Mathematics

a. Test the function f(x) for continuity at the point x = 0, given f(x) = x(1/x),\ x 0, and f(0) = 0 at x = 0. (3 marks) b. Examine for continuity and different…

a. Verify Rolle's theorem in the case of function f(x) = 2x3 + x2 - 4x - 2. (4 marks) b. If x2a2+u + y2b2+u + z2c2+u = 1, then prove that ux2 + uy2 + uz2 = 2(x…

a. If y1 = r1, y2 = r12, y3 = r123, …, yn-1 = r12n-2n-1 and yn = r12n-1, then find the value of (y1, y2, , yn)(r, 1, 2, , n-1). (5 marks) b. Discuss the maximum…

a. State and prove Rolle's mean value theorem. (3 marks) b. Show that v-u1+v2 < -1v - -1u < v-u1+u2. (2 marks)

a. Using – approach, show that (x,y)(0,0) xy\,x2-y2x2+y2 = 0. (2 marks) b. Check for the continuity of the following functions: (3 marks) (i) f(x,y) = cases 2x2…

a. Prove that the function f(x,y) = |xy| is not differentiable at origin. Do its partial derivatives exist at origin? Justify. (3 marks) b. Show that the functi…

a. Expand -1yx in the neighborhood of (1,1) by Taylor's theorem. Hence compute f(1.1, 0.9). (2 marks) b. If u = -1x3+y3x+y, then show that x u x + y u y = 2u an…

a. (i) Evaluate r3\,dr\,d over the area bounded between the circles r = 2 and r = 4. (ii) Find the area enclosed by the pair of curves y = 2 - x and y2 = 2(2 -…

a. State and prove Lagrange's Mean Value theorem. Discuss the geometrical interpretation of it. (2 marks) b. Using Cauchy Mean Value theorem, show that 2e x2-1x…

a. Define ε–δ definition of continuity of a function of two variables f : D R2 R. Using ε–δ definition, discuss the continuity of the function f(x,y) = cases x2…

a. Show that for the function f(x,y) = cases y(x2-y2)x2+y2, & if (x,y) (0,0) \\ 0, & if (x,y) = (0,0) cases, fxy(0,0) fyx(0,0). (2 marks) b. If u = -1x1/3+y1/3x…

a. If xyz - u = 0, x2 + y2 + z2 - v = 0, and x + y + z - w = 0, find (x,y,z)(u,v,w). (2 marks) b. Obtain a quadratic Taylor's series approximation to the functi…