©All rights reserved, 1982
by Martin E. Flashman
Revised June 11, 2002
Outline for The Sensible Calculus Book 
One Variable Component
Martin E. Flashman 
Humboldt State University 
Arcata, CA 95521 
Chapter 0. Preliminary Remarks.
        A. What is calculus?    Whys and wherefores.
        B. Backgrounds - Key Ideas:
                1. Numbers and Coordinates.
                2. Functions, Transformation Figures, and Graphs. 
                3. Lines
        C. Models and Mathematics
        D. Calculators, Computers, and Approximations
        E. (optional)  Least Upper Bounds: an important property of the real number system.

Chapter I. Introduction to the Derivative.
        A. Geometry - Tangent Lines
        B. Instantaneous Velocity
        C. Other Model Contexts: Probability, Economics, and Rates in Physical Sciences
              1. Comments for rates in physics and chemistry.
                        a. Rates of variables depending on time.
                        b. Rates of interdependent physical variables. 
                2. General comments on probability. Continuous random variables.
                        a. Outcomes and events.
                        b. Measuring probability.
                        c. Distribution and density functions.
                3. Comments on Economics 
                        a. Marginal Cost for indivisible commodities. 
                        b. Marginal cost for perfectly divisible (continuous) commodities.
                        c. Other marginal concepts for economics.
        D. The Definition of the Derivative:  A Number - A Function.
        E. Notation -The good, the bad, and the confusing
        F. Finding the Derivative.
                1. Numerical Estimation.
                2. Using the Definition.
                3. Some Geometric Insights.
                4. Functions Without Derivatives
        G. Using the Derivative - A Preview.[1]- recognition/discovery of some qualities of derivatives.
               1. First derivative for increasing, decreasing, extrema.
               2. Second derivative for acceleration, concavity,
               3. Reversing information about rates to discover/recover primitives. 
                        Discrete approaches to the continuous in reverse. 
        H. Limits in More Detail.
               1. What are we estimating and how.
               2. How to find limits when they exist--in brief...thinking.
               3. Some ways that limits can fail. Where thinking can avoid mistakes.
        I. Continuity.
              1. Continuity and the Derivative.
                2. Definition and Geometry/Approximations.
                3. The Intermediate Value Property
                4. The Extreme Value Theorem
(Optional)  Appendix:
             1. Alternative Views of Continuity, Sequences and Approximations.
             2. Proof of the Intermediate Value Theorem.
             3. Proof of the Extreme Value Theorem.

Chapter II. The Calculus of Derivatives.
        A. Sums and Scalar Products.  Polynomials.
        B. The Product and Quotient Rules.
        C. Application to Trigonometric Functions -
           an intuitive approach to trig derivatives followed by application of the various rules.
        D. The Chain Rule.
                1. Preliminary Discussion of Rates.
                2. "Proof" for Most Cases.
                3. Applications to Related Rates.
                4. (optional)  Proof of the Chain Rule.
        E. Implicit Differentiation.
                1. Implicit Functions.
                2. Implicit Differentiation.
                3. Application:  More Related Rates.
                4. Application:  Rational Powers.
                5. (optional)  Inverse Functions and Derivatives.
        F. Higher Order Derivatives.

Chapter III. Applications of the Derivative.
        A. Approximations Using the Derivative.
              1. Newton's Method for Approximating Roots.
                2. The Differential
        B. Extrema.
                1. Introduction to Extrema - Local and Global.
                2. Another Look at Continuity and the Extreme Value Theorem.
                3. A Necessary Condition:  The Critical Point (Fermat's) Theorem.
                4. A Complete Method for Finding Extreme of Continuous Functions on Closed Intervals.
                5. Word Problem Applications.
        C. Other Graphic Features of the Derivative.
              1. The Mean Value Theorem.
                2. Increasing and Decreasing Functions -
                   Including Reference to Use of the Intermediate Value Theorem.
                3. Extrema Revisited.
                4. Concavity with Graphical and Numerical Interpretations.
                5. The Second Derivative Test.
        D. Graphing in General.
                1. Intercepts and Symmetry.
                2. Asymptotes.
                3. A General Method for Graphing from Key Points.
        E. Another Look at Applications to Physics , Probability, and Economics.
                1. Velocity and Acceleration.
                2. Density, modes, and higher order features.
                3. Marginal Concepts and Elasticity.

