Continued Fractions - Analytic Theory and Applns by W. Jones, W. Thron

By W. Jones, W. Thron

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5 = 0 . 5 . Ay2-Ay- 9x2 -\2x-A 2 = 0. 2 6 · 9x - bx - Ay - Ay - 1 = 0. 7 · x2 — 2x = y2 + 2j>. 8· X2 + 2X=J 2 -2J. Write the equations of the following hyperbolas: 9 · Asymptotes y-3 = ±2(x + 1), through (0, 3). 10 · Asymptotes y + 2 = ± 3 ( * — l), through (1, 1). 11 · Asymptotes y - 3 = ±2(x + 1), through ( - 1 , 5). 12 · Asymptotes y + 2 - ± 3 0 - 1), through (2, - 2 ) . 13 · Center at the origin with directrix x = A and e = 2. 14 · Center at the origin with directrix x = — A and e = 2.

Out comes 8. Put in any number greater than 1, less than or equal to 2. Out comes 16. Put in any number greater than the integer n but less than or equal to n + 1. Out comes 8(/z -f 1). This machine would be useful in a post office. Call it M. | As in the preceding example, we often wish to talk about all the numbers x greater than a, less than or equal to b. We shall express this by writing, " x is a member of the interval (a, 6]," or more concisely yet, x e (a, b]. The left-hand parenthesis means a is not included; the right-hand square bracket means b is included.

See C. P. Nicholas, "A dilemma in definition," Amer. Math. Monthly 73 (1966), pp. ) With no desire to do battle in this arena, we have merely chosen a definition that lends itself to the spirit of our presentation. PROBLEMS A · Let H be the function H(x) = 3 + >/4 — x2. ) (a) What is the domain of ΗΊ (b) What is the range of //? B · Consider the set S of all pairs (x, y) for which (y — 2)2 = 4(x + 3). (a) Explain why no function G can be found so that the set S above will be the graph of (x, G(x)).

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