"Differential and Integral Methods". Faculty of Engineering. (Ya. Yakubov)

1. Real-valued functions, the domain, the range, graphs, shifting graphs, increasing and decreasing, inverse functions, composite functions.
2. Elementary functions: linear and quadratic, polynomials, power, exponential, logarithmic, trigonometric, hyperbolic, absolute value, integer.
3. Informal definition of limit of functions, continuous functions. Number e as a limit, the limit of Sin(x) divided by x. Continuity of a function using sequences and using epsilon-delta, one-sided limits and continuity, the intermediate value theorem, inverse function and its continuity. Existence of extremum. Continuity of elementary functions.
4. 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.
5. The chain rule, derivative of rational powers, derivatives of sinh(x), cosh(x), tanh(x), arcsinh(x), arccosh(x), arctanh(x). Derivative of a in power x using the chain rule. Parametrizations of plain curves and their derivatives.
6. Rolle theorem, the intermediate value theorems of Lagrange and Cauchy.
7. Linearization and differentials. Taylor's formula with a remainder and Taylor series, the proof of Taylor formula with Lagrange remainder. Taylor's formula of elementary functions. Application to l'Hopital's rule. Application of Taylor series to binomial series. Application of Taylor's formula to sufficient condition of an extremum. Investigation of a function.
8. Complex numbers, Euler's formula, complex representation of trigonometric functions.
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, improper integral. Integrals which depend on a parameter and their derivative with respect to the parameter (Leibniz's rule). Evaluating integrals using series.
10. Applications of integrals: area between curves, the length of curves, volumes of solids of revolution, moments and centers of mass.
11. Limit and continuity of functions of two variables, partial derivatives, gradient, tangent and normal planes to surface. The chain rule, differentials, implicit differentiation. Taylor's formula for functions of two variables. Extremum. Lagrange multiplier method.
12. Double and triple integrals, iterated integrals.
13. Line integral of scalar functions. Line integral of vector-functions. Work. Path independent line integrals (conservative fields). Green's theorem (in the plane).
14. Surface area and surface integrals. Theorems of Stokes and Gauss.