Calculus, Vector and Complex Numbers

Calculus Chapters

Switch to Volume 3, _Why slopes and More Math for starter lessons for differentiation, limits and integration. Switch back to this site folder to learn more.

This site folder offers groups of lessons (chapters) on the following.

[ Appetizers ] [ 1 Review ] [ 2. Limits ] [ 3. Derivatives ] [ 4. Applications of Derivatives ] [ 5. Definite Integrals - Preview ] [ 6. Integration Applications ] [ Proofs ]

The Calculus Starter Guide talks about common difficulties and ideas for easing or avoiding them - satisfaction not guaranteed. Every teacher learns to remain silent or give advice described as approximately correct, for some circumstance not all. Good luck.

Study Tip: Before you hand in an assignment, ask a fellow student or a calculus expert to to catch and correct all errors in your notation and presentation. Put the same request on your assignment as well. You want to catch all errors in your presentation. Do likewise for all the exercises that you do. Poor presentation and poor understanding are two sides of the same coin.
Study Tip: Volume 3, Why Slopes and More Lesson, includes starter lessons for differential calculus - see chapters 1 to 5, and for integral calculus - see chapters 17 & 18, if not 19.

Appetizers: Preview of Chapters 3 and 4 -Why Slopes Starter and Preview Lessons for Calculus: These starter and preview lessons provide a context for earlier studies of (i) slopes, (ii) polynomial factorization and (iii) maximums and minimums for functions. These previews or starter lessons insert more steps into the start of calculus in the hope of easing or avoiding algebra shock. Calculus in the first instance consists of slope related calculations, their interpretation and reversal. Calculus provides a language for discussing numbers and quantities, and the relations between them in accounting, investing, engineering and science. New: These starter lessons included Flash Videos

These Calculus Appetizers (3 & 4 Preview) place the easiest part of calculus first (informally) to provide a gradual introduction of the full strength use of algebra in calculus.

Chapter 1: High School Math Revisited - last minute preparation for calculus. The aim here is to catch common errors, improve reading skills and revisit some basic concepts in algebra - Most calculus text include a chapter reviewing high school material starting with Functions:

The online version of this chapter, see below, starts before that and points students to Arithmetic Review Problems with Hints of Algebra,

Chapter 2: Limits of Functions - Saying how to calculate a number directly or via the limit of approximations defines it. In the study of derivatives, Limits of approximations are used to provide "official" definitions of slopes to curves y = f(x), velocity, other instantaneous rates of change, and acceleration. Areas and volumes are further defined by the limit of approximations (Riemann sum approximations) and some of these limits may be evaluated via reversal of slope or derivative methods. Watch out for a twist: Limits are said to exist only if discrete or continuous quantities approach a finite limit. Then limits with values +oo (plus infinity) or -oo (negative infinity) are defined but said not to exist because they are not finite. This website re-introduces the self-sufficient decimal viewpoint of limits to make them accessible, or to serve as a stepping stone to the algebraically challenging, epsilon-delta, decimal-free, viewpoint - which almost all do not get.

Chapter 3: Derivatives - (1) Derivatives (the slopes of a function in the preview) at a point are defined or calculated via limits of approximations to what the slope of a tangent line should be. (2) The arithmetic viewpoint is easy to follow. But the dependence of derivatives (slopes) of y =f(x) on the x-coordinate requires the algebraic concept of keeping x constant while an h or dx in a secant approximation to the derivative varies towards to zero in a limiting process. That pattern depends on a full mastery of what is a variable. (3) Next, there is a further twist. Namely, differentiation rules give algebraic methods (justified by limit consideration) for calculating derivatives algebraically. (4) Then rules for differentiating (obtaining slopes) for polynomials, trig functions, logs and/or exponentials alone, or combined as in algebraic expressions or composed follow. Limits and properties of these functions play key role in justifying and implying algebraic rules. Your aim here is to master the algebraic rules, and be well aware of how limits were use to imply those rules. Your aim is to also to master the chain-rule. Differentiation rules will also be used alone and in combination. to develop more rules. Mathematical induction will appear in that development. There may be variation between calculus courses. Some will postpone the discussion of logs and exponentials to later. (5) There may also be a discussion of implicit differentiation - very few further topics depend on it. (6) In this or the next chapter, you may meet several theorems - patterns to apply when certain conditions are met. Your aim is to understand their statement and in that statement be aware of the difference between saying A if B and saying A if and only if B. That is where logic appears.

