Autoevaluaciones Calculo II, Ejercicios de Cálculo para Ingenierios

¡Descarga Autoevaluaciones Calculo II y más Ejercicios en PDF de Cálculo para Ingenierios solo en Docsity! CALCULUS II SELF-EVALUATION 1 - SOLUTIONS from the 20st to the 27th of February of 2020 Degree in Mobile and Space Communications Engineering. Time: 1 hour The mark obtained here is not part of the final mark of the course. The only purpose is to check the progress of the student. Problem 1 (2 points) a) Find the domain of f(x, y) = 1 log(x2 + y2 − 4)1/2 . b) Plot the domain and study if it is open, closed, bounded and compact and obtain its boundary, its closure and its interior. Solution: a) The argument of the square root must be positive or zero, and the argument of the logarithm strictly positive, also the logarithm cannot be zero, so: Dom(f) = {(x, y) : x2 + y2 − 4 > 0} \ {(x, y) : x2 + y2 − 4 = 1} = {(x, y) : x2 + y2 > 4, x2 + y2 6= 5} (outside of the circle of center 0 and radius 2 but for the circumference of radius √ 5). b) The domain is an open set, because it contains no part of its boundary. Then it is not compact. Also it is not bounded. Besides: ∂Dom(f) = {x2 + y2 = 4} ⋃ {x2 + y2 = 5}, (Dom(f))o = Dom(f), Dom(f) = {(x, y) : x2 + y2 ≥ 4}. Problem 2 (1 + 1 + 0.5 = 2.5 points) We consider the function g(x, y) =  x2 + y3√ x2 + y2 if (x, y) 6= (0, 0) 0 if (x, y) = (0, 0) . a) Study the continuity of g in R2. b) Compute the partial derivatives of g(x, y) at (0, 0) if it is possible. c) Can g be differentiable at (0, 0)? Solution: i) g is continuous at R2 \ {(0, 0)} directly, and at the origin: lim (x,y)→(0,0) x2 + y3√ x2 + y2 = 0 = g(0, 0), 1