53 lines
1.3 KiB
Org Mode
53 lines
1.3 KiB
Org Mode
* Taylor Series Approx.
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Suppose f has $\infty$ many derivatives near a point a. Then the taylor series is given by
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$f(x) = \Sigma_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
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For increment notation we can write
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$f(a + h) = f(a) + f'(a)(a+h - a) + \dots$
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$= \Sigma_{n=0}^{\infty} \frac{f^{(n)}(a)}{h!} (h^n)$
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Consider the approximation
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$e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |f'(a) - \frac{1}{h}(f(a + h) - f(a))|$
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Substituting...
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$= |f'(a) - \frac{1}{h}((f(a) + f'(a) h + \frac{f''(a)}{2} h^2 + \cdots) - f(a))|$
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$f(a) - f(a) = 0$... and $distribute the h$
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$= |-1/2 f''(a) h + \frac{1}{6}f'''(a)h^2 \cdots|$
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** With Remainder
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We can determine for some u $f(a + h) = f(a) + f'(a)h + \frac{1}{2}f''(u)h^2$
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and so the error is $e = |f'(a) - \frac{f(a + h) - f(a)}{h}| = |\frac{h}{2}f''(u)|$
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- [https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series]
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+ > Taylor's Theorem w/ Remainder
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** Of Deriviatives
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Again, $f'(a) \approx \frac{f(a+h) - f(a)}{h}$,
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$e = |\frac{1}{2} f''(a) + \frac{1}{3!}h^2 f'''(a) + \cdots$
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$R_2 = \frac{h}{2} f''(\xi)$
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$|\frac{h}{2} f''(\xi)| \leq M h^1$
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$M = \frac{1}{2}|f'(\xi)|$
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*** Another approximation
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$\text{err} = |f'(a) - \frac{f(a) - f(a - h)}{h}|$
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$= f'(a) - \frac{1}{h}(f(a) - (f(a) + f'(a)(a - (a - h)) + \frac{1}{2}f''(a)(a-(a-h))^2 + \cdots))$
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$= |f'(a) - (f'(a) + \frac{1}{2}f''(a)h)|$
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