Determine the limit of the given function
and plot it in the interval
Pg. 7-1
1) Expand
in Taylor Series w/ remainder:
2) Find Taylor Series Expansion and Remainder of f(x). eq. 4 of p 6-3.--Egm6341.s10.team2.niki 02:26, 26 January 2010 (UTC)
Given:
![{\displaystyle {\boldsymbol {P_{n}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{(1)}(x_{0})+...+{\frac {(x-x_{0})^{n}}{n!}}f^{(n)}(x_{0})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bef4f91cf9f89fe8357e8fbc96fc3687f3fc98c)
[equation 4 p 2-2]
[equation 1 p 2-3]
for the case that
, we get,
=
Using equation 1 p 2-3, we get the remainder as
for
, we get
finally,
dividing both sides by x we get,
and remainder becomes
since
, we have
where
Finally,
Pg 5-1.
Prove the Integral Mean Value Theorem (IMVT) p. 2-3 for w(.) non-negative. i.e
We have the IMVT as
For a given function
Let m be the minimum of the function
and M be the maximum of the same function
Then we know that,
multiplying the inequality throughout by
and integrating between
we get
writing
, we get
It is seen that when w(x) = 0, the result is valid. Consider the case when w(x) > 0
dividing throughout by
From the Intermediate Value Theorem, we know that there exists
such that
i.e
Hence Proved
--Egm6341.s10.team2.niki 02:28, 27 January 2010 (UTC)