Calculus I


At the end of this course the student will have acquired knowledge and skills that will enable him (i) to know and to use the concepts of majorant, minorant, supremum and infimum of a set, (ii) to deal with the absolute value function and to use triangular inequality to estimate errors, (iii) to know and to work with inverse trigonometric functions, in particular, identifying domains and solving equations and inequations involving these functions, (iv) to compute limits and to solve indeterminate forms, (v) to know Taylor’s formula and to use it to obtain polynomial approximations estimating
the errors, (vi) to compute antiderivatives and integrals, (vii) to know the concept of improper integral and to know methods to study convergence, (viii) to solve some first order differential equations, (ix) to conceive and to interpret algorithms to several of the topics considered.

The student will be able to a deeper study of infinitesimal analysis.

General characterization





Responsible teacher

Fábio Augusto da Costa Carvalho Chalub


Weekly - 5

Total - 70

Teaching language



High-school levem mathematics ("Matemática A").


The main reference is

Keith E. Hirst, Calculus of One Variable,  Springer undergraduate
mathematics series, 2006.

This book can be legally and freely download from the university networt at

Other reference books are:

1 - H. Anton, I. Bivens & S. Davis, Calculus, 8th edition, John Wiley & Sons, 2005.
2 - G. Simmons, Calculus with Analytic Geometry, 2nd edition, McGraw Hill, 1996.
3 - M. Spivak, Calculus, 3rd edition, Publish or Perish Inc., 2006.
4 - G. Strang, Calculus I, OpenStax, 2018.

Teaching method

Classes are theoretical/practical with oral presentation of concepts and results together with their proofs, whenever possible, and complemented with examples and applications. A list of exercises and problems is provided to the students to be solved independently. The student is encouraged to use and conceive computational programs related to the topics under study. Specific student difficulties will be addressed during classes or in individual sessions scheduled with the professor.
Students need to attend a minimum of two thirds of the classes in order to be evaluated. Continuous evaluation is based on two tests and possibly homework. If a student does not obtain approval through continuous evaluation, he can try it in an additional assessment.

Evaluation method

To pass in the continuous evaluation (épcoa normal) students are required to have a grade equal ou higher than 9.5. This value is the weighted average of

First test: 40%; second teste: 50%; project: 10%

Furthermore, students are required to attain a minimum of 7 points in the second test.

If a student do not pass in the continuous evaluation, he or she can do a "exame de recurso".

Grades are reounded to the nearest integer (n.5 -> n+1)

Further details can be obtained in the ocumentação de apoio, outros

Subject matter

1. Absolute value; errors; majorant, minorant, supremum and infimum.
2. Inverse trigonometric functions; Cauchy’s rule; Taylor’s formula.
3. Antiderivatives.
4. Integration (definite and indefinite integral, improper integral).
5. Differential equations (first order linear equations; equations with separable variables; change of variables).