Matemáticas 01 2018, Exámenes de Matemáticas

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8/1/2018 Name and surname(s): Mathematics |. Final exam ALGEBRA == E. » 5 Exercise 1. Check that (U, = (1,2, 1), 2 = (2, 9,9), ús = (3,3, 4)) is a basis of R* and compute the coordinates of u = (1, 1,0) with respect to the basis. Exercise 2. Find a basis and the dimension of the following vector subspace: S=((0,4y,2) €R:2—3y+2=0). Exercise 3. Study for what value (or values) of k is the distance between (2,4, —2) and (3,2 k — 1) equal to k - 4, ñ Exercise 4. Study for what value (or values) of a is (ú = (lb 4, 3), U2 a (22, a, a), U3= (4, 7, 1)) an orthogonal basis of 23 CALCULUS Ñ Exercise 5. Describe the topological properties of the domain of Í(z,y) = In(4— (2 - 1)? — (y + 1)). Can we apply the Extreme Value Theorem? Explain the consequences. Exercise 6. Find the equation of the tangent plane of Huy) = Y +2 Er al p = (1, —1). Use it to approximate the output of the function at (1, -1'1) á Exercise 7. The equation -? +27 + 07r+24+2 ayy defines 2. of point p = (0, 0,0). Find the gradient of z at p, VA Exercise g, l Classify the stationary points of Ocal extreme points (maximum or minimum), ;