BSc Mathematics SEM V 2016 17 2016-17 Maths I Real Ana. & Multivariable Calculas (Old) Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 All questions are compulsory
2. Q2 Figures to the right indicate marks
3. Q1 (a) Attempt any ONE of the following 8 marks
   • (i) Let R be a bounded function. If P,Q are partitions of [a, b] then show that (i) L(P, f) < U(P, f) ii) L(P, f) < f)
   • (ii) If f is Riemann integrable on [a,b] and a<c<b then show that f is Riemann integrable on [a,c] and [c,b] and further f =| f +{ f
   • (b) Attempt any TWO of the following Prove that if f :[a,b] > is Riemann integrable then |f| is Riemann integrable. Is the converse true? Justify 12
   • (ii) Using Riemann Criterion, show that f: [0,3] R defined by f(x) = [x] is Riemann integrable on [0,3] where [x] is the integral part of x
   • (iii) Let f:[0,1]— defined by f(x) {Pn} be a sequence of partitions, given by Calculate U( Pn, f) and show L(P,, and hence find | f (x)dx
   • (iv) Express the following a Riemann sum of a suitable function and
4. Q2 (a) Attempt any ONE of the following 8 marks
   • (i) triple integral of a bounded function f: Q IR where Q = is a rectangular box in Further show that where m, M are the infimum and the supremum of f on Q. Also evaluate State & prove Fubini’s theorem for a rectangular domain
   • (b) Attempt any TWO of the following Use suitable change of variables to show that } f (xy)dxdy = log 2 f (u)du where S is the region in the first quadrant bounded by the curve xy = 1, 12
   • (ii) the area enclosed by one loop of four leaved rose r =
   • (iii) Find the volume of the cylinder with base as the disc of unit radius in the xy-plane centred at (1,1,0) and the top being the surface
   • (iv) Evaluate | (x+ S is the parallelepiped bounded by the
   • Q.P.Code:05709
5. Q3 (a) Attempt any ONE of the following 8 marks
   • (i) | Let {fn} be a sequence of continuous real valued functions defined ona non-empty subset S of R .If {fn} converges uniformly to a function fon S then show that f is continuous on S. Further show that lim lim = lim lim for each p S
   • (ii) Let a sequence of Riemann integrable function on [a,b].If the series f, converges uniformly to f on [a,b] then show that f is Riemann integrable on [a, b] and
   • (b) Attempt any TWO of the following 12
   • (i) Show that the sequence = does not converge uniformly on [0, but converge uniformly on [0, a] where a > 0 Show that the series converges uniformly on [-1, 4] Examine whether dx = dx . Is the series x uniformly convergent on [0,1] ? Justify
   • (iv) (iv) If areal power series has the radius of convergence r, then show that it converges uniformly in [—s,s] where 0 < s < r. Further show that if = in then f is differentiable and
6. Q4 Attempt any THREE of the following 15 marks
   • (i) Ifa function f defined on [a, b] is continuous and non-negative. If f(c) for some c [a, b]. Show that 0
   • (ii) Riemann’s criterion for integrability of a bounded function defined on [a,b] and use it to prove that the function f (x) = is Riemann
   • (iii) Consider the triple integral ” dzdydx. Rewrite the integral as an equivalent iterated integrals in five other ways
   • (iv) Find the volume bounded by the cylinders x*= y and the planes
   • (v) Let = x" for x [0,1]. Find f(x) = lim Show that = lim dx but does not converge uniformly to f on [0,1] (vl) Show that the series converge uniformly,

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