- #1
Benny
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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 [tex]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 [/tex].
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 [tex]R_n = \frac{{f^{\left( {n + 1} \right)} \left( c \right)}}{{\left( {n + 1} \right)!}}\left( {x - a} \right)^{n + 1} [/tex] 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?
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 [tex]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 [/tex].
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 [tex]R_n = \frac{{f^{\left( {n + 1} \right)} \left( c \right)}}{{\left( {n + 1} \right)!}}\left( {x - a} \right)^{n + 1} [/tex] 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?