2. LIMITS AND CONTINUITY (3 weeks)
Introduction to the limits: informal definition of the limit. Cases of non-existence of the limit (Different behavior from the left and right-hand side of the point; Infinite values; Oscillation). Formal definition of the limit. One-sided limits. Infinite limits. Vertical asymptotes. Properties of the limits. Limit of a composite function. Limits of the trigonometric functions. Strategies for the computation of limits (Functions coinciding in all points except one; Sandwich theorem; Two special limits). Calculating the limit of a piecewise function. Indeterminate forms: k/0, ∞/∞, ∞-∞, 0/0, 0·∞, 1∞. Definition of continuity. Two types of discontinuity: removable and non-removable. Properties of the continuity. Intermediate value theorem
3. DERIVATIVES: CONCEPT AND COMPUTATION (2 weeks)
The tangent line problem. Derivation, definition. Lateral derivatives. Differentiability and Continuity. Derivative rules. Chain rule. Higher order derivatives. Derivatives of inverse functions. Implicit differentiation. Differentials. Linear approximations. L’Hopital rule
4. STUDY AND GRAPHICAL REPRESENTATION OF A FUNCTION (1 week)
Domain of the function. Symmetry. Periodicity. Intersections with the x-axis and y-axis. Asymptotes. Parabolic branches. Increase and decrease. Maxima and minima. Concavity and convexity. Inflection points.
5. INTEGRATION (3 weeks)
Concept, Indefinite integral. Properties of the indefinite integral. Fundamental formulas of integration. Integration by parts. Integration of rational functions. Integration of trigonometric functions. Integration by substitution or change of variable. Approximation of the area of a two-dimensional region. Definition of the definite integral. Properties of the definite integral. Fundamental theorem of calculus (the Barrow’s rule). Integral defined as a function. The second fundamental theorem of calculus. Improper integrals. Improper integrals with infinite discontinuities. Integration of even and odd functions. Area between a function and the x-axis. Area between two functions. Average function value theorem (mean value theorem)
6. SEQUENCES AND SERIES (2 weeks)
Sequences:
Pattern recognition. Operations with sequences. Limit of a sequence. Properties of limits of sequences. Squeeze (the sandwich) theorem for sequences. Bounded monotone sequences.
Infinite series and partial sums. Definition of a convergent and divergent series. Geometric series. Properties of infinite series. The n-th term divergence test. The integral test. The p-series and harmonic series. The comparison test. The limit comparison test. The alternating series test. Absolute and conditional convergence. The ratio test. The root test. Strategy to analyze convergence of a series. Summary on convergence.
Other subjects:
Calculus II
Contabilidad I
Contabilidad II
Contabilidad III
Probability and Statistics I
Probability and Statistics II
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