By Jean-Baptiste Hiriart-Urruty
This e-book features a number of workouts (called “tapas”) at undergraduate point, quite often 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 alternative mathematical topics which will be solved. the writer presents a few tricks and (partial) solutions and in addition places those rigorously selected workouts into context, provides info on their origins, and reviews on attainable extensions. With stars marking the degrees of hassle, those tapas exhibit or end up whatever attention-grabbing, problem the reader to unravel and examine, and should have astounding results.
This first quantity of Mathematical Tapas will attract mathematicians, prompted undergraduate scholars from science-based parts, and people quite often drawn to mathematics.
Read or Download Mathematical Tapas: Volume 1 (for Undergraduates) PDF
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Additional info for Mathematical Tapas: Volume 1 (for Undergraduates)
Example text
121. When the theorem of composition of differentiable functions (chain rule) does not apply Let f : R → R be defined as follows: f (x) = arcsin 2x 1 + x2 . This function looks nice, it results from the composition of two “very regular” func2x tions: x ∈ R → 1+x 2 with y ∈ [−1, 1] → arcsin(y). 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.
Concave functions and subadditivity Let f : [0, +∞) → [0, +∞) be a concave function. 1◦ ) Give examples of such functions. 2◦ ) Show that they are subadditive, that is to say: f (x + y) 136. f (x) + f (y) for all x, y in [0, +∞) . Uniform limit of convex functions Let fn : R → R be a convex function. We suppose that the sequence (fn ) converges pointwise towards a function f : R → R. 1◦ ) Show that f is convex. 2◦ ) (a) Prove that the convergence of (fn ) towards f is uniform on any interval [a, b] .
An and suppose that f (ai ) = 0 for all i = 1, 2, ... n. Show that for any x lying in the interval [a1 , an ], there exists a θ ∈ (a1 , an ) such that f (x) = (x − a1 )(x − a2 )... (x − an ) (n) f (θ). n! 34 133. Locally averaging a convex function Let f : R → R be a convex function. For h > 0, we define a new function fh : R → R as follows: x+h 1 fh (x) = f (t) dt, x ∈ R. 2h x−h Prove that fh is convex. 134. An involution operation on convex functions of the real variable Let f : (0, +∞) → R be a convex function.