Differential and integral calculus

Contents

• Real Numbers, Upper and lower bounds
• Three basic lemmas: Cantor, Heine-Borel, Bolzano-Weierstrass
• Sequences and their limits, Convergent sequences
• Cauchy’s sequences. Upper and lower limits. Extended convergence
• Subsequences and partial limits., Infinite series
• Rearrangement of the infinite series, Limits of functions. Basic properties
• The exponential function and the logarithm
• The second remarkable limit. The symbols “o small” and “∼”
• Continuous functions, I and II, The derivative
• Applications of the derivative, Derivatives of higher orders
• Basic theorems of the differential calculus: Fermat, Rolle, Lagrange
• Applications of fundamental theorems, Inequalities
• Convex functions. Jensen’s inequality
• The Taylor expansion, Taylor expansions of elementary functions
• The fundamental theorem of algebra and its corollaries
• Complex exponential function

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