Chapter IV. Differential Equations from an Elementary Viewpoint.
        A. Introduction to Differential Equations. - Key Concepts and Definitions.
                1. What is a Differential Equation?
                2. Antiderivatives and the Mean Value Theorem.
        B. The Calculus of Antiderivatives: A Beginning.
                1. x n.
                2. Linearity and Polynomials.
                3. The Chain Rule and Change of Variables.
                4. Integration by Parts.
                5. Reduction Formulae.
        C. Rates and Differential Equations -Another Look at Velocity and Acceleration.
        D. Geometry and Differential Equations -Tangent Fields and Integral Curves.
        E. Numerical Approximations.
                1. The Differential Revisited.
                2. Euler's Method and Difference Equations.
                3. Sequences and Convergence -An Initiation.
        F. Euler's Method Interpreted: Velocity and Area 
        G. The Geometry of Antiderivatives: Variation on A Theme.
        H. The Fundamental Theorem of Calculus / A Differential Equation Approach. 
                1. Computer programs for Euler's method.
                2. Proof of Fundamental Theorem of Calculus- Differential Equation Approach for positive functions

Chapter V. The Definite Integral. 
        A. Definition of the Definite Integral based on Euler sums. (Continuous functions are integrable)
        B. Key Properties of the definite integral with interpretations and justifications.
        C. The Fundamental Theorem of Calculus.
                1. Reformulation of the F.T.I.
                2. F.T. II.
        D. Other Methods for Approximating the Definite Integral.
                1. Partitions and Summation Notation.
                2. Midpoints.
                3. Upper and Lower Riemann Sums.
                4. Trapezoidal Rule.
                5. Parabolic (Simpson's) Rule.
        E. Riemann Sums and Definition of the Definite Integral. Continuous Functions are integrable.
        F. Remarks on Liebniz, the Differential, and  Infinitesmals.
        G. Applications of the Definite Integral -Two Views.
              1. A Riemann Sums View -The Area Between Curves.
                2. A Differential Equations View -The Area Between Curves.
        H. The Calculus of the Definite Integral.
                1. Linearity. [Done in V.B?]
                2. Additivity.[Done in V.B?]
                3. Change of Variables.
                4. Integration by Parts.
                1. Examples: Use of Riemann Sums to Find the Definite Integral.
                2. Theory in the Properties of the Definite Integral for Continuous Functions.
                3. Other Proofs of the Fundamental Theorem. MVT/Riemann Sum.
                4. An Example of a Nonintegrable Function.
                5. Another Kind of Integral.  An Intuitive View of the Lebesque Integral.

Chapter VI. Model Related Differential Equations: The Transcendental Functions.
        A. The Natural Exponential Function.
                1.A Population Growth Model.
                - DP/DT=P, P(0)=1 (by choice of units)
                - Preliminary discussion using tangent fields and Euler's Method after motivating the model  and suggesting the type of possible solution. 
                2. Basic Properties of a Solution. 
                -Demonstration that P behaves comparably to other exponential functions.
                -The approximation of  e=P(1)
                -The exponential function as  exp(x) = e x.
                -The integral of exp.
                3. Other Applications of Exp. 
                -Continuously compound interest.
                -Graphing functions involving exp. [esp.   exp(-x 2)]
                4. More General Growth and Decay Models 
                -DP/DT=KP, P(0)=A.
        B. The Natural Logarithm Function.
              1. A Learning Model 
                -DL/DT=1/T , L(1)=0
                -Preliminary discussion using tangent fields and Euler's Method after motivating the model and suggesting the type of solution. 
                2. L(t) = a definite integral : Solution by the Fundamental Theorem of Calculus. 
                3. Basic properties of L(t) = LN(t) = LOG(t) .
                -Demonstration of "inverse" relation between L  and exp. 
                4. Applications of  LN . 
                -The population model revisited. 
                -Logarithmic Differentiation. 
                -Functions with exponents:  a summary. 
                -Other bases for logarithms. 
        C. The Trigonometric Functions.
                1. The Inverse Trigonometric Functions and Their Derivatives.
                2. A Model for the Spring, a Model for Predator-Prey Populations: Simple Harmonic Motion.
                -D 2y/Dt 2 = -Y , Y(0) = 0 , Y'(0) = 1
                -Preliminary Discussion: Motivation of model and possible solution. 
                -Numerical solution with Euler's Method.  Feedback and the importance of  Y' . 
                3. Relation of  Y and  Y'  to the Unit Circle.
                    Y 2 + Y' 2 = 1 and the definition of sine and cosine. 
                     The derivative of the sine.
                4. The Trigonometric Functions and Their Derivatives. 
                5. A Final Look at the Spring, Populations, and Related Problems. 
                     Related Rates and Extreme Problems with Trigonometry. 
                6. Integration of Trigonometric Functions and Elementary Formulas. 
                1. Review of Trigonometry.  What You Need in Some Detail. 
                2. The Hyperbolic Trigonometric Functions. 

Chapter VII. Integration, Differential Equations, and the Trigonometric Functions.
        A. Summary of Elementary Integration Results.
        B. Applications of Substitution.
        C. Applications of Integration by Parts. -Reduction Formulae.
        D. Integration of Trigonometric Functions.
        E. Trigonometric Substitutions.
                1. A "polar" view of the area of the circle. 
                2. Trigonometric substitutions using sine. 
                3. Substitutions involving the tangent and secant. 
        F. Integration of Rational Functions.
                1. Simple examples.
                2. A general solution.
                3. Partial fractions.
        G. Integration by Table.
        H. Separation of Variables.
                1. A final look at DY/DX = KY and related problems.
                2. The method of separation of variables.
                3. Application to a model for limited growth.
                4. Application to rates in chemical reactions.
        I. Second Order Differential Equations.

