By Gabriel Klambauer (auth.)

This e-book is meant for college students accustomed to a beginner's model of differential and crucial calculus stressing in simple terms manipulation offormulas and who're now searching for a more in-depth research of uncomplicated strategies mixed with a extra inventive use of knowledge. The paintings is essentially aimed toward scholars in arithmetic, engineering, and technological know-how who locate themselves in transition from trouble-free calculus to rigorous classes in research. additionally, this ebook can also be of curiosity to these getting ready to coach a path in calculus. rather than exposing the reader to an far more than untimely abstractions that so simply can degenerate into pedantry, I felt it extra worthy to emphasize instruc tive and stimulating examples. The e-book includes a number of labored out examples and lots of of the workouts are supplied with valuable tricks or an answer in define. For additional workouts the reader should want to seek advice an issue e-book through the writer entitled difficulties and Propositions in research (New York: Marcel Dekker, 1979). For the historical past of calculus i like to recommend the e-book by means of C. B. Boyer, The innovations of the Calculus (New York: Dover, 1949).

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**Example text**

Tn we have (C1t 1 + C2t2 + ... + cntn)2 :::; (c 1 + C2 + ... + cn)(c1ti + C2t~ + ... + cnt;). 45). yfc;. 45) 1,2, ... 43), we 39 5. Miscellaneous Examples Putting REMARK. 45) yields (ttl + tt2 + ~t3)2 ~ ttf + tt~ + ~t~ for any real numbers t 1, t 2 , and t3. 10. We have the identity a + b(l + a) + c(l + a)(l + b) + d(l + a)(1 + b)(1 + c) + ... 46) = (1 + a)(1 + b)(1 + c)· ··(1 + q) - 1. Indeed, adding 1 to the left side, we can write [(1 + a) + b(I + a)] + c(I + a)(l + b) + d(I + a)(1 + b)(1 + c) + ...

The Exponential Function Since all terms in the last sum are positive, we have b- 1 O

Let b (delta) be larger than zero; then Ix - al < b means that a - b < x < a + band Ixl < b means that -b < x < b for in the latter case a = 0. Let a < b; then the inequality a < x < b can be expressed in the form Ix - AI < B, where A = (a + b)/2 and B = la - bl/2 (here A is the midpoint between the points a and band B is half the distance between the points a and b). If we know the curve y = f(x), then the curve y = If(x) I is easy to picture. To obtain the curve y = If(x) I from the curve y = f(x), we leave unchanged that portion of the curve which is above the x-axis, but reflect through the x-axis the portion which is below the x-axis.