What is Calculus?

The following sections are included:

• Dedication
• Contents
• Preface
• Notes for Instructors

The following sections are included:

• Introduction
• Tangents to Circles
  • Exercises
• Tangents to Parabolas
  • Exercises
• Motion with Variable Speed
  • Exercises
• Tangents to Graphs of Polynomials
  • Exercises
• Rules for Differentiation
  • Elementary Rules
  • Inverse Function Rule
  • Product Rule
  • Quotient Rule
  • Exercises
• More General Algebraic Functions
  • Exercises
• Beyond Algebraic Functions
  • Exercises

In this chapter we introduce and discuss in some detail the basic objects of study in calculus. Some of this material may be familiar to the reader, but other parts will be new, and particular attention should be given to the latter. As we realized at the end of the Prelude, the required foundations will include some material that is not part of typical high school courses. Consistent with the goals of this book, rather than aiming for technical completeness, we shall focus on the key concepts and ideas, and we will emphasize those aspects that are most important for an understanding of calculus.

As seen in the Prelude, derivatives—which were introduced by algebraic techniques based on double points and multiplicities—provide the solution to the ancient problem of finding tangent lines for large classes of curves. These elementary methods, however, fail in the case of non-algebraic functions, in particular for the important case of exponential functions. Motivated by the application to velocity that we considered in Section 4 of the Prelude, we recognized that derivatives can also be captured by an approximation process, e.g., the velocity at time t₀ is approximated by average velocities over decreasing time intervals containing t₀. Such average velocities, or more generally, average rates of change, are very easy to define. The main new difficulty thus concerns understanding the approximation process and developing a general setting where it leads to meaningful results. Starting with the elementary case of algebraic functions and guided by the example of exponential functions, we shall now investigate this approximation process in detail, culminating with a notion of differentiability that generalizes the algebraic formulation and that is equivalent to the classical version used in analysis. Along the way we shall highlight the interpretation of derivatives as instantaneous rate of change, a concept that is the foundation for the numerous applications of calculus to the natural sciences over several centuries and to many other disciplines in more recent times.

Before proceeding with this chapter, the reader is urged to review Sections 4 and 8 of the Prelude.

The following sections are included:

• Exponential Models
  • Growth and Decay Models
  • Radiocarbon Dating
  • Compound Interest
  • Exercises
• The Inverse Problem and Antiderivatives
  • Functions with Zero Derivative
  • The Mean Value Inequality
  • Antiderivatives
  • Solutions of y′ = ky
  • Initial Value Problems
  • Exercises
• “Explosive Growth” Models
  • Beyond Exponential Growth
  • An Explicit Solution of y′ = y²
  • Exercises
• Acceleration and Motion with Constant Acceleration
  • Acceleration
  • Free Fall
  • Constant Deceleration
  • Exercises
• Periodic Motions
  • A Model for a Bouncing Spring
  • The Solutions of y′′ + ω²y = 0
  • The Motion of a Pendulum
  • Exercises
• Geometric Properties of Graphs
  • Increasing and Decreasing Functions
  • Relationship with Derivatives
  • Local Extrema
  • Convexity
  • Points of Inflection
  • Graphing with Derivatives
  • Exercises
• An Algorithm for Solving Equations
  • Newton's Method
  • Numerical Examples
  • “Chaotic” Behavior
  • Exercises
• Applications to Optimization
  • Basic Principles
  • Some Applications
  • Exercises
• Higher Order Approximations and Taylor Polynomials
  • Quadratic Approximations
  • Higher Order Taylor Polynomials
  • Taylor Approximations for the sine Function
  • The Natural Exponential Function
  • Exercises

The following sections are included:

• The Inverse Problem: Construction of Antiderivatives
  • Antiderivatives and New Functions
  • Finding Distance from Velocity
  • Control of the Approximation
  • Geometric Construction of Antiderivatives
  • A Simple Example
  • The Definite Integral
  • Exercises
• The Area Problem
  • Approximation by Sums of Areas of Rectangles
  • Rectangles and Triangles
  • Area under a Parabola
  • Area of a Disc
  • Exercises
• More Applications of Definite Integrals
  • Riemann Sums
  • Areas Bounded by Graphs
  • Volume of a Sphere
  • Work of a Spring
  • Length of a Curve
  • Income Streams
  • Probability Distributions
  • Exercises
• Properties of Definite Integrals
  • Riemann Integrable Functions
  • Basic Rules for Integrals
  • Examples
  • Exercises
• The Fundamental Theorem of Calculus
  • The Derivative of an Integral
  • The Integral of a Derivative
  • Some Examples
  • Exercises
• Existence of Definite Integrals
  • Monotonic Functions
  • Functions with Bounded Derivatives
  • Exercises
• Reversing the Chain Rule: Substitution
  • Integrals that Fit the Chain Rule
  • Examples
  • Changing Integrals by Substitution
  • Exercises
• Reversing the Product Rule: Integration by Parts
  • Partial Integration
  • Some Other Examples
  • Partial Integration of Differentials
  • Remarks on Integration Techniques
  • A Word of Caution
  • Exercises
• Higher Order Approximations, Part 2: Taylor's Theorem
  • An Application of Integration by Parts
  • Taylor Series of Elementary Transcendental Functions
  • Power Series
  • Analytic Functions
  • Exercises
• Excursion into Complex Numbers and the Euler Identity
  • Complex Numbers
  • The Exponential Function for Complex Numbers
  • The Euler Identity
  • Exercises

The following sections are included:

• Epilogue
• Index