Bachelor of Science (B.Sc.) Sem IV 2022 2023 Mar 2023 MATHEMATICS I Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q4 Attempt any three. [each 5 Mks fy at (0, 0) if exists for f:R? is defined as f(x, y) = if(x, y) 0)
   • 2) For following function f, find the 1) if exists satisfying
   • 3) Using chain rule find the total derivative of f(x, y, Z) =xy?+yz?+zx’, x(t)
   • 4) Find directional derivative of fat a in direction of u f(x, y, Z)
   • 5) Let as f(x, y) =(f,, g(u, w) =(uvw,
   • 6) Locate all critical points of f(x, y) VCD/ SYBSC- SEM IV - MATHEMATICS I- 75MARKS NOTE : 1)For Q.1, Q.2 and Q. 3 attempt any one subquestion (each 8 marks) from part (a), and any two subquestions (each 6marks) from part (b).For Q.4 , attempt any three. (each 5
2. Q1 (a) Attempt any one. [each )Define a continuity of a vector valued function f:S subset of that for nonempty subset S of R", continuous at acS then f-g is continuous at
   • 2) Prove that sequence in R? converges to a limit w=(s,t) = R? iff (S,)
   • (b) Attempt any two. [each norm of x where = R" and prove that xy &R"
   • 2)Show that an open set
   • 3)Define f:R? +R defined by +y cos 1 for otherwise Find Lim _ f(x, y)if exists
3. Q2 (a) Attempt any one. {each 8Mks] and prove Euler's theorem for function of two variable
   • 2)Define a Differentiability of a scalar valued function f:S # @ subset of at point Prove that for nonempty subset S of be differentiable then Of(a) exists for
   • (b) Attempt any two. [each 6Mks] total derivative of f using definition at the mentioned point
   • 2)Let as f(x, y) =3sin x +ycos x. Find
   • 3)Let be non constant differentiable function, R, f(x, y) =k describes the curve C having tangent at each of ots points then prove that i)gradient vector V f is normal to Cii) the directional derivative of fis zero along
4. Q3 (a) one. [each 8Mks] open subset of a& S& bea scalar field. Let f be differentiable at a If a local maximum or local minimum at a then prove that V f(a) =0

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