BSc Mathematics SEM V 2018 19 2018-19 Maths I Real Analysis & Multivariable Calculus Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 From questions 1, 2 and 3 attempt any One from part (a) and any Two from part (b)
2. Q3 Attempt any Three from question 4 marks
3. Q4 Figures to the right indicate marks
4. Q1 (a) i) Let f:[a,b] — Rbe abounded function. If P, Q are partitions of Prove that 8 marks
   • ii) Let f,g:[a,b] — R be Riemann integrable on [a,b]. Prove that f + g is Riemann 8 integrable on [a, b] and hence show that +g) = f g
   • (b) i) Let f: [a,b] — R be acontinuous function on Show that f is Riemann integrable 6
   • i) if0<x<+ 6 — Ris defined by = , x [0,1]. Show that F is differentiable
   • iii) Let f:[0,1] R be defined by f(x) = x*. Using Riemann criterion show that f is 6 Riemann integrable on [0, 1]
   • iv) Express the sum sum of a suitable function and evaluate 6
5. Q2 (a) i) State and Prove Fubini’s theorem for a rectangular domain in 8 marks
   • ii) Define double integral of a bounded function f:Q — R where Q = [c,d] is a rectangle in IR*. show with usual notations 8
   • (b) i) State the change of variable formula for triple integrals. Stating clearly the conditions 6 under which it is valid. Express further, how will you use it to express the triple integral in Spherical coordinates
   • i) Evaluate f x dxdy by reversing the order of integration. Sketch the region of 6
   • iii) Introduce suitable change of variables and show that 6 Sf, f@y)dxdy f where S is the region in the 1“ quadrant bounded by
   • iv) the volume of the solid S by using triple integration where S is bounded by the 6 paraboloid z = + y? and the plane z = 2
   • Q.P.Code:21594 3 (a) i) be a sequence of real valued R-integrable functions on If converges. 8 uniformly to f on [a,b] then show that f is R-integrable on and
   • ii) Let be a sequence of continuously differentiable real valued functions defined on 8 [a, b]. If the series converges pointwise to f on [a,b] and the series converges uniformly on then show that =
   • (b) (i) Let be a sequence of real values functions defined on a non-empty 6 subset S of R. Show that } converges uniformly function f if and only if for positive integer n, such that | fin(x)| < = and
   • (ii) Discuss the pointwise and uniform convergence of the series of functions , 6
   • (iii) By a_ suitable power series. term by term show that 6 f,:[0, 1] R be defined by f,(x) = Check whether = Does > f uniformly? 4 i) Prove that if f:[a,b] IR is Riemann integrable then is Riemann integrable on [a, b]. 5 Is converse true? Justify
   • ii) R are Riemann integrable and have antiderivatives F and G on 5 then show that F(x)g(x) =
   • iii) Evaluate the following integral by using polar coordinates 5 Use spherical coordinates evaluate fff 5 dxdydz where S is the solid that lies between the spheres + y? + =1andx*+ y*+
   • v) areal power series has radius of convergence r, then show that it converges 5
   • vi) Let = < x < 1. Discuss the pointwise and uniform convergence of 5

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