First Year, First Part (I/I)

Lecture: 3
Tutorial: 2

It is assumed that incoming students have a good grounding in algebra, some knowledge of trigonometry and analytic geometry and previous to calculus. By the end of the course, students will have seen the development of all of the elementary functions, ranging from polynomials to the inverse hyperbolic functions. In parallel, the calculus will be developed, making use of the
increasing richness of the available functions. The student’s skills in differentiation and integration will thus be progressively improved. Simple applications of the calculus will be explored from time to time. The course will conclude with brief discussion of conic sections and coordinate transformations.

1. Review. (5 hours)
1.1 Limit, Continuity.
1.2 Derivability of functions of a single variable. Derivative rules and formulas.
1.3 Integration rules and standard integrals.

2. Derivative (9 hours)
2.1 Higher order derivatives.
2.2 Maxima and Minima.
2.3 Mean value theorems.
2.4 Taylor and Maclaurin series.
2.5 Tangent and Normal.
2.6 Curvature.
2.7 Asymptotes.
2.8 Curve tracing.

3 Antiderivatives. (12 hours)
3.1 Definite integrals.
3.2 Fundamental theorem of integral calculus.
3.3 Improper integrals.
3.4 Reduction formulae for integrals, Beta and Gamma functions,

4 Applications of Integral ( 8 hours)
4.1 Areas
4.2. Lengths
4.3 Volumes.
4.4 Surfaces

5 Ordinary differential equations ( 5 hours)
5.1 Differential equations of first and second orders.
5.2 Linear equations with constant coefficients.

6. Analytic Geometry of two dimensions (6 hours)
6.1 Translation and rotation of axes.
6.2 Parabola.
6.3 Ellipse.
6.4 Hyperbola.
6.5 Central conics.

1. E.W. Swokowski, “Calculus With Analytic Geometry”, Second Alternate Edition,
PWS-Kent Publishing Co., Boston.

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