Matemáticas I: C (1 point) Given the function f (x; y) = xy p 1 + x2 + y 2

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1 Matemáticas I: C -. ( oint) Given the function f (x; y) xy + x + y a) Calculate the directional derivative in the oint (; ) and in the direction of the vector v ;. b) In what direction is f increasing most raidly at (; )? Calculate a unitary vector in this direction. olution: The gradient of T in the oint (x; y) is y + x + y xy x (x; y) +x +y + x + y y + x + y x y y + y x + x + y xy y (x; y) +x +y + x + y x + x + y xy x + x rt (x; y) y + y i + x + x j In the oint (; ) we have (; ) () (; ) () Then the directional derivative is (;) f (; ) ; ; b) The direction in which we have to move the object is rf (; ) ; A unitary vector in this direction is v ; ; r ; + ; 9. (. oints) The equation x + xy sin y + x 6 de nes y (x) as an imlicit function of (x; y) in a neighborhood of the oint (; ), in which y (). a) Calculate dy b) Find the tangent straight line in (; ). c) Calculate d y. olution: Let f be the function f (x; y) x + xy sin y + x 6

2 The the gradient of f is rf (x; y) ; a) b) The tangent straight line is y + x + i + (x cos y sin y) j dy y + x + x cos y sin y dy (; ) + + cos sin y y (x ) y (x ) (x x ) y (x ) y (x ) c) d y d y + x + x sin y dy + x (x +) (y+x x sin y + x (x sin y) + y + x + cos (y) dy (x sin y) sin y) + y + x + (x sin y) cos (y) (y+x +) x sin y. (.) The roduction function of Cobb-ouglas for a certain manufacturer is given by f (x; y) x y being x the units of work ( euros each one) and y the units of caital ( euros each one). Find the maximum ermissible level of roduction for this manufacturer, if the maximum amount of money to send in the cost of work and caital is euros, i.e., olution: The Lagrangian function is x + y L (x; y; ) x y + (x + y ) Then the artial derivatives of L x y + (x + y ) L x y + (x + y ) L y L x + y + 7x y + x y

3 Obtaining the system and solving from the rst and second equation + 7x y + x y x + y therefore 7 x y x y 7 x y x y x y x y y x y x Finally, from the last equation, we obtain x + x and Therefore, the maximum level of roduction is x x u:o:w: y u:o:c: f (; ) () () 679 u:o::. ( oint) Prove that the force eld F y + yz i + sin z + xy + xz j + y cos z + xyz k is conservative and nd the otential function of F such that equals zero in A (8; ; ). Calculate the work by two di erent forms when we move the oint of alication of F along the helix when t [; ]. x (t) 8 cos (t) y (t) sin (t) z (t) t olution: First of all we have that the vector eld is conservative V i j @ y + yz sin z + xy + xz y cos z + xyz (; ; )

4 Then integral is I V dr U (8; ; ) U (8; ; ) ( + + ) The ath goes from oint (8; ; ) to oint (8; ; ). Calculating the y + yz U y + yz xy + sin z + xy + xz U sin z + xy + xz dy y sin z + xy + y cos z + xyz U y cos z + xyz dz y sin z + xyz therefore Then if, U (8; ; ), then U (x; y; z) y sin z + xy + xyz + C + C C.(.) Calculate the mass of the surface x + y z density function is (x; y; z) z limited above by the lane z 6 if the olution: In this case: Then M () z x + y + q (x; y; z) ds z + zx + zydy x + y + + x + y dy + + dd + + dd + dd d 76 6 d 6. ( oints) Calculate the work T done by a article under the in uence of the force eld F y i + y j + xz k along the closed curve C limited by (x ) + y 6 x + z 8 such that it moves from oint (8; ; ) to oint (; ; 8) in the rst octant.

5 olution: The force eld verify the hyothesis of toke s Theorem in all R. Therefore F I ds T rot F dr C being any surface with two sides limited by C, and being C and orientated by Maxwell s Rule. We will take as the ortion of x + z 8 limited by (x ) + y 6. In order to run C in the roer orientation, we should consider oriented such that its associated normal vector forms an acute angle with k. We are going to consider n as n, i.e., n n q z x i zy j + k + (z x ) + (z y ) In this examle therefore z 8 x and Therefore ince F rot i @ y y xz i z j k n q + ( ) + () ds z x z y i + k q + ( ) + () dy (x y (x z) i + (y) + (y) then T F dr rot ds F rot n ds dy z j k i + k ds Changing to olar coordinates x + cos () y sin () then T dy 6d d 8 d 8d 6 If we calculate the work using the line integral I I T F dr y + y dy + xzdz C C

6 we need to obtain the arametric equations of C. x (t) + cos (t) y (t) sin (t) z (t) 8 x (t) 6 cos t for t [; ]. Therefore sin (t) dt dy cos (t) dt dz sin (t) dt and T 6 sin (t) sin (t) + ( sin (t)) cos (t) + ( + cos (t)) (6 cos t) sin (t) dt 8 sin t + 6 cos t sin t 6 sin t + 6 cos t sin t 6 cos t sin t dt 6 7. (. oints) Calculate the ow of the vector eld i j F (x; y; z) x + arg cosh (y + z) + 7 y z + e sin(x) + k z through the ellisoid x + y + z olution: By Ostrogradski F F ds div dydz V x + y + z dydz 6 6 V cos sin x cos sin y sin sin z cos + sin sin + cos cos sin + sin sin + cos sin d sin d d d 6 sin d d d d d 6