Department of Mathematics and Statistics, I.I.T. Kanpur MTH101N -End-Semester Examination - 16 November, 2009
Maximum Marks: 100 Time: 8:00-11:00 hrs.
Note :
(A)Please write down the page numbers in the answer book.
(B)On the top cover of your answer book, write down the page number on which each question has been answered. For this purpose, make a table similar to the one given below.
(C) Answer all parts of a question together at one place.
Answer all questions.
(1) (a) Let f : R → R be such that f0(x) = 0 for every x ∈ R. Show that f is
constant. [2]
(b) Let f : R → R be such that f00(x) > 0 for every x ∈ R. Show that f(y)≥f(x) +f0(x)(y−x) for all x, y ∈R. [3]
(c) Let P∞
n=1an converge. Show that an →0. [2]
(d) Find limn→∞R1
0 ynsiny
(1+y)2dy. [3]
(e) Check whether the function f : R2 → R defined by f(x, y) = xx4+y2y2 when (x, y)6= (0,0) and f(0,0) = 0 is differentiable at (0,0). [3]
(2) (a) Sketch the graph ofr =−|sinθ|. [2]
(b) Find the y-coordinate ¯y of the center of gravity of the region bounded by the curves y =√
4−x2, y =√
9−x2 and the x-axis. [2]
(c) Find the equation of the tangent plane for z=x2+y2−2xy+ 3y−x+ 4 at
(2,−3,18). [3]
(d) Find three real numbers whose sum is 12 and the sum of whose square is as
small as possible. [3]
(e) Let R8
0
R2
√3
x
dydx =Rb
a
Rd c
dxdy. Finda, b, c and d. [3]
(3) (a) Let x0 = 1 and x1 = 2. Define xn = 12(xn−1+xn−2) for n ≥ 2. Show that (xn) converges. By observing that 2xn+xn−1 = 2xn−1+xn−2, find the limit
of (xn). [5]
(b) Let f :R →(0,∞) satisfy f(x+y) =f(x)f(y) for all x, y ∈R. Suppose f is continuous at x= 0. Show that f is continuous at all x∈R. [6]
1
(4) (a) Let f : [0,1] → R be a differentiable function such that f(0) = 0 and f(x) > 0 for every x ∈ (0,1]. Show that there exists c ∈ (0,1) such that
f0(1−c)
f(1−c) = 2ff(c)0(c). [5]
(b) Determine all the values of x for which the series P∞
n=2 xn
n(logn)2 converges
absolutely. [5]
(5) (a) Determine all the values of p for which the improper integral R∞
0 tp−1
1+t2dt con-
verges. [5]
(b) Prove that, for all x >0, 3Rx
0
u2 µu
R
0
f(t)dt
¶
du=Rx
0
f(u)(x3−u3)du. [5]
(6) (a) Calculate the area of the surface of revolution generated by revolving the cardioid x(θ) = 2 cosθ−cos 2θ, y(θ) = 2 sinθ−sin 2θ, 0≤θ ≤π, about the
x-axis. [6]
(b) Let f : [a, b]→R be an increasing function. Show that f is integrable. [5]
(7) (a) The position vector of a moving particle is given byR(t) = 2 costi+2 sintj+ 3tk. Find the principle normal at R(π4). Further show that the curvature of
this curve is constant. [5]
(b) Let f(x, y) = (x2+y2) sinx2+y1 2 if (x, y)6= (0,0) and 0 otherwise. Show that f is differentiable at (0,0). Find a sequence ((xn, yn)) such that (xn, yn) →
(0,0) but fx(xn, yn)9fx(0,0). [6]
(8) (a) Let f(x, y) = 5x4−6x2y+y2. Show that f has a local minimum at (0,0) along every line through (0,0). Does f have a minimum at (0,0)? Justify
your answer. [5]
(b) Evaluate RR
(x2+y2)dxdy over the region bounded between the circlesx2+
y2 = 2x and x2+y2 = 4x. [5]
(9) (a) Consider the surfaceS which is the intersection of the solid cylinderx2+y2 ≤ 1 and the plane y+z = 2. LetC be the boundary of S. Find a unit normal ˆ
n to S. Evaluate R
C
(2x−y)dx−y2dy −y2zdz where C is oriented in the direction of the orientation with respect to ˆn. [6]
(b) Let S be the unit sphere x2 +y2 +z2 = 1. Evaluate the surface integral RR
S
{(2x+ 3z)x−(xz+y)y+ (y2+ 2z)z}dσ. [5]
