Syllabus of Calculus I

Purpose and intended students

Three ancient problems led to calculus: the velocity of a moving body at an instant, the slope of the tangent to a curve and the area of a curved figure. In all three, direct substitution produces $\frac{0}{0}$ or a meaningless sum, and the correct answer is what the approximations approach. This course gives that idea of approaching an exact definition, the limit, and uses it to study the first two problems, which lead to the derivative; the third leads to the integral, the subject of Calculus II: Integral Calculus.

Its purpose is to found the differential calculus of functions of one real variable on the definition of limit and the completeness axiom, and to deduce from them, with complete proofs, what the derivative makes it possible to know about the change, the shape and the extrema of a function. If this is so, the course is not a repertoire of differentiation rules but a deductive edifice: every result that is used is stated and proved, or explicitly declared admitted when its proof belongs to another course, as happens with the theorem that guarantees the existence of implicitly defined functions or with the speed at which the Newton-Raphson method converges, which require later tools. Definitions are written as quantified statements, and the proofs with epsilon and delta are arranged so as to be reused: three lemmas of the lesson on the algebra of limits support the rules for limits, and the Carathéodory factorization supports the differentiation rules.

It is intended for anyone who has completed Algebra and Geometry II, or who masters its contents by another route: first-year university students of science, engineering, economics or mathematics teacher education, students in the last years of secondary school who are preparing for university entrance, and anyone who wants to rebuild rigorously a calculus learned as a set of rules.

Prior knowledge and prerequisites

The prerequisite of the course is Algebra and Geometry II, which in turn presupposes Algebra and Geometry I. From the two, the course uses above all the completeness axiom of the real numbers, with the supremum and the infimum, and calculation with inequalities and absolute values, which the lesson on the real line establishes and on which Bolzano’s theorem, the Weierstrass theorem and the mean value theorem rest; functions, their inverses, their transformations and their sign charts; powers with a real exponent, the number $e$ and the natural exponential, the logarithm and the hyperbolic functions; induction and the binomial theorem; and, from Algebra and Geometry II, the trigonometric functions, their inverses, the addition and sum-to-product formulas, and the inequality $\sin x < x < \tan x$ for $0 < x < \frac{\pi}{2}$, on which the derivative of the sine depends.

Now, many of those results were admitted there without proof because they required the limit: the continuity of the rational functions and of the trigonometric functions, asymptotes and the intermediate value theorem. This course proves them one by one. Anyone who has doubts about some part of the prior knowledge can go back to the synthesis of the corresponding unit of Algebra and Geometry; in addition, the entry assessment of each unit includes questions on what the unit takes as known, with the lesson worth reviewing after each error.

The course is designed, finally, to be followed in parallel with a course in predicate logic, Mathematical Logic II, planned on the site: the lessons write definitions as quantified statements and work with their negation without depending on that course. Anyone who wishes to acquire the habit of formal deduction beforehand can study Mathematical Logic I: Propositional Logic.

Where it leads

Two courses have this one as a prerequisite. Calculus II: Integral Calculus will define the integral as the limit of sums of areas of rectangles, will connect it with the derivative through the fundamental theorem, whose basis is the corollary according to which two functions with the same derivative differ by a constant, and will run through the differentiation rules in reverse: substitution reverses the chain rule, and integration by parts, the product rule. Mechanics I: Foundations of Mechanics will take velocity and acceleration as the first and the second derivative of position. After Calculus II, Calculus III: Multivariable Calculus will carry the limit, the derivative and extrema over to functions of several variables.

From this follows what this course leaves, by express decision, to the later ones. To Calculus II, uniform continuity, antiderivatives, sequences, which will open the way to the definite integral and, at the end, to series, and the Taylor polynomial, which will bound the error of the differential and prove the quadratic convergence of the Newton-Raphson method. To Calculus III, parametric curves and the implicit function theorem, which the implicit differentiation of unit 2 presupposes.

The results of the course also reappear in courses that do not require it. In Financial Mathematics, continuous compounding will be a standard limit, and the rate that equates the present value of a series of payments with a given amount will be computed with the Newton-Raphson method. In Data Analysis with R, growth models will be distinguished by their behavior at infinity, and the mean of a set of data will be obtained as the number that minimizes a sum of squares. In Programming in C++ I, bisection relies on the intermediate value theorem, the cost of an algorithm is measured with big $\operatorname{O}$ notation, and floating-point arithmetic explains the rounding errors with which a computer makes a table of values misleading, like the one in unit 1.

