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Pune University - SY BSc (Sem - I) MATHEMATICS - I, April 2010

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Total No. of Questions : 4] P258 [Total No. of Pages : 2 [3717] - 101 S.Y. B.Sc. (Sem. - I) MATHEMATICS - I (51111) Calculus of Several Variables (New Course) Time : 2 Hours] [Max. Marks : 40 Instructions to the candidates : 1) All questions are compulsory. 2) Figures to the right indicate full marks. Q1) Answer the following questions. lim [10] x 2 y2 , if exists. x4 + y4 x2 y2 a) Evaluate b) If the function f(x, y) = log (x2 + y2 2 x 2 y + 6) is continuous at the point (2, 1), find f(2, 1). c) State sufficient conditions for f(x, y) to be differentiable at (a, b). d) Find x e) If f(x, y)= | x | + | y |, show that fx(0, 0) doesnot exist. f) If f(x, y) = log (x + y) and x = t2, y = 1 t2, find g) Find Jacobian J = h) State Maclaurin's theorem for function of two variables. i) Show that (1, 1) is a stationary point of the function f(x, y) = xy + ( x , y ) ( 0,0 ) f f + y , where f(x, y) = sin 1 x y ( x, y) ( u, v ) x y x + y . df . dt , where x = u uv, y = uv. 11 +. xy 1x j) Evaluate e yx d y d x. 00 P.T.O. Q2) Attempt any two of the following: a) [10] Let f(x, y) be a function defined on a neighbourhood of a point (a, b). If f is differentiable at (a, b) then prove that i) ii) b) f is continuous at (a, b) and fx and fy both exist at (a, b). If f(x, y) = xy x 2 y2 ; (x, y) (0, 0) and f(0, 0) = 0. Show that x2 + y2 fxy(0, 0) fyx(0, 0). c) x 4 + y4 If e = , using Euler's theorem for homogeneous function show x y u 2 that x 2 2 2 u u 2 +y = 3 . 2 + 2 xy x xy y2 u Q3) Attempt any two of the following: [10] a) Using differentials, find approximate value of ( 5 .98)2 + (8 . 0 1 2 . ) b) Using Lagrange s method of undetermined multipliers, find the minimum value of f(x, y, z) = x2 + y2 + z2 under the condition x + y + z = 30. c) Show that f(x, y) = xy is not differentiable at (0, 0). Q4) Attempt any one of the following: a) [10] i) State and prove Schwarz s theorem for equality of second order mixed partial derivatives. ii) Evaluate dx dy over the region R bounded by the curves y = x2 R and y = x. axy b) i) Evaluate xy 23 z dzdydx . 000 ii) By changing to polar co-ordinates, evaluate Y [3717] - 101 2 ( x x2 + y 2 1 2 + y 2 ) dxdy . 72

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Additional Info : S.Y. B.Sc. Mathematics (Semister - I) : Calculus of Several Variables (Paper - I) (New Course), Pune University
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