By Hans Sagan

ISBN-10: 0395170907

ISBN-13: 9780395170908

By Hans Sagan

ISBN-10: 0395170907

ISBN-13: 9780395170908

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Extra info for Advanced calculus: Of real-valued functions of a real variable and vector-valued functions of a vector variable

Example text

11. 12. 10 vi) (right) Thus the inequalities x^ 0 b) x-1 2x - 3 X —2 X — 3 ~K 2x + 3 X+ 1 < X -\- 5 \x — 1\ f) 3 < \/x^ - 2x h) ^1x2-41 -X >0 l-lx \x\ x-l ^ X+ 1 2x- 1 V^r^ - 6x > X + 2 x+ 3 (x + i)2y^2rr3 >o Xy/jx^ - 4 | _ ^ >0 x^ - 4 2. Describe the following subsets of M; [a^l A = {x G M : x^ + 4x + 13 < 0} n {x G M : 3x2 + 5 > 0} b) 3x + l >0} B = {x G R : (x + 2)(x - l)(x - 5) < 0} n {x G M : x-2 c) C = {xe d) L) = { X G R x^ - 5x 4- 4 < 0} U {x G M : \/7x + l + X = 17} x2-9 : X - 4 > V:^2 _ g^ _^ 5} U { X G M : X + 2 > \ / ^ " ^ } 26 1 Basic notions 3.

3, bottom right) / : M -^ M, fix)= M{x) = X - [x] (the property of the floor function implies 0 < M{x) < 1). Let us give some examples of sequences now. ix) The sequence a„ = ^ is defined for all n > 0. 6, Its graph is shown in Fig. 4 (top left). 75 . x) The sequence an={l + iy is defined for n > 1. 37037, a^ =--= 4 27 256 Fig. 4 (top right) shows the graph of such sequence. 44140625. 4. 9). The graph of this sequence is shown in Fig. 4 (bottom left); as the values of the sequence grow rapidly as n increases, we used different scalings on the coordinate axes.

Thus for —y < x < 17, squaring yields 7x + 1 = (17 - xf , x^ - 41x + 288 = 0. The latter equation has two solutions xi = 9, X2 = 32 (which fails the constraint X < 17, and as such cannot be considered). The second set then contains only X = 9. Therefore C = (-3,1) U (3,4) U {9}. d) D=[1,+(X)). 3. Subsets of R^a) The condition holds if x and y have equal signs, thus in the first and third quadrants including the axes (Fig. 13, left). b) See Fig. 13, right. c) We have y — x'^ if 2/ > ^^ , 1 ^ - ^ 1= 1 2 -r / 2 y It y < x^ .