Global learning objectives

The objectives below describe what anyone who completes the course should be able to do. Each one is formulated as an observable performance, so that it can be checked with a concrete task; each lesson declares, in turn, its own objectives, and the closing lesson shows, lesson by lesson, to which of these seven each one was assigned. The objectives are numbered so that they can be cited there.

1. Objective 1. Prove the statements of differential calculus by explicit deductive reasoning: write definitions as quantified statements and negate them, build arguments with epsilon and delta that reuse the lemmas already proved, deduce the existence theorems from the completeness axiom, show with counterexamples that a hypothesis cannot be removed, and distinguish in each proof what is deduced from what is admitted.
2. Objective 2. Define and compute limits of functions of one real variable, finite, one-sided, at infinity and infinite, resolve indeterminate forms with the algebra of limits, the squeeze theorem, standard limits, equivalent infinitesimals and L’Hôpital’s rule, and use them to determine the asymptotes of a function and to sketch its graph.
3. Objective 3. Decide the continuity of a function and classify its discontinuities, prove the continuity of the elementary functions and of their inverses, and apply Bolzano’s theorem, the intermediate value theorem and the Weierstrass theorem, which rest on the completeness axiom, to prove the existence of solutions and of extreme values and to find the image of an interval.
4. Objective 4. Define the derivative as the limit of the difference quotient and characterize it by the linear approximation and by the Carathéodory factorization, prove the differentiation rules, the chain rule and the derivative of the inverse function, and use them to differentiate the elementary functions, implicitly defined functions and powers with a variable exponent and to compute higher-order derivatives, specifying the domain of each derivative.
5. Objective 5. Prove and apply Fermat’s theorem, Rolle’s theorem, Lagrange’s mean value theorem and Cauchy’s mean value theorem, deduce from them the tests for monotonicity, for local and absolute extrema and for convexity, and study with the first and the second derivative the shape of the graph of a function: increase and decrease, extrema, concavity, inflection and curvature.
6. Objective 6. Approximate values of functions and roots of equations with the differential, the bisection method and the Newton-Raphson method, bound the error of each approximation with the mean value theorem, monotonicity or convexity, and recognize when a method fails or converges slowly.
7. Objective 7. Model with the derivative situations from geometry, physics and the other sciences, such as rates of change, related rates and optimization problems, fixing the variables, their units and the domain precisely, and communicate in writing the setups, the calculations and the proofs with correct notation, including that of Leibniz, complete argumentation and a check of the plausibility of the results.

Objectives 2, 4 and 5 correspond, essentially, to the three content units: objective 2 to unit 1, which studies limits; objective 4 to unit 2, which studies derivatives; objective 5 to unit 3, which applies the mean value theorems. L’Hôpital’s rule, although it is proved in unit 3 from Cauchy’s mean value theorem, serves above all to compute limits, and for that reason belongs to objective 2. Objectives 3 and 6 are threads that run through the three units. Objective 3 has as its main lessons the lesson on limit and continuity in unit 1, the one on the intermediate value theorem in unit 2 and the one on the Weierstrass theorem in unit 3; the last two rest on completeness, which unit 1 already uses for the limits of monotonic functions. Approximation appears in the principal part of an infinitesimal, in bisection, in the differential and in the bounds given by the mean value theorem, and it has its own lesson in the Newton-Raphson method. The first and the seventh, on the other hand, are transversal: they do not belong to one unit, but are exercised in all of them. From this it follows that their achievement cannot be attributed to a particular lesson, and that it is advisable to cultivate them throughout the whole course. Now, both also have lessons that serve them in a principal way: the first, the three syntheses, whose core reconstructs the deductive chain of each unit; the seventh, the two lessons in which setting up the problem is the subject matter itself, the one on rates of change and related rates and the one on optimization problems.

Contents

The course is organized in five units. The first and the last frame the journey; the three intermediate ones develop the contents. Each of these opens with an introduction lesson, which presents its lessons, the historical background of its ideas, the problem that gave rise to them and their relevance, and closes with a synthesis lesson, which reconstructs the deductive chain of the unit, connects it with the rest of the course and with Algebra and Geometry, and proposes ten challenges of greater difficulty than the problems of the lessons.

