Differential Calculus
Program
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1. Real Variable Functions
1.1. The real line: Different kinds of numbers. Inequalities, absolute value and subsets. Elementary curves.
2.2. Elementary functions: First definitions. Properties. Trigonometric functions. Logarithm and exponential. Operations with functions. Inverse functions. Polar coordinates.
2. Limits and Continuity
2.1. Limits: Definitions and basic properties. Calculus of limits.
2.2. Continuity: Continuity at a point. Fundamental theorems. Uniform continuity.
3. Derivatives and their Applications
3.1. Differentiability: Definition and basic derivatives. Basic properties.
3.2. Meaning of the derivative: First and second derivatives. Strong theorems. Extrema.
4. Local study of a function
4.1. Graphic representation: Convexity. Asymptotes.
4.2. Other applications of the derivative: Implicit derivative. Logarithmic derivative.
4.3. Taylor polynomial: Construction. Properties. Applications.
5. Sequences and Series of Real Numbers
5.1. Sequences of numbers: Definitions and properties. Limits. Recurring sequences.
5.2. Series of numbers: Preliminaries. Series of non-negative terms. Absolute convergence of series.
6. Sequences and Series of Functions
6.1. Sequences of functions: Punctual and uniform convergence.
6.2. Series of functions: Punctual and uniform convergence.
6.3. Taylor series: Convergence set. Properties.