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Mangoes
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Homework Statement
Determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001.
[tex] e^x ≈ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}[/tex]
For x < 0
Homework Equations
Taylor's Theorem to approximate a remainder:
[tex] |R(x)| = |\frac{f^{n+1}(z)}{(n+1)!}(x-c)^{n+1}|[/tex]
Where z is some number between c and x, n is the degree of the approximating function, and c is where the function is centered at.
The Attempt at a Solution
From the Taylor polynomial given, c = 0 and n = 3. Since f(x) is e^x, the fourth derivative is simply e^x. If I want an error of less than 0.001,
[tex] |R(x)| = |\frac{e^z}{4!}x^4| < 0.001 [/tex]
Not too sure about this next part, but I think that since the function above increases as x increases, which means the error increases with an increasing x, I replaced z by x since the maximum error is given by the largest z value, and the largest z value is equal to x, and I'm interested in the error bound anyways.
[tex] |R(x)| = |\frac{(x^4)(e^x)}{4!}| < 0.001 [/tex]
...but I'm still left with the problem that I have to find a value that's stuck in an exponent and outside of one, so I'm assuming I'm doing this wrong and I can't figure out any other way to do it.
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