# Schaum's Outline of Advanced Calculus, Second Edition by Robert C. Wrede

By Robert C. Wrede

This best-selling guide--which has bought greater than 340,000 copies on the grounds that its first publication--has been completely up to date all through to correspons to present complex calculus classes. an entire and accomplished overview of the topic, this up to date version good points very important new chapters on topology and Laplace transforms and crucial new theorems, with explanatory proofs.

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Example text

3. With each time t after the year 1800 we can associate a value P for the population of the United States. The correspondence between P and t deﬁnes a function, say F, and we can write P ¼ FðtÞ. 4. For the present, both the domain and the range of a function have been restricted to sets of real numbers. Eventually this limitation will be removed. To get the ﬂavor for greater generality, think of a map of the world on a globe with circles of latitude and longitude as coordinate curves. Assume there is a rule that corresponds this domain to a range that is a region of a plane endowed with a rectangular Cartesian coordinate system.

This deviation from precise notation will appear in the text because of its value in exhibiting the ideas. The domain variable x is called the independent variable. The variable y representing the corresponding set of values in the range, is the dependent variable. Note: There is nothing exclusive about the use of x, y, and f to represent domain, range, and function. Many other letters will be employed. There are many ways to relate the elements of two sets. ] For example, given the equation y2 ¼ x, there are two choices of y for each positive value of x.

22 NUMBERS [CHAP. 107. (a) If x is a rational number whose square is less than 2, show that x þ ð2 À x2 Þ=10 is a larger such number. (b) If x is a rational number whose square is greater than 2, ﬁnd in terms of x a smaller rational number whose square is greater than 2. 108. Illustrate pﬃﬃﬃ Dedekind pﬃﬃﬃto deﬁnepﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ you would pﬃﬃﬃ use pﬃﬃﬃcuts pﬃﬃﬃ how (a) 5 þ 3; ðbÞ 3 À 2; ðcÞ ð 3Þð 2Þ; ðdÞ 2= 3. Sequences DEFINITION OF A SEQUENCE A sequence is a set of numbers u1 ; u2 ; u3 ; . . , a correspondence with the natural numbers) and formed according to a deﬁnite rule.