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,
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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?