Show that is an algebraic number.
An algebraic number is any number that is a root of a non-zero polynomial equation with integer coefficients. To show that (5 + \sqrt{2}) is an algebraic number, ________ ______ __________ _____ ___ ________.
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a) If the partition P2 is a refinement of the partition P1 of [a,b], then and
. Verify this result for the function
defined over the interval
and the partitions
and
.
b) Evaluate: .
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.
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.
Determine the local minimum and local maximum values of the function f defined by .
Use Cauchy's mean value theorem to prove that:
Using Weiestrass M-test, show that the following series converges uniformly.
Show that the equation has a real root other than