10.1D Taylor’s Theorem
Name___________________________
Taylors Remainder: 𝒇(𝒙) = 𝒑𝒏 (𝒙) + 𝑹𝒏 (𝒙)
𝑷𝒏 (𝒙) is the Taylor polynomial of order n centered at a.
𝑹𝒏 (𝒙) is the remainder
𝑹𝒏 (𝒙) =
𝒇𝒏+𝟏 (𝒄)
(𝒏+𝟏)!
(𝒙 − 𝒂)𝒏+𝟏 for some point c between x and a
Estimating the Remainder
When we use Pn  x  , the nth degree Taylor polynomial, to approximate f  x  , the error is
the difference:
En  x   f  x   Pn  x 
we are interested in finding a bound on the magnitude of the error, En ; that is, we want a
number which we are sure is bigger than En . In practice, we want a bound which is
reasonably close to the maximum value of En and after examination it turns out that we
can acquire the necessary information for most of the functions that one encounters in an
elementary calculus course. It is not an exaggeration to say that this is the real reason that
we study power series: Power series allow us to approximate the calculus of the function
f  x  by way of the calculus
of the Taylor polynomials. And it is the Lagrange error bound that often plays a central role
in establishing this fact for a given function f  x  .
The Lagrange Error Bound for Pn  x 
Suppose f and all its derivatives are continuous. If Pn  x  is the nth Taylor polynomial
for f  x  about a , then:
En  f  x   Pn  x  
where the maximum f
n1
M
x a
 n  1!
 n1 x  M
on the interval between a and x .
 
 Note that when you use a Taylor polynomial it is an
approximation to the value of the function even on the interval of
convergence since the interval of convergence is for Taylor series.
 The a above is the center of the interval of convergence and the x
is an input value within the interval of convergence.
 The error tells you how far off you are from the actual value of f
at that particular value of x .
 Notice this is very similar to the “first omitted term” in the
alternating series error!!
Example 1 Estimate the error for 𝒆.𝟒𝟓 using a degree 6 Taylor
Polynomial.
Example 2 Given 𝒑(𝒙) = −𝒙 −
𝒙𝟐
𝟐
−
𝒙𝟑
𝟑
−⋯
A. Find the bound on the error in approximating ln(1-x) by 𝑝3 (𝑥) for
values of x on [-.5, .5].
B. How many terms are needed to approximate values of
f(x) = ln (1-x) with error less than .001 on [-.5, .5]?
HW p678 49,51,55,59,63,69