### Aide Mémoire Analyse Mathématique by Gilbert Demengel PDF

By Gilbert Demengel

ISBN-10: 2040012648

ISBN-13: 9782040012649

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Extra info for Aide Mémoire Analyse Mathématique

Sample text

13, right. c) We have y — x'^ if 2/ > ^^ , 1 ^ - ^ 1= 1 2 -r / 2 y It y < x^ . Demanding y > x^ means looking at the region in the plane bounded from below by the parabola y — x^. 13. 14. The sets C and D of Exercise 3 that \s x^ x^ — \ ^ hence x^ — 1 < y < x^. Eventually, the required region is confined by (though does not include) the parabolas y = x'^ - I and y = x'^ + 1 (Fig. 14, left). d) See Fig. 14, right. / > —-. Thus we consider all points of the first and third quadrants above the hyperbola y = —-.

Its x~^ domain coincides with the domain of ^ = x~^ minus the origin. All maps are strictly decreasing on (0, H-oo), while on (—cx), 0) the behaviour is as follows: writing a = —^ with m odd, the map is strictly increasing if n is even, strictly decreasing if n is odd, as shown in Fig. 17. In conclusion, we observe that for every a 7^ 0, the inverse function oi y = x^^ where defined, is y = x^l^. 2 Polynomial and rational functions A polynomial function, or simply, a polynomial, is a map of the form P{x) — anX^ ^ h a i x -h ao with a^ 7^ 0; ^ is the degree of the polynomial.

2), to -2sy > x — 2 and defines the open half-plane above the line y — x — 2. What we get is therefore the strip between the two lines, these excluded (Fig. 11, right). v) By Pythagoras's Theorem, the equation x^-^y^ = l defines the set of points P in the plane with distance 1 from the origin of the axes, that is, the circle centred at the origin with radius 1 (in trigonometry it goes under the name of unit circle).