Prove that a strictly decreasing function is always one-one.
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Use the Fundamental Theorem of Integral Calculus to evaluate the integral
Show that , the lagrange's form of remainder in the Maclaurin series expansion of e4x, tends to zero as
. Hence obtain the Maclaurin's infinite expansion for e4x.
a) Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
">
Also check whether the function f is derivable at x = 1.
Verify the second mean value theorem for the function and
in the interval
.
Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function defined by
is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function defined by
is not integrable.
Find the following limit
Using Weiestrass M-test, show that the following series converges uniformly.