How Do I Use Taylor Polynomials for Approximation and Error Estimation?

[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?

 

[member 553083]

1. For your first question, you can use the Taylor polynomial to approximate e^(1/2). You can start by writing out the Taylor series for e^x: e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}Now substitute x = 1/2 into this equation and you have your Taylor polynomial approximation of e^(1/2). To get the value to four decimal places, you need to take the sum up to n = 4, which gives you an approximate value of 1.6487.2. For the second question, you need to use the remainder equation to determine the values of x for which the replacement is valid. The remainder equation is given by: R_n = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} where f(x) is the original function (in this case, sqrt(1+x)), c is a point between x and a (in this case, a = 0, since sqrt(1) = 1), and n is the order of the Taylor polynomial. To determine the values of x for which this replacement is valid, you need to find an n such that R_n < 0.01. To do this, you can set R_n = 0.01 and solve for n. This will give you the maximum allowable error of 0.01.