Taylor Notes 1, 2, and 3 Review
This review sheet discusses, in a very basic way, the key concepts from these sections. This review is
not meant to be all inclusive, but hopefully it reminds you of some of the basics. Please notify me if
you find any typos in this review.
1. General Comments - We are building up a theory of how to approximate functions with so-
called Taylor polynomials. Initially, the construction of these polynomials and the errors we find
may seem unmotivated. However, Taylor polynomials are an essential tool for exploring functions
in science and engineering. They also provide an introduction to infinite series which paves the
way for the study of Fourier Series (a vital tool for scientists). So I ask for you to bear with me
for the first couple lectures as we build up the necessary machinery to study Taylor series.
2. Essentials - After TN 1, 2, and 3, you should definitely know the following big four concepts. I
discuss each of these concepts, in order, below.
•Given f(x) and b, find the Taylor polynomials of f(x) based at b. That is, find T1(x), T2(x),
etc.
•Given f(x), band an interval I= [c, d] (usually this interval has bat the midpoint), give
a bound for the approximation error in using Tn(x). That is, find a number Bso that
|f(x)−Tn(x)| ≤ B(you have to use Taylor’s inequality).
•Given f(x), b, and an error bound, find an interval so that the approximation error is less
than the error bound. That is, find I= [c, d] so that |f(x)−Tn(x)| ≤ ‘The Given Error
Bound’.
•Given f(x), b, an error bound, and an interval I= [c, d], find an integer, n, so that the
approximation error is less than the error bound. That is, find nso that |f(x)−Tn(x)| ≤
‘The Given Error Bound in the Interval’.
3. Finding T1(x),T2(x),etc And the Corresponding Error Formulas
•For each Taylor polynomial, I give the expanded form, then I rewrite it in sigma notation.
T1(x) =
1
X
k=0
f(k)(b)
k!(x−b)k=f(b) + f0(b)(x−b).
T2(x) =
2
X
k=0
f(k)(b)
k!(x−b)k=f(b) + f0(b)(x−b) + f00(b)
2(x−b)2.
T3(x) =
3
X
k=0
f(k)(b)
k!(x−b)k=f(b) + f0(b)(x−b) + f00(b)
2(x−b)2+f000(b)
3! (x−b)3.
Tn(x) =
n
X
k=0
f(k)(b)
k!(x−b)k=f(b) + f0(b)(x−b) + f00(b)
2(x−b)2+· · · +f(n)(b)
n!(x−b)n.
•The error formulas (Taylor inequalities) below are given in the same order as the Taylor
polynomial that they correspond to above.
ERROR = |f(x)−T1(x)| ≤ M
2|x−b|2, where |f00(x)| ≤ Mon the interval.
ERROR = |f(x)−T2(x)| ≤ M
3! |x−b|3, where |f000(x)| ≤ Mon the interval.
ERROR = |f(x)−T3(x)| ≤ M
4! |x−b|4, where |f(4)(x)| ≤ Mon the interval.
ERROR = |f(x)−Tn(x)| ≤ M
(n+ 1)! |x−b|n+1 , where |f(n+1)(x)| ≤ Mon the interval.
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