Page 103 - Calculus for the Clueless, Calc II

4. The remainder R n(x) = f(x) - S n(x) for all x in I.

Then there is a point w in I—w is between a and x— such that

Let us give three examples worked out all the way.

Example 38—

2
Write a polynomial of degree 2. Write the remainder. Find the approximate value for e. and estimate the
2
maximum error from the actual value of e. .

x
This sounds like a lot of work, but, as we will see, this process for e (e doesn't stand for "easy," but it should) is
really quite short.

2
x
Therefore, S 2(.2) = 1 + .2 + (.2) /2 = 1.22 and R 2(.2) = e (.2) /6, where w is between 0 and .2. Because e is an
w
3
5
½
w
2
increasing function, e < e. < e. < (3) < 2 (being very wasteful). Therefore, , the
maximum error.
This is a pretty good approximation. Remember, we were really rough-estimating the error, and this is only a
polynomial of degree 2.
Example 39—
Let us do the same for In (1 + x), polynomial degree 3, a = 0, x = 1, estimate the error for In 1.1.