BSc Sem V 2015 2016 2016 Math 1 Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 All questions are compulsory
2. Q2 From Question 1 , 2, and 3 Attempt any one from part (a) and two from part (b). Be,
3. Q3 From Question 4. Attempt any three. &
4. Q4 Figures to the right indicate marks State and prove First Fundamental Theorem of calculus
   • (ii) If f is Riemann integrable on [a,b] then show that f is Riemann \ ) integrable on [a,c] and [c,b] and further J f= J
   • (b) State and prove Mean Value Theorem for integrals. 6
   • (ii) Prove that if f :[a,b] > is Riemann integrable then | f| is Riemang Is the converse true? Justify 6
   • (iii) Express the sum as a Riemann Sum of a suitable function and 6
   • (iv) Let f: R defined by f(x) = x Let of partitions, given by U( and show that 6
5. Q2 @ Define double integral ofa f:EOR where is a rectangle in .Further show notations 8 marks
   • (ii) State & prove Fubini’ for a rectangular domain Use suitable variables to show that = log where is gion in the first quadrant bounded by the curve 8
   • (ii) Using s ‘cal co-ordinate, find the volume of the solid S bounded by the sphere 6
   • (iii) SketsA the region and evaluate the integral by reversing the order of integration (x+ y+ z)dxdydz where S is the parallelepiped bounded by the (6) planes & xty+z=2 , & x-y+z=3 and x-y-z=3 & 6
   • (a) @ Let { f,} be a sequence of continuous real valued functions defined on a non- Sot If converges a function f on S then show that f is continuous on Further show that Gi) Let {f,} be a sequence of Riemann integrable function on [a,b].If the series (8) f, converges uniformly to f on [a,b] then show that f is Riemann integrable Define radius of convergence of a power series. If power series radius of convergence r > 0 then show that it converges s| for 8
   • (i) Show that the series converges uniformly on . uniformly convergent on [0,1] ? 6
   • (iv) Discuss the pointwise and of { f,} where on case { does uniformly on , check whether it converge uniformly on or where a> 0 be F(x)= f (x)dx, x Discuss (i) continuity of f of F when exists & check whether Riemann’s criterion for integrability of a bounded function defined on [a,b] (5) and use it to prove that the function f is Riemann integrable Evaluate xyzdxdydz , E is the region bounded by x=0, y=0, z=0 & (5) 6
   • (d) Find the volume bounded by the cyli = & ounded by the cylinders x, and the planes (5) iscuss the pointwise & uniform convergence of the series on [0,1]. O Let f, :[0,1] > be defined by f,(x)= for x the limit f of { f,} that | = Does {fn} uniformly to f on [0,1] ? Justify your answer

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