Unit 0. Course overview

It consists of this lesson, which sets out the purpose, the global objectives, the contents, the workload and the design of the course.

Unit 1. Limits and continuity

Its objective is to define the limit with epsilon and delta, to prove its rules and to compute it in all its forms, finite, one-sided, at infinity and infinite; to define asymptotes by means of limits; to define continuity and prove it for the elementary functions; and to compare functions by means of equivalent infinitesimals and infinitely large functions.

1. Limits and continuity: introduction.
2. Limit of functions of a real variable.
3. Algebra of limits.
4. The squeeze theorem and trigonometric limits.
5. One-sided limits.
6. Limits as the variable tends to infinity.
7. Infinite limits.
8. Asymptotes and curve sketching techniques.
9. Limit and continuity.
10. Continuity of the elementary functions and standard limits.
11. Equivalent infinitesimals and infinitely large functions.
12. Limits and continuity: synthesis and challenges.

Unit 2. Derivatives

Its objective is to define the derivative as a limit and to characterize it by the linear approximation and by the Carathéodory factorization; to prove the differentiation rules and the chain rule; to prove the intermediate value theorem, from which follows the continuity of the inverse that its derivative requires; to differentiate the elementary functions, their inverses, implicitly defined functions and powers with a variable exponent; and to apply higher-order derivatives, the differential and rates of change.

1. Derivatives: introduction.
2. The derivative as the limit of a function.
3. Differentiation rules.
4. Derivatives of the elementary functions.
5. The chain rule for the derivative of a composition of functions.
6. The intermediate value theorem.
7. Derivative of the inverse function and implicit differentiation.
8. The variable in the exponent and logarithmic differentiation.
9. Higher-order derivatives and the differential.
10. Rates of change and related rates.
11. Derivatives: synthesis and challenges.

Unit 3. Applications of derivatives

Its objective is to prove, starting from the completeness axiom, the theorems of Weierstrass, Fermat, Rolle, Lagrange and Cauchy, and to deduce from them L’Hôpital’s rule and the tests for monotonicity, for extrema and for convexity; to study the curvature of a graph, to solve optimization problems and to approximate roots with the Newton-Raphson method.

1. Applications of derivatives: introduction.
2. The Weierstrass extreme value theorem.
3. The theorems of Fermat, Rolle, Lagrange and Cauchy.
4. L’Hôpital’s rule.
5. Monotonicity and the first derivative test.
6. Maxima and minima of a function.
7. Concavity, convexity and inflection points.
8. Curvature and radius of curvature.
9. Optimization problems.
10. The Newton-Raphson method.
11. Applications of derivatives: synthesis and challenges.

Unit 4. Course closing

It consists of the lesson Final review of Calculus I, which recalls the seven global objectives, shows in a figure to which of them each lesson of the course was assigned and which specific objectives each unit achieved, and gathers by objective the first attempt at the exit assessments.

Workload and credits

The workload of a course is not measured by hours of lecturing, which do not exist in an asynchronous course, but by the total work of the person studying it: reading, following the proofs, solving problems and reviewing. The estimate follows the same method in all the courses of the site: it counts the reading time of the text of each lesson at one hundred words per minute, counting each formula as one word, twenty minutes per solved problem, thirty-five per proposed problem, forty-five per challenge and twelve per entry or exit assessment, and adds a margin of one tenth for review.

Results

Taken together, the lessons of units 1 to 3 contain $140$ solved problems, $140$ proposed problems and $30$ challenges, as well as six unit assessments. The estimated workload per unit, rounded to the hour, is as follows:

• Unit 0, course overview: less than $1$ hour.
• Unit 1, limits and continuity: $66$ hours.
• Unit 2, derivatives: $61$ hours.
• Unit 3, applications of derivatives: $63$ hours.
• Unit 4, course closing: less than $1$ hour.

The three content units have a similar load: unit 1 gathers more lessons, and unit 3, longer lessons. The total is about $191$ hours of work. This figure should be read for what it is: an estimate for someone who studies all the proofs and solves all the problems and challenges. Whoever has already studied calculus, or restricts themselves to the proposed problems without tackling the challenges, will need considerably less; whoever studies the subject for the first time and without the habit of proving may need more, above all in unit 1, where one learns to reason with epsilon and delta.

