 By Jean-Baptiste Hiriart-Urruty

This publication includes a selection of routines (called “tapas”) at undergraduate point, in most cases from the fields of actual research, calculus, matrices, convexity, and optimization.
Most of the issues awarded listed below are non-standard and a few require large wisdom of other mathematical topics on the way to be solved. the writer presents a few tricks and (partial) solutions and likewise places those conscientiously selected workouts into context, provides details on their origins, and reviews on attainable extensions. With stars marking the degrees of trouble, those tapas exhibit or end up anything fascinating, problem the reader to unravel and study, and should have unbelievable results.

This first quantity of Mathematical Tapas will entice mathematicians, stimulated undergraduate scholars from science-based parts, and people as a rule drawn to mathematics.

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Mathematical Tapas: Volume 1 (for Undergraduates)

This ebook incorporates a choice of routines (called “tapas”) at undergraduate point, in general from the fields of genuine research, calculus, matrices, convexity, and optimization. many of the difficulties awarded listed here are non-standard and a few require huge wisdom of alternative mathematical topics on the way to be solved.

Extra info for Mathematical Tapas: Volume 1 (for Undergraduates)

Example text

Symmetric matrices whose k-th power is the identity Let A ∈ Mn (R) be symmetric. We suppose that there exists a positive integer k such that Ak = In . 1◦ ) Show that, necessarily, A2 = In . 2◦ ) Show that if k is odd then A = In . 167. Bounding the square roots of In For a positive integer n, let Xn = A ∈ Mn (R) A2 = In . Is the set Xn closed? Is it bounded ? 168. Commuting with all the permutation matrices A matrix Mσ = [mij ] ∈ Mn (R) is called a permutation matrix when mij = 1 if σ(j) = i, mij = 0 if σ(j) = i, 43 where σ : {1, 2, ...

Question: is f diﬀerentiable on R? 31 The limit of an integral 122. Let f : [0, 1] → R be a continuous function. We then deﬁne: 1 For all x > 0, F (x) = 0 x f (t) dt. x2 + t 2 Prove that F (x) has a limit as x → 0 and determine this limit. The intermediate value property for derivatives 123. Let I be an open interval of R and let f : I → R be a diﬀerentiable function. We note A = {(x, y) ∈ I × I ; x < y} . 1◦ ) Check that A is a convex subset of R2 . 2◦ ) For (x, y) ∈ A, deﬁne g(x, y) = f (y)−f (x) .

2◦ ) Let E be the ellipse in the plane whose Cartesian equation is x2 y 2 + 2 = 1, a2 b with a b > 0. The center of the ellipse is the origin denoted by O. Consider two points M and N on the ellipse such that the triangle OM N is rightangled, the right angle being at O; in other words, the two radiuses OM and ON of the ellipse are perpendicular. Prove that the line M N is tangent to a circle centered at O and whose radius is to be determined. 155. A tetrahedron with faces of equal areas Let us consider a tetrahedron whose four faces have the same area.