# Calculus: Newton Meets Technology

## A textbook emanating from

# Project DIRACC: Developing and Investigating a Rigorous Approach to Conceptual Calculus

## Patrick W. Thompson, Mark Ashbrook

## (c) 2016 All rights reserved

A Work in Progress

## NSF Grant DUE-1625678

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### Textbook:

Preface for Students

Preface for Instructors

Chapter 1:
Infinitely Small and Infinitely Large Quantities

- 1.0 Installing Graphing Calculator
- 1.1 Large and Small Quantities
- 1.2 Large and Small are Relative
- 1.3 Large Changes are Made of Tiny Changes
- 1.4 Infinitesimal variations and zooming a graph
- 1.5 Constant rate of change

Chapter 2:
Trigonometric Functions

- 2.1 A Graph You've Surely Seen
- 2.2 The Meaning of
*x* in *y* = sin(*x*)
- 2.3 Positive and Negative Angle Measures
- 2.4 Angle Measures Greater than 2π or less than -2π
- 2.5 The Meaning of
*y* in *y* = sin(*x*)
- 2.6 The Tangent Function in Trigonometry
- 2.7 The Meaning of
*x* and *y* in *y* = tan(*x*)
- 2.8 Properties of Trigonometric Functions

Chapter 3:
Functions, Variables, and Graphs

- 3.1 Variables, Constants, and Parameters
- 3.2 Graphing Calculator (GC)
- 3.3 Using Graphing Calculator
- 3.4 Typing Mathematical Statements in GC
- 3.5 What is a Graph?
- 3.6 How GC Draws Graphs
- 3.7 Understanding GC's DNA
- 3.8 (To Be Rewritten)
- 3.9 Coordinate Systems
- 3.10 Functions
- 3.11 Function Notation
- 3.12 Using Function Notation
- 3.13 Function Notation and Mathematical Models
- 3.14 Independent Variable vs. Argument of Function
- 3.15 Constant Rate of Change & Linear Functions
- 3.16 Operations on Functions
- 3.17 Properties of Functions
- 3.18 Inverse of a Function
- 3.19 Transformations of Graphs: An Application of Inverse Functions

- 4.1 Constant Rate of Change
- 4.2 Differentials as Linear Functions
- 4.3 Differentials in Action
- 4.4 Rate of Change at a Moment
- 4.5 How Small is "Small Enough"?
- 4.6 Functions Having a Value At Which There Is No Rate of Change at a Moment
- 4.7 Wrinkly Functions
- 4.8 Infinitely Wrinkly Functions Having No Rate of Change at Any Moment
- 4.9 Exact Rate of Change Functions

Chapter 5:
Accumulation from Rate of Change

- 5.1 Introduction to Accumulation Functions
- 5.2 Approximate Net Accumulation Functions from Exact Rate of Change Functions
- 5.3 Exact Net Accumulation Functions from Exact Rate of Change Functions
- 5.4 Overview of Accumulation from Rate

Chapter 6: Rate
of Change from Accumulation

- 6.1 Approximate Rate of Change in Open Form from Exact Accumulation in Closed Form
- 6.2 Closed Form Rate of Change from Closed Form Accumulation
- 6.3 A Garden of Exact Rate of Change Functions
- 6.4 Big Assumptions That We Made

- 6.5 The Mean Value Theorem

- 7.1 Properties of Rate of Change Functions and What They Tell Us
- 7.2 Higher Order Rate of Change Functions
- 7.3 Optimization
- 7.4 Related Rates
- 7.5 Local Linearity, Indeterminate Forms, and L'Hospital
- 7.6 More Fundamental Theorem of Calculus -- in Contexts

Chapter 8:

More on Integrals
- 8.0 Review of Terms and Meanings
- 8.1 The Nature of Integral Problems and Ways to Approach Them
- 8.2 Regions and Their Signed Areas
- 8.3 Volumes of Regions in Space
- 8.4 Arc Length and Surface Area
- 8.5 Applications of Integrals in the Sciences and Social Sciences

Chapter 9:Integration Techniques

- 9.0 Motive for Integration Techniques
- 9.1 Rate of Change Functions, Accumulation Functions, and Antiderivatives
- 9.2 Undoing the chain rule
- 9.3 Undoing the product rule
- 9.4 Antiderivatives of expressions involving trigonometric functions

Chapter 10:

Approximating Functions' Values
- 10.0 Approximating Rate of Change and Accumulation Function's Values
- 10.1 New Perspectives on Approximating Accumulation Functions from Rate of Change Functions
- 10.2 Polynomial Approximations, Taylor Series, and Convergence at a Moment
- 10.3 Sequences, Series, and Convergence Over an Interval

Chapter 11: Polar Coordinates

- 11.0 Review of Polar Coordinates
- 11.1 Graphs in Polar Coordinates and Their Properties
- 11.2 Coordinate Conversions
- 11.3 Signed Areas of Regions Bounded by Graphs in Polar Coordinates