Chapter 4: Application of Derivatives - The calculus starter lessons are previews of these applications. In the previews, formulas for slopes or derivative functions are given, the application here require their calculation and analysis to locate maximums and minimumss of height and also slope functions. Exercises include graphing functions and identifying interior and end point maximums and minimums of functions or their derivatives. Interior maximums and mininumss of derivatives (slopes) are called inflection points. You will also met first (slope) and second derivative (slope of slope) tests for interior maximums and minimums. Graphing may also involve vertical, horizontal and slanted asymptotes. The calculus preview included the first test. Further application include word max-min problems in which you will define a function y = f(x) and have find its max or min. Here velocity, rates of changes appear as derivatives while acceleration appears as a second derivative.

Chapter 5: Integration: The Riemann sum approximation (whatever that may be) of areas under curves y = f(x) between say x = a and x = b (b>a) in the limit, when the limit exists, leads to a definite integral. The first fundamental theorem gives conditions for the existence of that limit with and without the area interpretation. Then the second fundamental theorem of calculus says how to calculate the limit or definite integral with the aid of functions F(x) whose derivative or slope function is f(x). The net result is a Riemann sum approximation and limt process that yields a definite integral involving f(x) which can be calculated by finding (if you can) an anti-derivative of f(x). In the foregoing envelope, you will meet (a) summation notation for sums, see the algebraic properties of sums, derive those properties via mathematical induction, (b) a finer discussion of the Riemann sum approximation process - the requirement for the common or maximum width of rectangles in the Riemann sum approximation process to tend to zero; and (c) ad hoc antidifferentiation methods for finding from f(x) an anti-derivative F(x) with the property that f(x) is the derivative of F(x). Here all the rules for differentiation are applied in reverse. The introduction of indefinite integrals provides a context for this independent of the interpretation of definite integrals as limits of the Riemann sum approximation process. Most likely, you will meet indefinite integrals and anti-derivatives first, along with algebraic properties inherited from those for differentiation.

Chapter 6. Integration Applications:In essence, the applications consist of identifying physical quantities which can be approximated by and calculated in the limit via Riemann sums and the definite integral representation of the limit. Area under a curve is the application met in the introduction and motivation of the Riemann Sum Approximation and Limit process, the process that leads to a definite integral. This Riemann sum approximation and limit process yields definite integral representations and even definitions for areas between two curves via vertical, horizontal or slanted slicing, volumes of solids via vertical, horizontal or slanted slicing, volumes of solid of revolutions via slicing in planes perpendicular to the axis of revolution in the so called disk or washer methods; volumes of solid of revolutions via slicing into cylindrical shells around the axis of revolution. Here the convergence of the Riemann sum implies what an area or volume should be, and so provides a definition of area or volume for regions in 2D and solids in 3D for which area and volume were not previously defined. Further applications of the Riemann Sum Approximation and Limit process yields formulas (definite integrals) for work, fluid pressure, arc-length, moments, center of mass and so on - quantities need in geometry of engineering and physics - and college level statistics.

Appendices: Proofs : Here the proofs and concepts normally omitted or not seen in first and further courses in calculus. The treatment here provide a simpler, but not a simple path through the proofs with a few variations - pointers to an alternative calculus program & a context for ideas that gifted and talented students, or students who insist on having proofs may appreciate. The One Sided Range Theorems appear to be site Eurekas - a publishable paragraph perhaps. First time readers should scan the theorems and skip the proofs on first reading.