Download A Course in Calculus and Real Analysis (Undergraduate Texts by Sudhir R. Ghorpade, Balmohan V. Limaye PDF

By Sudhir R. Ghorpade, Balmohan V. Limaye

ISBN-10: 0387305300

ISBN-13: 9780387305301

This ebook presents a self-contained and rigorous creation to calculus of services of 1 variable. The presentation and sequencing of themes emphasizes the structural improvement of calculus. whilst, due significance is given to computational options and functions. The authors have strived to make a contrast among the intrinsic definition of a geometrical suggestion and its analytic characterization. through the booklet, the authors spotlight the truth that calculus offers an organization origin to numerous innovations and effects which are usually encountered in highschool and authorised on religion. for instance, you will discover right here an evidence of the classical end result that the ratio of the circumference of a circle to its diameter is identical for all circles. additionally, this ebook is helping scholars get a transparent knowing of the concept that of an attitude and the definitions of the logarithmic, exponential and trigonometric capabilities including an explanation of the truth that those should not algebraic features. a couple of subject matters which may were inadequately coated in calculus classes and glossed over in genuine research classes are handled right here in substantial element. As such, this publication presents a unified exposition of calculus and genuine analysis.

The in simple terms necessities for studying this e-book are issues which are often coated in highschool; even if, the reader is predicted to own a few mathematical adulthood and a capability to appreciate and enjoy proofs. This booklet can be utilized as a textbook for a significant undergraduate path in calculus, whereas components of the e-book can be utilized for complex undergraduate and graduate classes in genuine research. every one bankruptcy includes a number of examples and a wide collection of workouts, in addition to "Notes and Comments" describing salient positive aspects of the exposition, similar advancements and references to proper literature.

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Additional resources for A Course in Calculus and Real Analysis (Undergraduate Texts in Mathematics)

Example text

F is said to be bounded below on D if there is β ∈ R such that f (x) ≥ β for all x ∈ D. Any such β is called a lower bound for f . 3. f is said to be bounded on D if it is bounded above on D and also bounded below on D. Notice that f is bounded on D if and only if there is γ ∈ R such that |f (x)| ≤ γ for all x ∈ D. Any such γ is called a bound for the absolute value of f . Geometrically speaking, f is bounded above means that the graph of f lies below some horizontal line, while f is bounded below means that its graph lies above some horizontal line.

Give an example of a nonconstant function f : (−1, 1) → R such that f has a local extremum at 0, and 0 is a point of inflection for f . 32. Let I be an interval and f : I → R be any function. (i) If f is monotonically increasing as well as monotonically decreasing on I, then show that f is constant on I. (ii) If f is convex as well as concave on I, then show that f is given by a linear polynomial (that is, there are a, b ∈ R such that f (x) = ax + b for all x ∈ I). 33. Let I be an interval and f : I → R be any function.

Likewise, if S is bounded below, then show that the set LS = {β ∈ R : β is a lower bound of S} is bounded above, max LS exists, and inf S = max LS . (iii) i3 = Exercises 33 6. Let S be a nonempty subset of R. If S is bounded above and M ∈ R, then show that M = sup S if and only if M is an upper bound of S and for every ϵ > 0, there exists x ∈ S such that M − ϵ < x ≤ M . Likewise, if S is bounded below and m ∈ R, then show that m = inf S if and only if m is a lower bound of S and for every ϵ > 0, there exists x ∈ S such that m ≤ x < m + ϵ.

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