1. Historical survey 2. Convergence of Sequences 2.1 Definition of Convergence 2.2. The Basic Property of Real Numbers. 2.3. Real Line 3. Continuous Functions 3.1 Continuous Functions and Their Limits 3.2 Properties of Continuous Functions. 3.2.1 The Intermediate Value Theorem 3.2.2. Maxima and Minima of Continuous Functions 3.3. The graph of a function 4. Differential Calculus 4.1 Derivative 4...

1. INTRODUCTION. Everyone who has taught the error estimate for Simpson's rule in a few freshman calculus courses knows that the chances are very good that someone who hasn't quite thought things through will look at the error estimate and then ask " Yes, but how can we find the error exactly? " The questioner is usually thunderstruck by the observation that if we knew how to do so in general, ...

We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural numbers whenever s > 2k + 2k − 3.

The point (M,f(M)) is called an absolute maximum of f if f(x) ≤ f(M) for every x in the domain of f . The point (m, f(m)) is called an absolute minimum of f if f(x) ≥ f(m) for every x in the domain of f . More than one absolute maximum or minimum may exist. For example, if f(x) = |x| for x ∈ [−1, 1] then f(x) ≤ 1 and there are absolute maxima at (1, 1) and at (−1, 1), but only one absolute mini...

We present a new proof of generalized Flett’s mean value theorem due to Pawlikowska (from 1999) using only the original Flett’s mean value theorem. Also, a Trahan-type condition is established in general case.