Fundamental of Calculus — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

3. This question paper has six questions. x-x does not exist. (i) Show that lim,→ (i) Evaluate limy→l Tx -1 b) Examine the continuity at x = 1 and x = 2 of the function f(x) defined below Also, the type of discontinuity, if any. c) Discuss the differentiability at x = 0 of the function f(x) defined below f(x) = x e1/x-e-1/x 2. a) Find the n" differential coefficient of tan 1(2x/(1 - x2)). b) If y = cos(m sin 1x), evaluate yn (O): x 1/2 +y/2), prove that " + 2ХУ ахду - tan u (13 + tan?u). 3. a) State Rolle's theorem and verify the same for the function f(x) = (x-a)(x -b)3 for all x E [a, b]. b) State and prove Lagrange's mean value theorem. c) Verify the Cauchy's mean value theorem for the following pair of functions: (i) f(x) = sin x and g(x) = cos in the domain [-T / 2,0]. (i) (x) = V and and = Jo in the domain domain 5]. 4. a) State Taylor's theorem and prove that sinax = ax - (ax)° + (ax)' ....+ (ax*- - sin where a is a nonzero real number and 0<0<1. Find the Maclaurin's series of log(l+ x) V-1<x≤1, with Cauchy's form of the remainder. Evaluate the following: (i) lim,→o a) Trace the curve r = 2 + 3cose, 0≤0 ≤ 21 b) Sketch the graph of the function y= x3 - 3x+2 6. a) Determine the intervals of concavity and points of in flexion of the function f(x) =x3- 3x2+1. b) Use the second derivative test to find the points of local maxima and minima of the function f(x) = 12x5 - 45x* + 40x3 + 6. Also determine the intervals on which f(x) is increasing and decreasing.