.1
Related Rates
2.7
Inderminate Forms in Limits
2.7.1
Variability of Indeterminate Forms
2.7.2
Indeterminate Forms
\(0/0\)
and
\(\infty/\infty\)
2.7.3
More Indeterminate Forms
3
Using Derivatives
3.1
Absolute and Relative Extrema
3.1.1
Absolute Extrema
3.1.2
Identifying Relative Extrema
3.2
Interpreting Derivatives
3.2.1
Rolle’s Theorem
3.2.2
Mean Value Theorem
3.2.3
Increasing and Decreasing Intervals
3.2.4
1st Derivative Test
3.2.5
Concavity
3.2.6
2nd Derivative Test
3.3
Finding Extrema
3.3.1
Finding Relative Extrema
3.4
Analyzing Curves and Functions Using Derivatives
3.4.1
Graphing Curves
3.4.2
Interpreting Functions and Derivatives in Context
3.5
Samples of Numeric Mathematics
3.5.1
Linearization
3.5.2
Newton’s Method
4
Integrals
4.1
Integrals
4.1.1
How Square Pegs Fill Round(ed) Holes
4.1.2
Constructing an Area Approximation
4.2
Interpreting Integrals
4.2.1
Interpret an Integral
4.3
Properties of Riemann Integrals
4.3.1
Properties to Break up Intervals of Integration
4.3.2
Properties to Break up Functions Integrated
4.3.3
Month One of Calculus 2
4.3.4
What Functions can be Integrated?
4.4
Fundamental Theorem of Calculus
4.4.1
Another Motivation for Integration
4.4.2
Related Concepts
4.4.3
Statement of the Fundamental Theorem of Calculus
4.4.4
Interpreting Integrals
4.5
Calculating Integrals
4.5.1
Using the FTC
4.5.2
Using Multiple Integral Properties
4.6
Integration using Substitution
4.6.1
Using the Chain Rule backwards
4.6.2
Examples of integral substitution
4.7
Calculating Area Using Integrals
4.7.1
Restricting Integrals for Areas
4.7.2
Setting up Area Calculations
5
Special Functions
5.1
A Calculus Definition for Logarithms
5.1.1
Old and New Definitions
5.1.2
Proving Log Properties
5.1.3
More Trig Anti-Derivatives
5.2
Inverse Functions and Exponential Functions
5.2.1
Function Inverses
5.2.2
Slopes on Inverse Functions
5.3
Evaluating Integrals of Inverse Trigonmetric Functions
5.3.1
Inverses of Trigonometric Functions
5.3.2
Differentiate Inverse Sine
5.3.3
Algebra to the Rescue: Inverse Trig Anti-Derivatives
5.3.4
Applications
5.4
Deriving Inverse Hyperbolic Trigonmetric Functions
5.4.1
Derive Inverse Hyperbolic Sine
Preface
Preface
This set of lessons is intended for pre-class and in-class activities for a first semester calculus course covering limits, derivatives, and integrals.
