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

- 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.
- Infinite series, convergence and divergence of series, convergence tests of series.
Absolute and conditional convergence.
- Real-valued functions, the domain, the range, graphs, shifting graphs, increasing and
decreasing, inverse functions, composite functions.
- Elementary functions: linear and quadratic, polynomials, power,
exponential, logarithmic, trigonometric, hyperbolic, absolute value.
- 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.
- 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.
- 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.
- 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.
- 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.
- Applications of integrals: area between curves, the length of curves, volumes of solids
of revolution, moments and centers of mass. Improper integrals.
- Power series:
Cauchy-Hadamard theorem, differentiation and integration
of power series, Taylor and McLaurin series.

Books:
Ben Zion Kun, "Heshbon Diferenziali ve Integrali 1 ve 2", Bak, Haifa, 1994 (in Hebrew).
Protter and Morrey, "A First Course in Real Analysis", UTM Series, Springer-Verlag, 1991.
Thomas and Finney,"Calculus and Analytic Geometry", 9-th edition, Addison-Wesley, 1996.