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I am attempting to an equation to determine the lowest value of $\lambda$ for which $f(x) = \lambda \sin ( \pi x)$ and $y = x$ intersect outside of 0 on the interval $[0,1]$ for some numerical analysis I am doing. Would anyone be able to show me how to write such a function? Thank you very much for your help

update: Thanks for the reply! I'm just trying to find this value for a paper I am writing. So solving the taylor series centered at $\pi$ should give me what I want? That is, $\frac{x}{\pi} = \sin( \pi) + \pi \cos (x- \pi) + \pi^2 \sin(x-pi)/2$ should give me what I want?

sarah
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  • Think about solving $\frac{x}{\lambda}=sin(\pi x)$. As $\lambda$ gets smaller, the line will eventually become too steep. Are you seeking an approximation, or some type of an algorithm to solve this system? I have found that second degree Taylor expansions are reasonably accurate in this situation. – reluctant mathematician Oct 24 '14 at 00:54
  • Thanks for your quick reply! I have edited the main comment with my response – sarah Oct 24 '14 at 01:06
  • I guess it would start as a third taylor, which reduces to a second degree polynomial. For values near $x=0$, this approximation is quite good. – reluctant mathematician Oct 24 '14 at 01:36

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Consider the following approximation:

$$\frac{x}{\lambda}=sin(\pi x)\approx (\pi x)-\frac{1}{6}(\pi x)^3$$ Then $$\frac{6}{\pi^3}(\pi-\frac{1}{\lambda})\approx x^2$$ This requires $\pi-\frac{1}{\lambda}>0$, which is to say, $\lambda >\frac{1}{\pi}$.

reluctant mathematician
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