1. The problem statement, all variables and given/known data
Determine the polynomial p of degree at most 1 that minimizes
[tex]\int_0^2 |e^x - p(x)|^2 dx[/tex]
Hint: First find an orthogonal basis for a suitably chosen space of polynomials of degree at most 1
3. The attempt at a solution
I assumed what I wanted was a p(x) of the form
[tex]
p(x) = \frac{<e^x, 1>}{<1,1>} + \frac{<e^x, x>}{<x,x>}x
[/tex]
where the inner product is
[tex]<f, g> = \int_0^2 f(x)\bar{g(x)} dx[/tex]
But this fails just for the first term, ie
[tex]\frac{<e^x, 1>}{<1,1>} = \frac{e^2-1}{2}[/tex] does not coincide with the correct answer
Correct answer:
[tex]p(x) = 3x + \frac{1}{2}(e^2 - 7)[/tex]
Determine the polynomial p of degree at most 1 that minimizes
[tex]\int_0^2 |e^x - p(x)|^2 dx[/tex]
Hint: First find an orthogonal basis for a suitably chosen space of polynomials of degree at most 1
3. The attempt at a solution
I assumed what I wanted was a p(x) of the form
[tex]
p(x) = \frac{<e^x, 1>}{<1,1>} + \frac{<e^x, x>}{<x,x>}x
[/tex]
where the inner product is
[tex]<f, g> = \int_0^2 f(x)\bar{g(x)} dx[/tex]
But this fails just for the first term, ie
[tex]\frac{<e^x, 1>}{<1,1>} = \frac{e^2-1}{2}[/tex] does not coincide with the correct answer
Correct answer:
[tex]p(x) = 3x + \frac{1}{2}(e^2 - 7)[/tex]
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