BSc Mathematics SEM III 2018 2019 Oct 2019 MATHEMATICS PAPER I Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate marks for respective parts

1. Q1 Choose correct alternative in each of the following: 20 marks
   • i. The set S = {(x,y) 1 y? <2} is
   • (a) An open set (b) set
   • (c) Neither open nor closed (d) None of these Let f: R* > R defined as f(x,y) = . Then,
   • (a) f(x,y) exist. is continuous at (0,0)
   • (c) < ; (d) None of these
   • ii. Let f(x,y) = |x| + for y) then
   • (a) = 0, f,(0,0) = 0 and f,(0,0)do not exist
   • (c) = 1, f,(0,0) = 1 None of these Iv. Let A: Gradient of a scalar field is a scalar B: Every differentiable scalar field is continuous Then which of the following is true?
   • (a) A is true, B is false. (b) A is false, B is true
   • (c) Both A & B are true. Both A & B are false Let R be such that and x exist and are bounded .Then
   • (a) f may or may not be (b) continuous at all points continuous at all points
   • (c) f is differentiable at all (d) None of these Vi. The total derivative T, of a scalar field is
   • (a) a constant (b) transformation The linear approximation to e*cos(y + Z) near the origin is
   • (a) independent of x. (b) independent of y
   • (c) independent of z (d) 1 If z (x,y) is differentiable and x = g(u,v), y = are also differentiable functions then is
   • ix. y? then
   • (a) (1,0) is acritical (1,0) is acritical point of f which is a which is a local minima. local maxima
   • (c) (1,0) is a saddle point of f. (d) None of these Stationary point is a point where function f (x, y) have
   • (c) Both (a) and (b). (d) None of these
2. Q2 a) Attempt any ONE question from the following: 8 marks
   • i. Let f: R” be a vector valued function and let a IR”. Prove that f is continuous at a if and only if each f; > R is continuous at a li. State and prove mean value theorem for derivatives of scalar fields
   • b) Attempt any TWO questions from the following: 12
   • i. Using definition of limit, check f(x,y) exists, where li. Define limit of a function S — R where at point a, and show that the limit of function of several variables is uniquely determined Define directional derivative of a scalar field f(x, = (=) the point a = (1,1,1), in the direction of u = (2,1, —1) Find the directional derivative of the following functions at the indicated point in the direction If sin (x,y,z) # (0,0,0). Then prove that
3. Q3 a) Attempt any ONE question from the following: 8 marks
   • i. Let R bea function and a Define the total derivative Df (a) in terms of a linear transformation and show that Df (a) when exist, is uniquely defined li. State and prove chain rule for the derivative of a scalar field
   • b) Attempt any TWO questions from the following: 12
4. Q1 Find total derivative as linear transformation T for the function
   • ii. Find directional derivative of f(x, y,z) = 3x —5y + 2z at (2,2,1) in the direction of outward normal to the sphere + y2 +2? = 9 Find the equation of tangent plane and normal line to the surface Iv. Show that, for each of the following functions, the second order mixed partial derivatives are equal
5. Q4 a) Attempt any ONE question from the following: 8 marks
   • i. Define Df (a), the total derivative at a IR" for a function f: R” > in terms of a linear transformation. Show that if f is differentiable at a then f is continuous at a. Is the converse true ? Explain il. Let f:5 > R bea scalar field. Let (a, b) S be a stationary point of f Suppose f(x,y) possesses continuous second order partial derivatives in some neighbourhood of (a,b). Let A = B = C = fyy(a,b) and A= AC Then prove that
   • (1) if A> 0,A > 0, then local minimum at (a, b)
   • (2) if A> < 0, then fhas local maximum at (a, b)
   • (3) if A< 0, then f has saddle point at (a, b) if A= 0, the test is inconclusive (show this by example)
   • b) Attempt any TWO questions from the following: 12
   • i. Let U is open set in :U R” is differentiable ata U. Show li. Find the Taylors polynomial of degree 2 at p = (1,72) for the function Find the critical points, saddle points and local extrema if any for the Iv. Find the greatest area that a rectangle can have if the length of its diagonal is
6. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) Show that for the following functions the limit does not exists
   • b) A if xy =0 Let f: R defined by f(x,y) = 1 if xy £0 Show that f is not continuous at (0, 0) but both partial derivatives exist at (0,0) Find the direction in which function f(x,y) = 9x3 + 5y? increases most rapidly and the direction in which decreases most rapidly at point (2,1)
   • d) Evaluate the total derivative of z = + where x = 4+ 4t* and
   • e) Compute the matrices and in each of the following and verify that
   • f) Find the Hessian matrix of > R given by

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