"Differential and Integral Calculus 1B". School of Mathematical Sciences. (Ya. Yakubov)

1. Basic notions on sets. Limits: definition of limit of infinite sequences, Cauchy condition, limit of monotone sequences, divergence, uniqueness of the limit, the sandwich theorem, subsequences, Bolzano-Weierstrass theorem.
2. Infinite series, convergence and divergence of series, convergence tests of series. Absolute and conditional convergence.
3. Real-valued functions, the domain, the range, graphs, shifting graphs, increasing and decreasing, inverse functions, composite functions.
4. Elementary functions: linear and quadratic, polynomials, power, exponential, logarithmic, trigonometric, hyperbolic, absolute value.
5. Informal definition of limit of functions and using epsilon-delta, continuous functions. Number e as a limit, the limit of Sin(x) divided by x. One-sided limits and continuity, the intermediate value theorem, inverse function and its continuity. Existence of extremum. Continuity of elementary functions.
6. Derivative as a tangent slope and a velocity, tangent and normal lines to functions. Calculating derivatives of polynomials, negative powers, Sin(x), Cos(x). Differentiation rules, derivative of tan(x) and inverse functions.
7. The chain rule, derivative of log_a(x) and of sinh(x), cosh(x), tanh(x). Derivative of a in power x using the chain rule. The intermediate value theorems of Rolle and Lagrange.
8. Taylor's formula with a remainder, the proof of Taylor formula with Lagrange remainder. Taylor's formula of elementary functions. Application to l'Hopital's rule. Application of Taylor's formula to sufficient condition of an extremum. Investigation of a function.
9. Indefinite integral, integral formulas, definite integral and area, Darboux integrals. The fundamental theorem of calculus, evaluating integrals. Substitution, integral of rational functions, integration by parts, trigonometric substitutions. Evaluating integrals using Taylor's formula.
10. Applications of integrals: area between curves, the length of curves, volumes of solids of revolution, moments and centers of mass. Improper integrals.
11. Power series: Cauchy-Hadamard theorem, differentiation and integration of power series, Taylor and McLaurin series.