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Topos Uranos
English

Final review of Calculus I

Summary

This entry is not one more lesson, but the final account of the course: it recalls the seven learning objectives that the syllabus set at the outset, shows in a single figure to which of them each lesson of each unit was assigned, explains how each objective was worked on throughout the three content units, distinguishes the objectives that have a unit of their own from the threads that run through all three and from those that run through the whole course, teaches how to read the self-assessment by objective, which gathers the first attempt at the exit assessments, and points out what the course leaves to Calculus II and to Calculus III.

Learning objectives

  1. State the seven global objectives of the course and recognize in which units and lessons each one was worked on.
  2. Distinguish the objectives that have a unit of their own from those that run through several, such as completeness and approximation, and from those that run through the whole course, deductive reasoning and modeling, and explain how the latter are exercised.
  3. Explain how the definition of limit and the completeness axiom underpin, through the derivative and the mean value theorem, what the course knows about the shape of a function, and what the course leaves to Calculus II and to Calculus III.
  4. Decide, on the basis of the self-assessment by objective, of the figure and of the challenges of each synthesis, which units and which lessons are worth reviewing before moving on to Calculus II.

What the course set out to do

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. The course took up the first two, which lead to the derivative, and was born with a general aim: 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. The lesson on the syllabus unfolded that aim into seven global learning objectives, which should be kept in view with the numbering by which the figure cites them:

  1. Prove by explicit deductive reasoning: write definitions as quantified statements and negate them, argue with epsilon and delta, deduce the existence theorems from the completeness axiom, show with counterexamples that a hypothesis is not superfluous, and distinguish what is deduced from what is admitted.
  2. Define and compute limits, finite, one-sided, at infinity and infinite, resolve indeterminate forms, also with L’Hôpital’s rule, and use them to find the asymptotes and the graph of a function.
  3. Decide the continuity of a function, classify its discontinuities and apply Bolzano’s theorem, the intermediate value theorem and the Weierstrass theorem, which rest on the completeness axiom.
  4. Define the derivative, characterize it by the linear approximation and by the Carathéodory factorization, prove its rules and use them to differentiate all kinds of functions, specifying the domain of each derivative.
  5. Prove and apply the theorems of Fermat, Rolle, Lagrange and Cauchy, and use them to study the monotonicity, the extrema, the convexity, the inflection and the curvature of a graph.
  6. Approximate values and roots with the differential, bisection and the Newton-Raphson method, bound the error of each approximation and recognize when a method fails.
  7. Model with the derivative rates of change, related rates and optimization problems, and communicate in writing setups, calculations and proofs.

Each lesson of the course also stated its own specific objectives. The lessons do not say which global objective each of them serves; that correspondence is a reading made from outside, with the same method as in the other courses of the site. Each specific objective is assigned to one main global objective and, if it touches them, to other secondary ones; each counts in full toward its main objective and by half toward each secondary one, and the lesson is joined as main to the global objective that gathers the most weight and as secondary to the others that receive some. If two objectives tie, the one with the lower number prevails. The objectives of the introductions that only describe the route of the unit are not assigned to any. The syntheses follow a criterion common to all the mathematics courses of the site: their main objective is objective 1, because their core reconstructs the deductive chain of the unit. Finally, the lesson on optimization problems has objective 7 as its main objective, although by a narrow margin the rule would give objective 5, because its declared subject is the setting up: translating the problem, fixing the domain and justifying that the extremum found is absolute. If this is so, the whole course takes the form of a spine: the head is the general aim; the column, the order of the units, each with its segment; each lesson is a node of that segment, joined to its main objective, and the seven objectives float around the column, held up by the lessons that feed them.