Braver S. Full Frontal Calculus An Infinitesimal Approach

Vector Vectorum Books, 2023. — 204 p.Acknowledgements
Preface for Teachers
Changes in the 2nd Edition
Preface for Students
Differential Calculus
The Basic Ideas
Calculus: From A to The
Resurrection
The World of Calculus: An Overview
Differential Calculus: The Key Geometric Idea
Rates of Change
The Derivative of a Function
A Gift from Leibniz: d-Notation
An Infinitesimal Bit of an Infinitesimal Bit
The Derivative of y=x^2
Derivatives of Polynomials
One Last Example
The Differential Calculus Proper
The Derivatives of Sine and Cosine
The Product Rule
The Quotient Rule
Derivatives of Exponential Functions
The Chain Rule
Applications - Contrived and Genuine
Concavity and the Second Derivative
Optimization
Light on Light: Optics and Optimization
Implicit Differentiation
Related Rates
Chapter Pi: Limits
A Different Dialect: Limits
Zero Over Zero (and Other Indeterminate Forms)
Integral Calculus
The Basic Ideas
Integrals: Intuition and Notation
Positives and Negatives
Preparation for the FTC: The Antiderivative Lemma
Preparation for the FTC: Accumulation Functions
The Fundamental Theorem of Calculus (Stage 1: The Acorn)
The Fundamental Theorem of Calculus (Stage 2: The Oak)
The Integral Calculus Proper
Antiderivatives
Substitution
Interlude with Elementary Functions
Algebraic Trickery (Rational Functions)
Trigonometric Trickery (Part 1)
Trigonometric Trickery (Part 2)
Integration by Parts
Odds and Ends
Arclength
Surface Area
Improper Integrals
Riemann Sums
Escape From Flatland
Special Topics
Taylor Polynomials and Series
Taylor Polynomials
The Errror in a Taylor Approximation
Taylor Series
Infinite Series in General, Geometric Series in Particular
The Divergence Test and the Harmonic Series
Comparison Tests and P-Series
Absolute Series, Absolute Convergence
The Ratio Test
Power Series
(Five Terms in) A Series of Stories of Series
Polar Coordinates and Parametric Equations
Polar Coordinates
Polar Equations and their Graphs
Conversions
Areas in Polar Coordinates
Slopes in Polar Coordinates
Parametric Equations
Calculus and Parametric Equations
Selected Answers to Exercises
Index
About the Author