Chapter VIII. More Applications of the Definite Integral.
        A. Refresher on Riemann Sums, Differential Equations and the Definite Integral.
        B. Arc Length: Several Views.
                1. Approximation using a chord. 
                2. Approximation using the tangent line. 
                3. A differential equation for arc length. 
                4. Parametric equations and arc length. 
                5. Speed and velocity. 
        C. Volumes.
                1. Drawing in three dimensions. 
                2. Volumes in general: An approach via approximation. 
                3. Volumes of solids of revolution. 
                        a. The disc method. 
                        b. The shell method. 
                4. Volumes of solids with cross sectional analysis. 
        D. Improper Integrals: Extending the Concepts of Integration.
                1. Integrals with noncontinuous functions. 
                2. Integrals with unbounded intervals. 
                3. The theory of convergent integrals: 
                -Comparison tests for integrals.
        E. Physics and Engineering Applications.
                0. General comments on models for physical science. 
                1. Mass and density. 
                2. Pressure and force. 
                3. Force and work. 
                4. Moments and the center of gravity. 
        F. Probability and Calculus.
                0. General comments on probability. 
                1. The mean of expected value. 
                2. Variance, the standard deviation, and  Tchebyscheff. 
                3. Appendix: Integration with Respect to a "Measure." 
                     A Further Glimpse at the Lebesque Integral. 
        G. Economics and Calculus.
                0. General comments. 
                1. Consumer savings. 
                2. Compound interest and present value. 

Chapter IX. Differential Equations and Polynomials: Taylor's Theorem.
        A. Very Simple Differential Equation Involving Higher Order Derivatives: Taylor Polynomials.
        B. Using Taylor Polynomials to Approximate:  Error  Estimation.
                0. The geometry of Taylor's Polynomials. 
                1. Taylor's Theorem. 
                        a. Derivative form of the remainder. 
                        b. Integral form of the remainder. 
                2. Approximating known functions. 
                3. Approximating solutions to differential equations using Taylor's theorem. 
        C. The Question of Convergence: A Prelude for Infinite Series.

Chapter X. Sequences and Series: Fundamental Properties.
        A. Sequences.
                1. Simple examples and definitions: visualizing sequences. 
                2. Special examples: 
                        a. Euler's method, Riemann sums and other integral approximations. 
                        b. Taylor approximations. 
                3. Elementary convergence. 
                        a. Definition. 
                        b. Examples: How to find limits. 
                4. Key theory of convergence 
                        a. The algebra of convergence. 
                        b. Convergence for monotonic sequences. 
                        c. The sandwich property. 
        B. Infinite Series: Introduction.
               1. Simple examples and definitions: Visualizing  series.
                2. Special examples:
                        a. Geometric series.
                        b. Harmonic series.
                        c. Taylor approximations.
                3. Elementary convergence (series).
                        a. Definition.
                        b. Key examples: Geometric and harmonic series.
       C. Key theory of convergence (series).
                1. The algebra of convergence. 
                2. The divergence test. 
                3. Positive series. 
                        a. Bounded convergence tests.
                        b. Comparison tests.
                        c. Integral tests.
                        d. Root tests. (optional)
                        e. Ratio test (Part I).
                 4. Alternating series.
                 5. Absolute convergence.
                         a. Definition and examples of conditional convergence. 
                         b. Absolute convergence implies convergence.
                         c. Ratio test (Part II).

Chapter XI. Power Series: Polynomials and Series.
        A. Definitions and Examples.
                1. The radius and interval of convergence.
                2. Functions and power series.
                        a. Functions defined by power series.
                        b. Power series defined by functions.
                        c. Taylor's Theorem revisited.
                3. Key examples.
                        a. Geometric series.
                        b. The transcendental functions.
                        c. The binomial theorem.
        B. Manipulating Power Series.
               1. Substitutions.
                2. Differentiation.
                3. Integration.
        C. Solving Differential Equations with Power Series.
                1. Finding the coefficients of the solution.
                2. Convergence of the solution.
                3. The transcendental functions: A final visit to functions defined by differential equations.

General Appendix.
        1. Review of Elementary Geometry and Trigonometry.
                A.The Pythagorean Theorem.
                B. Similar Triangles. 
                C. The Trigonometric Functions. 
                D. Basic Trigonometric Identities. 
        2. Mathematical Induction and Indirect Proofs.
        3. Polar Coordinates.
              A. Introduction to Polar Coordinates. 
               B. Differentiation in Polar Coordinates. 
               C. Integration in Polar Coordinates. 
                        1. The area problem in polar coordinates. 
                        2. Arc Length in polar coordinates. 
        4. Parametric Equations.
                A. Introduction to Parametric Curves. 
                B. Differentiation. 
                C. Area. 
        5. Analytic Geometry, the Conic Sections, and Elementary Differential Geometry.
                A.The Conic Sections as Loci.
                B. Coordinate Transformations. 
                C. Calculus and the Conics: The Differential View. 
                D. Curvature from an Elementary Viewpoint. 
                E. Total Curvature of Simple Closed Curves. 

[1] (The following sections may be deferred for later treatment)