Now, the figure is nominal: it is the one that sets the credits of the course, and it assumes reading at a constant pace and three quarters of an hour per challenge. Until it is measured with real students, it should be accompanied by a realistic range. If reading proceeded at half the assumed pace, as happens when each proof is rewritten with pencil and paper, and each challenge required seventy-five minutes instead of forty-five, the same method would give about $231$ hours. Therefore, the workload to be expected lies between about $191$ and about $231$ hours. The upper end is a prudent bound, not a measurement; the range will be corrected once the time spent by those who study the course is known, and until then the credits are calculated with the nominal figure.

Suggested pace

To name the paces, this plan uses the same criterion as the other plans of the site: full-time study is that of an academic year of $60$ credits, about $1650$ hours spread over some forty weeks, that is, about $40$ hours a week, and half-time is half of that, about $20$ hours a week. From this follows a recommended pace of one sixteen-week semester, with about $12$ hours of study a week, less than one third of full-time study; with the realistic range, from $12$ to $14$ hours a week, it remains below half-time. Thus Calculus I can occupy the semester that follows Algebra and Geometry II, and Calculus II the one after that.

An intensive pace, of about $24$ hours a week, somewhat more than half-time, completes the course in eight weeks, which is equivalent to the workload of someone who is taking no other demanding course in the same period. A more relaxed pace, of about $6$ hours a week, spreads it over two semesters, which can be divided as follows: units 0 to 2, limits, continuity and derivatives, take about $128$ hours, and units 3 and 4, the applications of derivatives and the closing, about $63$.

Credits

The credits of this site measure the total workload that a course requires of the person studying it (reading, problems, challenges and review), not class hours. One credit is equivalent to about $27.5$ hours of work, and a year of full-time study to $60$ credits, that is, to about $1650$ hours; this is the same criterion with which the preceding subsection names the paces of study.

Now, dividing the estimated $191$ hours by $27.5$ gives 6.95, which is rounded to $7$. If a band of $25$ to $30$ hours of work is allowed for the credit, the number $C$ of credits of the course is bounded between 6.4 and 7.6:

Therefore, the course has $7$ credits, somewhat less than one eighth of a year of full-time study. Anyone wishing to study only a portion of the course can use the list of hours per unit: units 0 to 2, for example, add up to about $128$ hours, that is, about $5$ credits.

Their use is practical: they make it possible to compare the workload of the courses on the site with one another and to decide how many to study at the same time without exceeding the time available. Someone who can devote half-time study, for example, to a sixteen-week semester, that is, about $320$ hours, can take on courses adding up to about $12$ credits; with full-time study over the same semester, about $23$.

These credits are a measure specific to this site, for planning study: they do not correspond to any official credit system, nor do they entail academic recognition by any institution.

Course design

Mode of study

The course is asynchronous, online and self-paced: it has no dates, live sessions or deadlines, and each student advances at the speed that their preparation and their time allow. It is offered in the fourteen languages of the site, with the same mathematics in all of them: the formulas and the steps of the proofs are identical in each version, and only the text is translated. Each lesson declares which lesson it is advisable to have studied beforehand, and the first ones refer to Algebra and Geometry, so that the recommended order is explicit and can be verified.

The structure of a lesson

All topical lessons follow the same structure. They begin with a summary and a list of their own learning objectives, formulated as observable performances. Then comes the theory, with the definitions, the theorems stated precisely and their proofs, and the examples. The proofs and the examples are presented as step-by-step players: each step writes one line of the reasoning, highlights the terms involved and, when it relies on an earlier result, goes back to it and highlights it as it is read, so that the memory of the reasoning stays in view; many are accompanied by a figure that changes with each step. The lesson closes with five solved problems, with their step-by-step solution, and five proposed problems, each with a folded hint and, after it, the final answer, so that the student can check their result without the path being revealed; some proposed problems, above all those that ask for a proof, add, folded after the answer, a worked solution, which should be opened only after writing one’s own.

The introduction and synthesis lessons have a structure of their own. The introduction presents the unit, its history, the problem that gave rise to it and its relevance; the synthesis reconstructs in a figure the deductive chain of the unit, with the results it reuses and the links it proves, connects it with the earlier units and with Algebra and Geometry, and proposes ten challenges, each with a hint, a final answer and a folded worked solution, which combine several lessons and demand more invention than the proposed problems.

The voice reader

Every lesson can be listened to aloud with the site’s reader, which uses the device’s own voices: the text is read just as it is written and the formulas are pronounced in words. This makes it possible to review a lesson away from the screen, to follow a proof by listening to it while watching the step-by-step player, and to study with visual impairments.

