Therefore,
g satisfies Rolle’s theorem, and consequently, there exists
c between
a and
x such that
We now calculate
Using the product rule, we note that
Consequently,
Notice that there is a telescoping effect. Therefore,
By Rolle’s theorem, we conclude that there exists a number
c between
a and
x such that
Since
we conclude that
Adding the first term on the left-hand side to both sides of the equation and dividing both sides of the equation by
we conclude that
as desired. From this fact, it follows that if there exists
M such that
for all
x in
I , then
□
Not only does Taylor’s theorem allow us to prove that a Taylor series converges to a function, but it also allows us to estimate the accuracy of Taylor polynomials in approximating function values. We begin by looking at linear and quadratic approximations of
at
and determine how accurate these approximations are at estimating
Using linear and quadratic approximations to estimate function values
Consider the function
Find the first and second Taylor polynomials for
at
Use a graphing utility to compare these polynomials with
near
Use these two polynomials to estimate
Use Taylor’s theorem to bound the error.
For
the values of the function and its first two derivatives at
are as follows:
Thus, the first and second Taylor polynomials at
are given by
The function and the Taylor polynomials are shown in
[link] .
The graphs of
and the linear and quadratic approximations
and
Using the first Taylor polynomial at
we can estimate
Using the second Taylor polynomial at
we obtain
By
[link] , there exists a
c in the interval
such that the remainder when approximating
by the first Taylor polynomial satisfies
We do not know the exact value of
c , so we find an upper bound on
by determining the maximum value of
on the interval
Since
the largest value for
on that interval occurs at
Using the fact that
we obtain
Similarly, to estimate
we use the fact that
Since
the maximum value of
on the interval
is
Therefore, we have
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A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
