Question

2.

a) Find Ts(x), the third degree Taylor polynomial about x -0, for the function e2

b) Find a bound for the error in the interval [0, 1/2]

3. The following data is If all third order differences (not divided differences) are 2, determine the coefficient of x in P(x). prepared for a polynomial P of unknown degree P(x) 2 1 4

I need help with both. Thank you.

Answer 1

Solution:2 (a) Let .

The general form of a Taylor expansion centered at of a function x is

. where is the nth derivative of f(x).

We have to find the third degree Taylor polynomial about , for the function .

Therefore, substituting a=0 in (i), we have

Since ,

Therefore

Therefore, (ii) becomes

Therefore, the required third degree Taylor polynomial about , for the function .

is given by

b) Bound for the error in the interval [0, 1/2].

The error term, is given by for some .

Now .

In the given case, n=3 and according to Lagrange's error term, we need n+1=3+1=4 derivative of

.

That is .

Therefore, the error is given by

for some .

Now we want to find the largest possible value of this error on the interval [0,1/2]. Looking at the equation of we see this will happen when the numerator is maximized and the denominator is minimized, i.e. x=1/2 and c=0. Thus an upper bound for is:

.

Therefore upper bound for the error in the interval [0, 1/2] is 1/24.