Kakatiya University (KU) 2010-2nd Year B.Sc Mathematics B.A./ (ABSTRACT ALGEBRA

b) Establish the convergence of


51. a) Establish the convergence of by comparison test.

b) Establish the convergence of by ratio test.





52 obtain the value of k such that f(x) is continuous at x = 2.


a) Let K > 0 and let f : R ® R satisfy the condition | f(x) - f(y) } < K | x - y |, for all x, y I R. Show that f is continuous at every point c I R.


b) Suppose f : R ® R is continuous on R and f (r) = 0 for every rational number r. Prove that f(x) = 0 for all x I R.




53 A function f : R ® R is stated to be additive if f(x+y) = f(x) + f(y) for all x, y in R. Prove that if f is continuous at a few x0, then it is continuous at every point of R.



54 Let I = [a, b] and let f : I ® R and g : I ® R be continuous on I. Show that the set E={ x I I : f(x) = g(x) } has the property that if (xn) K E and xn ® x0, then x0 I E.






UNIT-IV



55 a) Show that f(x) = x1/3, x I R, is not differentiable at x = 0.

b) Let f : R ® R be described by f(x) = x2 for x rational

0 for x irrational.

Show that f is differentiable at x = 0, and obtain f ' (o).



56 a) obtain the points of relative extrema, the intervals on which the function

f(x) = x + 1/x, (x ¹ 0) is increasing or decreasing.

b) Use the mean value theorem to show that | sin x - sin y | < | x - y |, x, y I R.




57 a) Let a1, a2 …… an be real numbers and let f be described on R by for x I R. obtain the unique point of relative minimum for f.


b) Use the mean value theorem to show that




58 a) Let f(x)= x2 sin 1/x for x ¹ 0, let f(o) = 0, and let g(x) = sin x for x IR. Show that = 0, but does not exist.


b) obtain



59 a) obtain


b) obtain




60 a) obtain


b) obtain



61 Show that if x > 0, then .



62 Use Taylor's theorem with n = two to find more right approximations for Ö1.2 and Ö2.



63 a) If f(x) = ex, show that the remainder term in Taylor's theorem converge to zero as n ® µ, for every fixed x0 and x.


b) Determine whether or not x = 0 is a point of relative extremum of .



64 a) If f(x) = sin x, show that the remainder term in Taylor's theorem converges to zero as n ® µ for every fixed x0 and x.


b) Determine whether or not x = 0 is a point of relative extremum of .



65 Let f(x ) = two if 0 < x < 1

one if one < x < 2.

Show that f I R [0, 2] and evaluate its integral.



66 a) If f I R [a, b] and | f (x) | < M for all x I [a, b], show that

b) Let g(x) = 0 if x I [0, 1] is rational.

1/x if x I [0, 1] is irrational.

discuss why g L R [0, 1]. However, show that there exists a sequence (Pn) of tagged partitions of [a, b] such that || Pn || ® 0 and exists.




67 a) Consider the function h described by h(x) = x + one for x I [0, 1] rational, and h(x) = 0 for x I [0, 1] irrational. Show that h is not Riemann integrable on [0, 1].


b) Suppose that f is continuous on [a, b], that f(x) > 0 for all x I [a, b] and that . Prove that f(x) = 0 for all x I [a, b].



68 Suppose that a > 0 and that f I R [-a, a].

a) If f is even for all x I [0, a], show that

b) If f is odd, for all x I [0, a], show that .

69 a) If n I N and for x I [a, b], show that the fundamental theorem implies that . What is the set E here ?

b) If g(x) = x for | x | > 1

-x for | x | < 1

and if G(x) = ½ |x2 - one |, show that .




70 a) Let B(x) =


Show that .

b) Let f I R [a, b] and describe .

Then evaluate in terms of F.




71 a) obtain F' (x) when F is described on [0, 1] by : .

b) If f : R ® R is continuous and c > 0, describe g : R ® R by

Show that g is differentiable on R and obtain g' (x).



72 a) obtain F' (x) when F is described on [0, 1] by :

b) If f : [0, 1] ® R is continuous and for all x I [0, 1], show that f(x) = 0 for all x I [0, 1].

Earning:   Approval pending.