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Posted Date: 11 Jun 2008 Posted By: arunkumar Member Level: Gold
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2004 Calculus I Question paper
Calculus I
Mid-Semester Examination, Semester I
September, 2004
Answer any 5 questions. All of them carry equal marks marks:25
• If x is rational, x = 0 and y is irrational, prove that x + y , x -
1.
y
y , xy , x , x are all irrational.
y
• Is the sum or product of two irrational numbers always irrational?
Justify your answer.
v
• Prove that 2 is not a rational number.
2.
• If x and y are 2 real numbers, x < y, show that there is an irrational
number z satisfying x < z < y.
3. Let r be any real number. For any integer n = 1 prove the following:
• r2 + 3r2 + · · · + (2n - 1)r2 = (n · r)2
(n+1)5 ·r 4
• r4 + (2r)4 + · · · + (n · r)4 < 5
• Let
4.
S = {x ? R|x irrational and < 1}
What is sup S and inf S ? Justify your answer.
• Let n and p denote positive integers. Show that
(n + 1)p+1 - np+1
np < < (n + 1)p
p+1
• Let f : [0, 1] ? R be a continuous function and f (0) = f (1). Assume
5.
f is not a constant function. Then show that there are infinitely
many r ? R for which the set f -1 (r) = {x ? [0, 1]|f (x) = r} has
more than one element.
• Let n = 1 be an integer. Show that the function on R definied by
the polynomial xn + nxn-1 + ... + 2x - 1 = 0 takes the value 0.
• Find all real x such that sin(2x) = cos(2x).
6.
• Find all real x such that sin(x) + cos(x) cos(p) = 1 where p is teh
real number defined as the area of the unit circle.
1
• Prove that if A and t are given real numbers, there exists real numbers
7.
B and C such that
A sin(x + t) = B sin(x) + C cos(x)
for all x.
• Show that tan(x - y) can be expressed as a function of tan(x) and
tan(y). What exception one has to make in order that this expression
is valid?
• Let f : [0, 1] ? R be a continuous function. Let m = inf {f (x)|x ? [0, 1/2]}
8.
and n = sup {f (x)|x ? [1/4, 1]}. Show that there are x, y ? [0, 1] such
that f (x) = m and f (y) = n.
• Let f : [-1, 1] ? R be a continuous function. Assume
f (-1/2) < f (1/2)
. Show that for every y ? (f (-1/2), f (1/2)), there is a x ? (-1/2, 1/2)
such that f (x) = y.
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