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I have an exercise and i would like to ask if someone has any idea what i have to use to solve it.( what i about the notes : absolute and relative error, general formula of error propagation). Any idea,is being accepted :) thanks

Given the function $f(x) = e^{2x}\cos(3x)$, we want to compute its value at any irrational number $x \in [0, \pi]$ using an approximation $x^*$. Find how many digits should be correct in $x^*$ in order that the value $f(x^*)$ have at least four correct digits.

Ruzayqat
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Irene
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1 Answers1

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For $y=fg$, we have $\Delta y=fg\left(\frac{\Delta f}{f}+\frac{\Delta g}{g}\right)=g\Delta f+f\Delta g$. If $f=\cos 3x$, then $\Delta f=3\Delta x\sin 3x$. If $g=e^{2x}$, then $\Delta g=2\Delta x\, g$. Thus, $|\Delta y|=|3g\Delta x\sin 3x+2fg\Delta x|=|\Delta x|\,|3g\,\sin 3x+2fg|$. Thus $|\Delta x|<\frac{10^{-4}}{|3g\,\sin 3x+2fg|}$. You have to find a lower bound on $|3g\,\sin 3x+2fg|$.

Ruzayqat
  • 834