Assessment

The course neither grades nor certifies: its assessment on the site is formative. The proposed problems make it possible to check the achievement of the objectives of each lesson, and the challenges, that of the objectives of each unit. Each of the three content units also has an entry assessment in its introduction and an exit assessment in its synthesis, before the challenges: short self-assessments, corrected in the browser without grading. The entry assessment has nine or ten questions about the unit, plus five or six about what the unit takes as known from the earlier ones or from Algebra and Geometry, and the exit assessment, eleven or twelve; both indicate which lesson to review after each error and are compared with each other to show progress. The final review gathers the first attempt at each exit assessment into a profile by global objective, which is stored, like them, only in the student’s browser. The validation of skills, with certification value, will be done later and outside this site, on another platform.

In addition, the following evidence of achievement is suggested for the global objectives, which each student can produce on their own:

• Objective 1: write out, without consulting the lesson, the proof of a theorem of the course, such as the chain rule with the Carathéodory factorization or Lagrange’s mean value theorem, and show with a counterexample that none of its hypotheses is superfluous.
• Objective 2: compute an indeterminate limit in two ways, with equivalent infinitesimals and with L’Hôpital’s rule, and find all the asymptotes of a function.
• Objective 3: prove that an equation has exactly one solution in an interval, with Bolzano’s theorem and monotonicity, and find the image of a closed and bounded interval under a continuous function.
• Objective 4: differentiate a power with variable base and exponent and an implicitly defined function, specifying where each derivative exists.
• Objective 5: study the monotonicity, the extrema, the convexity and the inflection points of a function, and use them to sketch its graph.
• Objective 6: approximate with the Newton-Raphson method a root of an equation that cannot be solved explicitly, choose the starting point and bound the error of the approximation obtained.
• Objective 7: write out the complete solution of an optimization or related-rates problem, with setup, domain, working, justification of the extremum, answer and plausibility check, so that another person can follow it without additional explanations.

How to study this course

The course is designed to be studied in order. Each unit presupposes the earlier ones and the two Algebra and Geometry courses, and many proofs cite results from earlier lessons; skipping a lesson forces one, sooner or later, to go back to it. This holds above all for unit 1: the definitions with epsilon and delta and the lemmas of its first lessons support all the later proofs.

Each unit has two checkpoints. The entry assessment, at the beginning of the introduction, is the starting point: it should be answered before studying the unit, without consulting anything, noting the lessons to which the errors refer, which are the ones that call for more care. The exit assessment, in the synthesis and before the challenges, is the contrast: it is answered after finishing the lessons, also without consulting, and is compared with the entry assessment to see what changed and which lessons should be reviewed. In both, the first attempt is what counts, being the one that is compared and the one that the profile of the final review gathers; later attempts are made with the feedback already seen. The challenges complete the check, because they require combining and writing out what the short questions only touch on. From this it follows that whoever believes they already master a unit should not decide it by a single test: if their exit assessment is solid and they solve the challenges of the synthesis, they may move on to the following unit; if not, it is advisable to go through the lessons that the one and the others point to.

In each lesson, it is advisable first to read the summary and the objectives, which say what is sought; then to study the theory with pencil and paper, trying to anticipate each step of the proofs before the player writes it; to attempt the solved problems before looking at their solution, and to solve the proposed ones without help, opening the hint only when the attempt has been exhausted, and the answer only at the end, to check. A problem whose answer was consulted before attempting it has not been solved, but read. In a calculus course, moreover, it is advisable to distrust tables of values and graphs drawn by a calculator: they suggest a limit or an extremum, but they do not prove it, and unit 1 shows tables that mislead.

The challenges are harder than the problems of the lessons, and it is normal for some to require several attempts on different days. It is not advisable to dwell indefinitely on one: noting it down, continuing to advance and coming back to it later is often the shortest path. Now, the estimated workload assumes that all of them are attempted seriously, and in that effort much of what the course can teach is at stake.

Finally, it is advisable to write. Writing out a proof or a complete solution, with its premises and its conclusions, is the only way to know whether one has understood, and it is also the seventh of the course objectives. A study notebook, in which the definitions and the theorems of each unit are rewritten in one’s own words, is the best review material the course can leave behind, and it will be all the more so in Calculus II, which will return to almost everything that is proved here.