Calculus Without Derivatives (Graduate Texts in Mathematics, by Jean-Paul Penot

By Jean-Paul Penot

Calculus with no Derivatives expounds the rules and up to date advances in nonsmooth research, a robust compound of mathematical instruments that obviates the standard smoothness assumptions. This textbook additionally presents major instruments and techniques in the direction of functions, specifically optimization difficulties. while so much books in this topic concentrate on a selected concept, this article takes a normal method together with all major theories.

In order to be self-contained, the publication contains 3 chapters of initial fabric, each one of which might be used as an self sufficient direction if wanted. the 1st bankruptcy bargains with metric homes, variational rules, reduce rules, tools of mistakes bounds, calmness and metric regularity. the second offers the classical instruments of differential calculus and features a part concerning the calculus of adaptations. The 3rd encompasses a transparent exposition of convex research.

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44 CHAPTER 1. 1 a. True. For example, f (x) = x2 is such a function. b. False. For example, cos(π/2 + π/2) = cos(π) = −1 = cos(π/2) + cos(π/2) = 0 + 0 = 0. c. False. Consider f (1 + 1) = f (2) = 2m + b = f (1) + f (1) = (m + b) + (m + b) = 2m + 2b. ) d. True. f (f (x)) = f (1 − x) = 1 − (1 − x) = x. e. False. This set is the union of the disjoint intervals (−∞, −7) and (1, ∞). f. False. For example, if x = y = 10, then log10 xy = log10 100 = 2, but log10 10 · log10 10 = 1 · 1 = 1. g. True. sin−1 (sin(2π)) = sin−1 (0) = 0.

42 Θ (Using the figure from the previous solution) Note that c sec θ = = csc(π/2 − θ). b c b Thus, csc−1 (sec θ) = csc−1 (csc(π/2 − θ)) = π/2 − θ. 43 Let θ = sin−1 (x). Then sin θ = x and note that then sin(−θ) = − sin θ = −x, so −θ = sin−1 (−x). Then sin−1 (x) + sin−1 (−x) = θ + −θ = 0. 44 Using the hint, we have sin(2 cos−1 (x)) = 2 sin(cos−1 (x)) cos(cos−1 (x)) = 2x 1 − x2 . 45 Using the hint, we have cos(2 sin−1 (x)) = cos2 (sin−1 (x))−sin2 (sin−1 (x)) = ( 1 − x2 )2 −x2 = 1−2x2 . 46 Let N be the north pole, and C the center of the given circle, and consider the angle CN P .

Copyright c 2011 Pearson Education, Inc. Publishing as Addison-Wesley. 38 CHAPTER 1. 52 sec−1 (−1) = sec−1 1 1/2 1 −1 −1/2 √ 3/2 = 2π/3, since sin(2π/3) = = π/3, since sec(π/3) = = π, since sec π = 1 cos π 1 cos(π/3) = 1 −1 = √ 3/2 and cos(2π/3) = −1/2. 1 1/2 = 2. = −1. 53 tan−1 (tan(π/4)) = tan−1 (1) = π/4. 54 tan−1 (tan(3π/4)) = tan−1 (−1) = −π/4. 55 Let csc−1 (sec 2) = z. Then csc z = sec 2, so sin z = cos 2. Now by applying the result of problem 44, we see that z = sin−1 (cos 2) = π/2 − 2 = π−4 2 .

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