By Sudhir R. Ghorpade, Balmohan V. Limaye
This ebook presents a self-contained and rigorous advent to calculus of services of 1 variable. The presentation and sequencing of themes emphasizes the structural improvement of calculus. even as, due value is given to computational strategies and purposes. The authors have strived to make a contrast among the intrinsic definition of a geometrical proposal and its analytic characterization. during the e-book, the authors spotlight the truth that calculus offers an organization beginning to a number of strategies and effects which are in general encountered in highschool and accredited on religion. for instance, you'll find the following an evidence of the classical consequence that the ratio of the circumference of a circle to its diameter is similar for all circles. additionally, this booklet is helping scholars get a transparent figuring out of the concept that of an perspective and the definitions of the logarithmic, exponential and trigonometric services including an evidence of the truth that those will not be algebraic features. a couple of issues that could were inadequately lined in calculus classes and glossed over in genuine research classes are handled the following in enormous aspect. As such, this e-book presents a unified exposition of calculus and actual analysis.
The simply necessities for analyzing this booklet are issues which are usually coated in highschool; even though, the reader is anticipated to own a few mathematical adulthood and a capability to appreciate and have fun with proofs. This ebook can be utilized as a textbook for a major undergraduate direction in calculus, whereas components of the ebook can be utilized for complicated 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 gains of the exposition, comparable advancements and references to suitable literature.
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Extra info for A Course in Calculus and Real Analysis (Undergraduate Texts in Mathematics)
Given any ℓ(x), p(x) ∈ R[x] with ℓ(x) ̸= 0, use induction on deg ℓ(x) to prove that there are unique polynomials q(x) and r(x) in R[x] such that p(x) = q(x)ℓ(x) + r(x), and either r(x) = 0 or deg r(x) < deg ℓ(x). 59. Given any p(x) ∈ R[x] and α ∈ R, show that there is a unique polynomial q(x) ∈ R[x] such that p(x) = (x − α)q(x) + p(α). Deduce that α is a root of p(x) if and only if the polynomial (x − α) divides p(x). 60. Show that a nonzero polynomial in R[x] of degree n has at most n roots in R.
Ii) If f and g have a local maximum at x = c, then so does f g. What if f (x) ≥ 0 and g(x) ≥ 0 for all x ∈ R? (iii) If c is a point of inflection for f as well as for g, then it is a point of inflection for f + g. (iv) If c is a point of inflection for f as well as for g, then it is a point of inflection for f g. Part B 37. Given any ℓ, m ∈ Z with ℓ ̸= 0, prove that there are unique integers q and r such that m = ℓq + r and 0 ≤ r < |ℓ|. ) 38. Given any integers m and n, not both zero, a positive integer d satisfying (i) d | m and d | n and (ii) e ∈ Z, e | m and e | n =⇒ e | d Exercises 37 is called a greatest common divisor, or simply a GCD, of m and n.
Let D be a bounded subset of R and f : D → R be a polynomial function. Prove that f is bounded on D. 27. Let I be an interval and f : I → R be a monotonically increasing function. Given any r ∈ R, show that rf : I → R is a monotonically increasing function if r ≥ 0 and a monotonically decreasing function if r < 0. (iii) D = (−1, 1] and f (x) = x2 − 2x − 3, (iv) D = R and f (x) = 36 1 Numbers and Functions 28. Let I be an interval and f : I → R be a convex function. Given any r ∈ R, show that rf : I → R is a convex function if r ≥ 0 and a concave function if r < 0.
A Course in Calculus and Real Analysis (Undergraduate Texts in Mathematics) by Sudhir R. Ghorpade, Balmohan V. Limaye