Calculus, Vector and Complex Numbers

Welcome. This second site calculus section offers advice, directions and some worked examples for a first course in calculus. Each of the following links except the starter and warm up lessons corresponds to chapters in a typical North American calculus text.

[ Starter & Warm Up Lessons ] [ 1. Usual Review/Starter Lessons ] [ 2. Limits [13] ] [ 3. Differentiation Rules[28] ] [ 4. Applications of Derivatives [5] ] [ 5. Definite Integrals - Preview [5] ] [ 6. Integration Applications [6] ] [ Advanced Material ]

Algebra is required at full strength in calculus. The advice and directions in this section are aimed at easing or avoid difficulties by providing a progressive review and development of skills and concepts. Exploring this site section and online volumes 2 and 3 will provide you insights in to how to learn or teach a first course in calculus with greater ease. Good luck.

The online book 3 .Why.Slopes.&.More.Math.1995 may be regarded as the first calculus section in this site. This second section exists to further support calculus learning and teaching, and to avoid overwhelming the Volume 3 section with postscripts.
If sharing ideas with fellow instructor on how to ease or avoid difficulties in learning and teaching mathematics is impolite, this site is extremely impolite. Oops and Ouch.

Starter and Warm-up Lessons (Early Calculus or Late Precalculus Students)

This geometric preview and chapters 2 to 6 in Volume 3, Why Slopes and More Math, give a context for the senior high school level study of slopes and of factored polynomials. The same material may be employed at the start of calculus to make it easier. Calculus asks students to calculate derivatives (slopes for straight lines) and to do sign analysis, that is, to say identify interval where derivatives or slopes are zero, positive or negative. While calculus upto the calculation of derivatives is algebraically challenging, the sign analysis and interpretation as introduced in geometric preview and chapters 2 to 6 is very simple. Moreover, it develops algebraic skills in a way that makes the calculation of derivatives and before that limits, much easier.

Vol 2, Three Skills for Algebra covers many topics in algebra and logic that students starting calculus should have mastered or will have to master sooner or later. Also includes arithmetic review problems to catch common mistakes.

Vol. 3, Why Slopes & More Maths, gives starter lessons for differential and integral 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 site treatment is comprehensives.

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 and Differentiation Rules - (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.

Two common gaps to fill or avoid