Calculus — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

and hence prove that (b) If y=ein": prove that ""'; (c) State the Taylor's theorem for a function „defined on some open interval l of real: numbers. Also, find the Taylor's series expansioh ofe for *ěR about'zerö. (d) Sketch a graph of the equation y=- (c) Examine the nature of origin on the curve. (a) Trace the curve de 1-cose r. (NEP-UGCF 2022) « (c) The parts of each question are to be attempted together. 1. (a) Let fAiR and let c be a cluster point of A. Show that If =L if and only if for every that 2,zc for all nęN, the sequence sequence (x.) -in A that converges tố c such (b) Show that limsin- does not exist in l (c) Using &-8 definition, show that lim =* - =- (d) Find lim 1+:2x#v1+3x where x > 0. 2. (a) Suppose that the functions f and g have . S. limits: in R as →∞ and that f(x) ≤g(x) for. all x€(0,00), Prove that lim sling. (b) Prove by verifying the e-s property that the runction to so continuou a (e) State the Intermedrate value Theorem and = drine show that*5.cos x-for, some xin (d) Show that the function f(x) =2 is not uniformly continuous on (0,1)...... 3. (a) Proye that every uniformly continuous " "the "converse třue? justify? function, on a set A is continuous on A. Is (b) If I and g are functions that are *-adifferentiable at the point a, then prove: (c) Let f(x) =|x|+/x+1| for xeR. Determine the set of points at which the function f is not differentiable. Justify your answer. (d) Pröve cosi-cosy|s|x-y| for all *yeR. 4. (a) Suppose that 1: 10,2) → is continuous on 102), diferentiable on 10, 2), and that (i) Show that f'(c,) =1 for some x€ (0,1). (ii) Show that f'(c.) =0 for some-x e (1,2). it how that (= 1 or some x (0,2). (b) State and prove Mean Value Theorem. "(e) Show that e 2i+* for all XeR its non (a) Prove that if f'and g are differentiable on R, if f(0) € glõ) and if f(x)≤g" foř' all xER, then f(x) ≤g(x): for all x≥0.