B.Sc-B.Sc Mathematics 1st Sem Calculus(University of Pune, Pune-2013)
F.Y. B.Sc.
MATHEMATICS
Calculus
SEAT No. :
[Total No. of Pages : 3
(Paper - II) (2008 Pattern)
Time : 3 Hours] [Max. Marks :80
Instructions to the candidates:
1) All questions are compulsory.
2) Figures to the right indicate full marks.
Q1) Attempt each of the following: [16]
- Determine the set A ={xÎR 2x -3 <5}.
b) State completeness property of R.
c) Show that the series n= +
Show that the sequence⎜ n⎟ converges to 0. ⎝⎠ n =1
e) f) å1 nn 1 is divergent. ¥
Show that the function f (x) = x is continuous on R.
Use definition of derivative to find f′ (x), where f (x) = x2.
g) log x .. Find lim x®¥
h) If y = (ax + b)m, then find nth derivative yn.
Q2) Attempt any four of the following: [16]
If xÎR, then show that there exists n0ÎN such that x < n0 .
Show that between any two distinct real numbers there exists a rational
number.
Prove that x + y £ x + y , for all x, yÎR .
P.T.O.
d) If a > 0, then show that
lim
n®¥
(a )=1. 1
n
e) If sequence (xn )¥=1 converges to x and sequence (yn )¥=1 converges to
f) y, then show that sequence (xn yn )¥=1 converges to xy. n
Let x1 =1, xn+1 = 2 + xn for nÎN, show that sequence (xn )¥=1 is n
convergent, also find its limit.
Q3) Attempt any two of the following: [16]
a) Show that a sequence of real numbers is convergent if and only if it is a
Cauchy sequence.
b) Test the convergence of following series. 2n + 1
i) n= å1 3n -1 .
¥
ii) n= å1 ( n 4 + 1 - n 4 -1 . )
c) Use definition of limit of a function to show,
lim
x®0
lim
x®c
2x2 + 3 = 3
x +5 5 .
x2 =c2 for any cÎR .
d) Let AÍ R, f , g :A ® R and cÎR be a cluster point of A. If
lim
x®c f (x) = L and
lim
x®c g (x) = M then show that x®c⎜
f⎞(x) = L , provided M¹ 0 ⎟
Q4) Attempt any four of the following:
- Finda, b if the function f(x) is continuous on (–3 , 5) where,
⎧ x +a , - 3 < x <1
f (x) = ⎪ 3x + 2, 1£ x <3
⎩ b + x, 3£ x< 5
2
Let I = [a,b] be a closed bounded interval and f :I® R be continuous
on I. Then show that f has an absolute maximum on I.
If function f is continuous at x = c and function g is continuous at f (c),
then show that the composite function gof is continuous at c.
Let f : [0,1]® R be continuous, suppose 0£ f (x)£1, " xÎ[0,1], show
that there exists a point cÎ[0,1] such that f (c) =c .
Determine whether the function h(x) = x x is differentiable at x =0 . Find
h'(0) if it exist.
1
Let f (x) = x2 sin x , x ¹ 0 and f(0) = 0. Show that f′ (x) exists for all
xÎR but f′ (x) is not continuous at x = 0.
Q5) Attempt any two of the following: [16]
a) State and prove Rolle’s theorem. Also give its geometrical interpretation.
- Separate the intervals in which the polynomial 2x3-15x2 + 36x +1 is increasing or decreasing.
ii) If a < b then show that b- a <sin-1 b - sin-1 a < b - a
i) a < 1.
Evaluate lixm 0 xsin x . ® 1- a2
1- a2 where
d) If y = (sin-1 x)2 , then prove that
(1- x2) yn+2 -(2n +1) x y n+1- n2 yn =0.
Find series expansion of tan-1x in powers of x.
1+ x
Find series expansion of log 1 - x in powers of x.
| Earning: Approval pending. |
