# Error estimates for fuctions

1. Sep 24, 2005

### Benny

I'm not understanding questions where I need to determines values of x for which one function can be replaced by another with a specified maximum allowable error. I'm also having trouble trying to approximate a number to a specified number of decimal places. Here are two questions that I would like some help with.

1. Use Taylor polynomials to approximate the given number to four decimal places. (You may assume that 2 < e < 3)

e^(1/2) is the number.

2. For what values of x can we replace sqrt(1+x) by 1 + (x/2) with an error of less than 0.01?

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1. If I recall correctly then $$e^x = \sum\limits_{n = 0}^\infty {\frac{{x^n }}{{n!}}} \Rightarrow e^{\frac{1}{2}} = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}} \left( {\frac{1}{2}} \right)^n$$.

Just to give myself an idea of the numbers I'm working with I used a calculator to obtain e^(1/2) = 1.648721271.

So I need to find a value of n which will give me a number which is within 4 decimal places of e^(1/2).

I have the remainder equation $$R_n = \frac{{f^{\left( {n + 1} \right)} \left( c \right)}}{{\left( {n + 1} \right)!}}\left( {x - a} \right)^{n + 1}$$ for some c between x and a.

I'm not sure how to apply it to this question. I think question 2 requires the same thing but again I'm not sure how to apply the remainder equation. Can someone please help me out?