If a function
$f$ has a power series at
a that converges to
$f$ on some open interval containing
a , then that power series is the Taylor series for
$f$ at
a .
To determine if a Taylor series converges, we need to look at its sequence of partial sums. These partial sums are finite polynomials, known as
Taylor polynomials .
Visit the MacTutor History of Mathematics archive to read brief biographies of
Brook Taylor and
Colin Maclaurin and how they developed the concepts named after them.
Taylor polynomials
The
n th partial sum of the Taylor series for a function
$f$ at
$a$ is known as the
n th Taylor polynomial. For example, the 0th, 1st, 2nd, and 3rd partial sums of the Taylor series are given by
respectively. These partial sums are known as the 0th, 1st, 2nd, and 3rd Taylor polynomials of
$f$ at
$a,$ respectively. If
$x=a,$ then these polynomials are known as
Maclaurin polynomials for
$f.$ We now provide a formal definition of Taylor and Maclaurin polynomials for a function
$f.$
Definition
If
$f$ has
n derivatives at
$x=a,$ then the
n th Taylor polynomial for
$f$ at
$a$ is
The
n th Taylor polynomial for
$f$ at 0 is known as the
n th Maclaurin polynomial for
$f.$
We now show how to use this definition to find several Taylor polynomials for
$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}x$ at
$x=1.$
Finding taylor polynomials
Find the Taylor polynomials
${p}_{0},{p}_{1},{p}_{2}$ and
${p}_{3}$ for
$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}x$ at
$x=1.$ Use a graphing utility to compare the graph of
$f$ with the graphs of
${p}_{0},{p}_{1},{p}_{2}$ and
${p}_{3}.$
To find these Taylor polynomials, we need to evaluate
$f$ and its first three derivatives at
$x=1.$
We now show how to find Maclaurin polynomials for
e
^{x} ,
$\text{sin}\phantom{\rule{0.1em}{0ex}}x,$ and
$\text{cos}\phantom{\rule{0.1em}{0ex}}x.$ As stated above, Maclaurin polynomials are Taylor polynomials centered at zero.
Finding maclaurin polynomials
For each of the following functions, find formulas for the Maclaurin polynomials
${p}_{0},{p}_{1},{p}_{2}$ and
${p}_{3}.$ Find a formula for the
n th Maclaurin polynomial and write it using sigma notation. Use a graphing utilty to compare the graphs of
${p}_{0},{p}_{1},{p}_{2}$ and
${p}_{3}$ with
$f.$
Since
$f\left(x\right)={e}^{x},$ we know that
$f(x)={f}^{\prime}\left(x\right)=f\text{\u2033}\left(x\right)=\text{\cdots}={f}^{\left(n\right)}\left(x\right)={e}^{x}$ for all positive integers
n . Therefore,
Since the fourth derivative is
$\text{sin}\phantom{\rule{0.1em}{0ex}}x,$ the pattern repeats. That is,
${f}^{\left(2m\right)}\left(0\right)=0$ and
${f}^{\left(2m+1\right)}\left(0\right)={\left(\mathrm{-1}\right)}^{m}$ for
$m\ge 0.$ Thus, we have
Since the fourth derivative is
$\text{sin}\phantom{\rule{0.1em}{0ex}}x,$ the pattern repeats. In other words,
${f}^{\left(2m\right)}\left(0\right)={\left(\mathrm{-1}\right)}^{m}$ and
${f}^{\left(2m+1\right)}=0$ for
$m\ge 0.$ Therefore,
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?