SCC0692 - CALCULUS 1 WITH EXERCISE SESSIONS

Number Sets: Natural numbers and the principle of induction, summation notation, and the sum of the first n integers. Factorial and binomial coefficient. Binomial theorem. Rational numbers and the irrationality of the square root of two. Order properties and the supremum property; axiomatic definition of real numbers. Decimal expansions. Complex Numbers: Algebraic form and geometric interpretation. Conjugate and modulus. Trigonometric and exponential form. n-th roots. Equations in ℂ. Fundamental Theorem of Algebra. Introduction to the Topology of the Real Line. Real sequences and the epsilon-delta definition of the limit. Convergent, divergent, and irregular sequences. Properties of limits: uniqueness, monotonicity, sign permanence, the Squeeze Theorem. Arithmetic of limits and indeterminate forms. Limits and subsequences. Limits of monotone sequences and the number e. Ratio test for sequences. Fundamental limits. Infinite and infinitesimal sequences. The hierarchy of infinities. Landau symbols (Big-O, little-o, etc.). Subsequences and Their Properties: Sequences and topology. Bolzano–Weierstrass Theorem. Sequential compactness. Heine–Borel Theorem. The Cauchy condition and its equivalence with the existence of a finite limit. Numerical Series: Convergent, divergent, and irregular series. Necessary condition for convergence. Geometric series, the Mengoli series, and telescoping series. Comparison and asymptotic comparison tests for series with eventually constant sign terms. Harmonic series. Root and ratio tests with examples. Series with arbitrary sign terms: absolute convergence test. Leibniz’s test. Overview of Functions of a Real Variable: Domain, image, injectivity and surjectivity, invertibility, monotonicity, and boundedness. Review of elementary functions and their graphs. Transformations, symmetries, and graphing. Limits of Functions: Epsilon-delta definition and sequential limits. Properties and computation of limits. Landau symbols, infinitesimals, and infinities. Continuity: Epsilon-delta definition and sequential continuity. Arithmetic properties. Continuity of composite functions. Continuity of piecewise-defined functions and extensions by continuity. Classification of discontinuities. Continuity of piecewise functions and continuous extensions. Global Properties of Continuous Functions: Intermediate value and zero theorems, Weierstrass theorem. Uniform continuity and the Heine–Cantor Theorem. Invertible continuous functions. Introduction to Derivatives: Differentiability and continuity, differentiation rules: sum, product, quotient, and composition. Derivative of the inverse function. Local Maxima and Minima: Fermat’s Theorem, Lagrange’s Theorem, monotonicity test, characterization of constant functions on an interval. L'Hôpital's Rule. Higher-order derivatives. Taylor’s Theorem (with Peano and Lagrange remainders). Second-order conditions for determining the nature of stationary points. Convexity. Study of the graph behavior of a function. Introduction to the Riemann Integral: Step functions and their integrals. Definition and geometric interpretation of the Riemann integral. Necessary and sufficient conditions for integrability. Properties of the Riemann integral. Classes of integrable functions. Antiderivatives and a table of elementary antiderivatives. Integration techniques: substitution, integration by parts, integration of rational functions. Mean Value Theorem for integrals and the Fundamental Theorem of Calculus. Improper integrals and integral functions.