Calculus is the college or senior high school mathematics subject required for college or university studies in accounting, business, money matters, science, engineering and health. 

Calculus  employs at full strength most earlier elements of high school mathematics:-  functions, trig, algebra (including polynomials), mathematical induction, more logic,  geometry and exact arithmetic met briefly or fully in high school mathematics.  

Calculus is very, very demanding.  Half the students who take college calculus fail.    

Step (III) below is very long.  So it and step IV are switched.

Step (I)  Starter Lessons or Inserts for the first days of Calculus

Following these inserts, resume your normal methods for calculus instruction as is

From 1983-89, before I left college instruction for five years of work in applied mathematics, I tried to make calculus easier for engineering and non-engineering students at the junior college and university level.  Calculus itself requires the algebraic way of writing and reasoning met in high school at full strength and beyond that introduces further demands, very, very suddenly in a way that leads to algebraic shock.  That being said,  easy max-min analysis of functions based on the relatively easy, algebraic, sign analysis of derivatives appears in the middle of a calculus course, and not at the start. By giving students formulas for derivatives or slopes in factored form at the start of a calculus course (even for students who have met calculus upto the level of the chain-rule), we can develop algebraic reasoning skills more slowly by  providing a preview of differential calculus, geometric and algebraic.  That will ease a few fears and difficulties.  Site lessons on why slopes (Geometric  Preview) , Two logic puzzles (Implication Rules) and Three Skills for Algebra were all developed to ease or avoid fears and difficulties of students entering calculus in fall 1983 in an effort to support inductive principles for instruction. 

Calculus students may start with weak arithmetic,  weak algebra and weak or imprecise reading and writing skills.  Before any short or complete review of high school mathematics, or further development of calculus, try the following geometric and algebraic calculus previews  

The previews here provides a context for slope or derivative calculations, while developing algebraic skills slowly.   The algebraic preview here in fact takes the easy elements of later chapters on application of derivatives and put them first. That re-arrangement delays or slow the full-strength use of algebraic ways of writing and reasoning in calculus, while preparing students for it. 

To tackle weak or imprecise reading and writing skills, digress from calculus and present the two logic puzzle from this site's online chapter 2 on Implication Rules. Logic mastery (seeing the difference between one and two way implication rules  B if A (one way) and B if and only if A (two-way)  is a must for precision in reading and writing for calculus and studies in general.

For a first assignment, give arithmetic review problems with hints of algebra like the following to capturer and correct common arithmetic errors and to further develop or reinforce student algebra skills.  

 Arithmetic & Algebra Review  Exercises  to catch and correct common mistakes made by students entering calculus. 

Problem Set	Answers
A Arithmetic Problems	Solutions I
B Caclulator Problems	Solutions II
C More Arithmetic Problems	Solutions III
D Problems with hints of Algebras	Solutions IV

 Markers for my assignments would be told to catch and correct all errors in notation and comprehension, so that my students have the chance to learn from their mistakes. 

That assignment could also include slope and further sign analysis problems following the algebraic, calculus preview models. 

(II). Insert to develop the algebraic viewpoint of limits  
or their dependence on parameters variables

The section Limits via Algebra in chapter 15 of the online volume Why Slopes & More Math include examples to help student make the transition from compute limits of the Newtonian quotient

lim        f(x+h) - f(x)
h-->0           h

for given values of x (say 2, 3, 5 and 7) to any value of x. That shows how the slope or derivative of a function y = f(x) may regarded as a function of x.

(IV) Employ as an Integral Calculus Preview 
        or starter lesson.
        

(III). Insert to develop or avoid epsilons and delta view of limits, etc

The decimal perspective of limits, continuity and convergence

More sections from online volume Why Slopes & More Math

The epsilon-delta viewpoint of limits, continuity and convergence requires a  high level of algebraic reasoning skills, too high for all students, even those who will entering the study of pure mathematics.  The epsilon-delta viewpoint may be dropped without injury to students knowledge of mathematics or preceded by a decimal viewpoint that requires the assumption of decimal representation for real numbers.  That assumption is needed in  present-day mathematics course designs or curricula which inherit the decimal-free axiomitc viewpoint of real numbers present for the sake of purity, if not practicality, in modern mathematics curricula of the mid-1950s and onward.  The latter curricula failed the general population in mathematics by emphasizing an axiomatic structure for real numbers and the development of algebra and calculus disjoint from common practices - use of decimals especially - and without any sanction for those practices.  As a student following the modern mathematics curricula too literally, I sensed that separation and was disappointed by a lack of sanction for mathematical practices which appear in daily life as well as in the numerical examples met in calculus. The remedy I think is to provide an applied mathematics curricula in which axioms for decimal representation of real numbers and coordinates in 1, 2 and 3D are explicitly assumed to sanction common practices.  That sanction would go beyond the reach of pure mathematics, but it would provide an axiomatic codification of applied mathematics practices  from arithmetic to the end calculus.  In that, axioms could be classified as pure or as applied, so that axiomatic structure of modern mathematics is preserved. To learn more, see the text Calculus by Lipman Bers (Holt, Rinehart and Winston 1969, SBN 03-065240-5).

Pure mathematics introduces a decimal-free or independent theory of  real numbers into studies for undergraduate mathematics degrees and into axioms for real numbers met in high school and calculus. But first courses on calculus today may use decimals to introduce and illustrate the concept of what is a finite limit in examples while employing a decimal-free definition of limits and continuity, and may call latter a more precise viewpoint. But in the practical study of numerical methods and error control in calculations, decimal appear and provide a more concrete, applied mathematics, approach to the definition and discussion of limits, continuity and convergence.  Chapter 14 in site volume  show or suggest how to make the definition, evaluation and properties of limits, continuity and convergence easier to learn and teach with greater simplicity and precision, modulo assumptions about the representation of real numbers by decimal expansion and about the convergence of the latter.  The decimal viewpoint can stand alone, be sufficient for most students. It can also be presented before the epsilon-delta view to make the latter more accessible in enriched calculus courses for gifted students, or in real analysis courses for undergraduate students specializing in a quantitative discipline or mathematics.