A Course in Calculus and Real Analysis
Springer Science & Business Media, Jun 5, 2006 - Mathematics - 432 pages
Calculus is one of the triumphs of the human mind. It emerged from inv- tigations into such basic questions as ?nding areas, lengths and volumes. In the third century B. C. , Archimedes determined the area under the arc of a parabola. In the early seventeenth century, Fermat and Descartes studied the problem of ?nding tangents to curves. But the subject really came to life in the hands of Newton and Leibniz in the late seventeenth century. In part- ular, they showed that the geometric problems of ?nding the areas of planar regions and of ?nding the tangents to plane curves are intimately related to one another. In subsequent decades, the subject developed further through the work of several mathematicians, most notably Euler, Cauchy, Riemann, and Weierstrass. Today,calculus occupies a centralplacein mathematics and is an essential component of undergraduate education. It has an immense number of app- cations both within and outside mathematics. Judged by the sheer variety of the concepts and results it has generated, calculus can be rightly viewed as a fountainhead of ideas and disciplines in mathematics. Real analysis, often called mathematical analysis or simply analysis, may be regarded as a formidable counterpart of calculus. It is a subject where one revisits notionsencountered in calculus, but with greaterrigor and sometimes with greater generality. Nonetheless, the basic objects of study remain the same, namely, real-valued functions of one or several real variables. This book attempts to give a self-contained and rigorous introduction to calculusoffunctionsofonevariable.
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absolutely convergent algebraic antiderivative arctan assume Bolzano–Weierstrass Theorem bounded function calculus Cauchy Cauchy sequence centroid Chapter concave Consider the function continuous function Corollary curve given decreasing defined by f(x deﬁnition denote derivative differentiable function endpoints equal equation example Exercise exists f a,b f(t)dt is convergent f(xn fc=i ﬁnd ﬁnding ﬁrst ﬁxed point function f geometric graph Hence Hint improper integral indeterminate forms inequality inﬁnite series integrable functions interior point interval L'Hopital's Rule Lemma Let f limit line segment monotonically increasing Newton sequence nonnegative Note obtain partition point of inﬂection polynomial function positive real numbers Proof properties Proposition radius rational number result revolving Riemann condition Riemann integral sinx solid strictly increasing subinterval subset Suppose tangent Taylor series Taylor's Theorem Test Theorem uniformly continuous unique x-axis zero