18.01 Section, October 28, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
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More Taylor approximation error terms
1. Go to http://mathlets.org/mathlets/taylor-polynomials. For the function you’re
assigned, observe the following things and explain your observations using Taylor’s theorem:
1 (n)
f (x) = f (a) + f 0 (a)(x − a) + 2!1 f 00 (a)(x − a)2 + · · · + n!
f (a)(x − a)n +
(n+1)
1
(ξ)(x
(n+1)! f
− a)n+1
for some ξ between x and a.
• First, make sure you know what x, a, x − a, n, and ξ correspond to in the picture.
• Just stick to linear approximations but change a. Roughly speaking, which a’s correspond to better linear approximations (i.e. ones that are decent approximations in a
bigger range around a)?
• Now stick to a single a, and increase n. What happens to the range in which it’s a good
approximation?
• Try this with a few different a’s. Given a random pair of x and a, how confident are
you that the Taylor series centered at a will eventually approximate x accurately, given
enough terms?
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2. Someone told you that eπ − π = 20. Actually, it’s 19.99909999. . . . It’s taking too long to
calculate this using the Taylor series centered at 0, so you calculate this using the Taylor
series centered at ln 23. Using the fact that |π − ln(23)| < 0.01, how many terms of the
series do you need to calculate before you’re sure it’s not 20?
Problem inspired by xkcd #217.
3. Suppose that f 0 (x) > 0 on some interval. Use the Mean Value Theorem to show that f (x)
is strictly increasing on that interval.
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