function approximation

T

tremor

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Sep 22, 2010
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Hello - I THINK this belongs in the diff eq section so here goes. I'm having a little trouble with the following problem...namely I'm not sure where to get started. I have to approximate the following function:

x(t) = e^t

over the interval (0,1) using a second order polynomial

then, from the set of linearly independent functions [1,t,t^2], form an orthonormal set of functions.

Any help would be appreciated...sorry to not include any work but I need a clue where to get started.

Thanks!!
 
What is the topic - from which this problem was chosen?

Have you covered Taylor's series (expansion)?
 
An orthogonal set of non zero vectors can always be made into an orthonormal set by normalizing each of its vectors.

An orthonormal set is an orthogonal set with each vector having a norm of 1.
 
I understand the orthonormal part ok it's the approximating the function over an interval using a second order polynomial that has me lost.
 
From what I see, the second order poly is just the first three terms in the Taylor expansion for e.

\(\displaystyle 1+t+\frac{t^{2}}{2}\)

That's because \(\displaystyle e^{x}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\)
 
so would the approximation be the series at 0 plus the series at 1?

so 1+t+t^2/2 (approx at zero)

plus

1+1+1/2 (approx at 1)
 
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