Content GuideIntroductory (easy) materialThis Why Slopes appetizers was posted online before Volume 3 as an appetizers for mathematics and for the paperback version of Volume 3. It is not (yet) part of the paperback version. Chapter 1 to 5 offer a calculus preview in which a student with a minimal command of algebra may visualize why slopes (or derivatives) are computed and see how slope (or derivative) sign analysis allows for the identification of high points and low points on a curve or 2D skill hill y = f(x). These fivie chapters develop algebraic skills while providing motivation for calculus. Chapter 6 (poster material) points to the existence of jumps and discontinuities, discusses vertical shifts (or cliffs) in curves or trails y = f(x). Chapter 7 introduces velocity and the notion that slope may have units. In graphs of distance versus distance, units cancel in the ratios defining slopes, but in distance or position versus time, the ratios defining slope or velocity, a rate of change of one quantity with respect to another, the unit do not cancel. Here I am following the physical science convention of representing quantities by a number time a unit of measurement instead by the number alone. Unit of measurement can be carried through computations, algebraically, a convention employed in applied mathematics, but not always taught nor sanctioned in mathematics classes -- Oops! Chapter 8 Review offers a preview of calculus and calculus courses. It describes what to expect. Familiarity with previous chapters assumed. Chapter 9 On Calculus Courses describes a first, second and thirds courses in the subject. This description is lighter than that of Chapter 8. Chapter 10 Units & Slopes talks further about the appearance and removal of units in slope computations - this aids in the interpretation and comprehension of slopes or derivatives for linear and nonlinear functions in the presence and absence of units. Chapter 11 Slope of Slope asks for a slope to a 2D trail y = f(x) to graphed against position x. That leads to another graph and the slope to a slope (a hint or taste of the second derivative) Chapter 12 More on Units gives the physicists viewpoint of how units appear in first and second derivatives. The concept of dimension is introduced. Chapter 13 Acceleration graphs velocity ( a slope or first derivative) versus time. That leads to acceleration being viewed as the slope to a slope (or second derivative). Advanced Material -Chapter 14 Limits & Error (poster material)takes a technical turn. It starts with the decimal viewpoint of Limits, Error Control and Continuity. Sections cover the limits of functions, jumps in functions, Cauchy Sequences, Significant Digits, and limits of sequences from the decimal error control viewpoint of if there a guarantee that a computation will give a limit to finite or unlimited number of decimal places. Motivation is thus provided for the standard decimal-free ed viewpoint of limits and continuity.
Chapter 15 What is Slope (poster material)provides motivation for the standard approximation for the slope (or derivative) of a nonlinear function and then says if the approximation converge to a limit, that limit is the slope (or derivative). That is a twist of the typical kind in calculus or its interpretations. Saying how to compute or approximate a quantity, here the slope (or derivative) in the limit, defines it if there is convergence.
Chapter 16 What is Velocity (poster material) provides motivation for the standard approximation for the velocity non-constant speed motion, and then says if the approximation converges to a limit, that limit is the velocity (a slope or derivative). That is a second twist of the typical kind in calculus or its interpretations. Saying how to compute or approximate a quantity, here the velocity in the limit, defines it if there is convergence. Chapter 17 What is Area (poster material) provides motivation for the standard approximation for the area of a region and then says if the approximation converges to a limit, that limit is the area. That is a third twist of the typical kind in calculus or its interpretations. Saying how to compute or approximate a quantity, here the area in the limit, defines it if there is convergence. Area here is approximated by covering with squares. The first section of Chapter 18 Integration (poster material) moves from approximating areas under a curve y = f(x) > 0, a special region, with squares to a covering with rectangles and thus introduces the Riemann Sum approximation. The second second shows or suggest how to compute the limit, assuming it exists through anti-differentiation, that is a reversal of the slope computation process. The anti-differentiation process is justified in some cases by the Second fundamental theorem of calculus. The assumption that the limit of area approximations exists is justified by the first fundamental theorem of calculus. The proof of the latter is given in the last appendix as an appetizer for advanced students. The middle section of Chapter 18 introduces a function F(x) by defining it as the area under a curve y = f(x) between two points a and x. The last section of chapter 18 describes five more ways to define functions, ways often met in mathematics before, in and after calculus. Chapter 19 Logs & Powers introduces or defines the natural logarithm, another function, as the (signed) area under a curve. This definition is similar to that which appears in the middle section of Chapter 18, the discussion or derivation of the second fundamental theorem of calculus. Further logarithms are defined (briefly, too briefly) in terms of the natural logarithm, while the exponential function and one non-negative real number raised to a power (another real number) are introduced ro defined with the aid of the inverse to the natural logarithm. Vectors and Complex Numbers Revisited.Chapter 20 What's Next describes the next chapters. Chapter 21 Add Vectors shows how to add vectors in the plane, and attempts to provide motivation for it. The presentation here is an exploration of ideas which I wanted to clarify for myself, if not the reading. Chapter 22 Complex Numbers (Basic Ideas) introduces this numbers geometrically.
Chapter 23 Complex Numbers (Links to Trig) first points to the well-known simplification of trigonometry given by the use of complex number function cis(A) = cos(A) + i sin(A) = exp(iA) in higher mathematics and the mathematical disciplines, engineering and physics included. The last part of Chapter 23 Complex Numbers (Links to Trig) explores different ways to establish the complex number, distributive law for multiplication over addition. The simplified treatment of complex numbers (& trig), posted online at this site after the completion of this book, Volume 3 is simpler. But I am still puzzled regarded the optimal way to develop complex numbers and trigonometry impurely from a mix of assumptions about arithmetic and geometry. In pure mathematics, trigonometric functions may be defined without reference or dependence on geometric diagrams. However, novices need diagrams of one kind or another for their first comprehension of trig functions. Chapter 24 Complex Logs etc states formulas for logarithms, exponentials and powers of complex numbers, and formulas for the hyperbolic functions. It provides no more information about the functions. Decimal View of Real Analysis
The Appendices -- Beyond Calculus - Real Analysis, a Decimal Intro and ContextAppendix A. What's Next points to two references that could be read besides this work. Appendix B (poster material) gives finite and infinite version of the Pigeon Hole Principle, and then employs the infinite version to give a decimal-based proof of the Bolzano-Weierstrass Theorem. Here we assume that a real number is defined by an infinite decimal expansion or an infinite (recursive) process that in principle, if not practice, determine more and more digits in that expansion without stopping. Appendix C (poster material) covers the Triangle Inequality as is and Generalized Appendix D defines and shows the existence of greater lower bound and least upper bounds for bounded Sets & Sequences. Then it considers the limits of bounded Monotonic Sequences. Appendix E gives or states Error Control Inequalities needed or used to derive the Algebraic Properties of limit. Appendix F begins with a theorem to identity What Functions are Continuous. The theorem is to be used recursively. For continuous functions, appendix F continues with statements and proofs of theorems
Appendix G covers the properties of differentiable functions:
Appendix H cover integration theorems
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