By Serge Lang

This 5th version of Lang's e-book covers all of the issues normally taught within the first-year calculus series. Divided into 5 elements, every one portion of a primary direction IN CALCULUS includes examples and purposes with regards to the subject coated. furthermore, the rear of the publication comprises specified options to a great number of the routines, letting them be used as worked-out examples -- one of many major advancements over past variations.

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**Example text**

Then extremizers for considered isoperimetric problem have to satisfy the following conditions: α α k ????3 0 ????k x(k) 3 ∑ = λx(k), k = 1, 2, (13) (x(k))2 = 1, (14) k=1 x(0) = 0, x(3) = 0. (15) Note that solutions to the system (13)–(15) depend on λ. Table 2 presents examples of solutions for diﬀerent values of α and for chosen values of λ. Table 2 Examples of solutions to to system of Eqs. 8871928247 1 Acknowledgments Research supported by Bialystok University of Technology grant S/WI/02 /2011 (A.

13) In a wide class of the FODE one can extract from (13) the FOBE of the highest order. Then an equivalent form is as follows GL (np ????) ???? y(k) k0 k =f [ (n ????) GL (np−1 ????) ???? y(k),GL ????k p−2 y(k), … , y(k), k0 k k0 × ] (m ????) (m ????) ????k q u(k), … ,GL ????k q−1 u(k), … , u(k), k . × GL k k 0 0 (14) Deﬁning a set of equations x1 (k) = y(k), (????) x2 (k) ????k x1 (k) =GL ????(????) y(k), k0 k x3 (k) =GL ????(????) x (k) =GL ????(2????) y(k), k0 k0 k 2 k =GL k0 ⋮ xnp (k) =GL k0 ????(????) x (k) k np −1 =GL k0 (15) ((p−1)????) ????k y(k).

The Calculus of Variations. Universitext, Springer, New York (2004) 16. : Noethers second theorem for variable order fractional variational problems. , Stanisławski, R. ) Advances in Modelling and Control of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol. 320, pp. 59–68. Springer-Verlag, Berlin (2014) 17. : Diﬀerence Equations. An Introductions with Applications. Academic Press, INC [Harcourt Brace Jovanovich Publishers], New York (1991) 18. , Sturm-Liouville eigenvalue problems on time scales, Appl.