6.4 Working with taylor series  (Page 8/11)

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[T] Suppose that a set of standardized test scores is normally distributed with mean $\mu =100$ and standard deviation $\sigma =10.$ Set up an integral that represents the probability that a test score will be between $90$ and $110$ and use the integral of the degree $10$ Maclaurin polynomial of $\frac{1}{\sqrt{2\pi }}{e}^{\text{−}{x}^{2}\text{/}2}$ to estimate this probability.

The probability is $p=\frac{1}{\sqrt{2\pi }}{\int }_{\left(a-\mu \right)\text{/}\sigma }^{\left(b-\mu \right)\text{/}\sigma }{e}^{\text{−}{x}^{2}\text{/}2}dx$ where $a=90$ and $b=100,$ that is, $p=\frac{1}{\sqrt{2\pi }}{\int }_{-1}^{1}{e}^{\text{−}{x}^{2}\text{/}2}dx=\frac{1}{\sqrt{2\pi }}{\int }_{-1}^{1}\sum _{n=0}^{5}{\left(-1\right)}^{n}\frac{{x}^{2n}}{{2}^{n}n\text{!}}dx=\frac{2}{\sqrt{2\pi }}\sum _{n=0}^{5}{\left(-1\right)}^{n}\frac{1}{\left(2n+1\right){2}^{n}n\text{!}}\approx 0.6827.$

[T] Suppose that a set of standardized test scores is normally distributed with mean $\mu =100$ and standard deviation $\sigma =10.$ Set up an integral that represents the probability that a test score will be between $70$ and $130$ and use the integral of the degree $50$ Maclaurin polynomial of $\frac{1}{\sqrt{2\pi }}{e}^{\text{−}{x}^{2}\text{/}2}$ to estimate this probability.

[T] Suppose that $\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $f\left(x\right)$ such that $f\left(0\right)=1,{f}^{\prime }\left(0\right)=0,$ and $f\text{″}\left(x\right)=\text{−}f\left(x\right).$ Find a formula for ${a}_{n}$ and plot the partial sum ${S}_{N}$ for $N=20$ on $\left[-5,5\right].$

As in the previous problem one obtains ${a}_{n}=0$ if $n$ is odd and ${a}_{n}=\text{−}\left(n+2\right)\left(n+1\right){a}_{n+2}$ if $n$ is even, so ${a}_{0}=1$ leads to ${a}_{2n}=\frac{{\left(-1\right)}^{n}}{\left(2n\right)\text{!}}.$

[T] Suppose that $\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $f\left(x\right)$ such that $f\left(0\right)=0,\phantom{\rule{0.5em}{0ex}}{f}^{\prime }\left(0\right)=1,$ and $f\text{″}\left(x\right)=\text{−}f\left(x\right).$ Find a formula for ${a}_{n}$ and plot the partial sum ${S}_{N}$ for $N=10$ on $\left[-5,5\right].$

Suppose that $\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $y$ such that $y\text{″}-{y}^{\prime }+y=0$ where $y\left(0\right)=1$ and $y\prime \left(0\right)=0.$ Find a formula that relates ${a}_{n+2},{a}_{n+1},$ and ${a}_{n}$ and compute ${a}_{0},...,{a}_{5}.$

$y\text{″}=\sum _{n=0}^{\infty }\left(n+2\right)\left(n+1\right){a}_{n+2}{x}^{n}$ and ${y}^{\prime }=\sum _{n=0}^{\infty }\left(n+1\right){a}_{n+1}{x}^{n}$ so $y\text{″}-{y}^{\prime }+y=0$ implies that $\left(n+2\right)\left(n+1\right){a}_{n+2}-\left(n+1\right){a}_{n+1}+{a}_{n}=0$ or ${a}_{n}=\frac{{a}_{n-1}}{n}-\frac{{a}_{n-2}}{n\left(n-1\right)}$ for all $n·y\left(0\right)={a}_{0}=1$ and ${y}^{\prime }\left(0\right)={a}_{1}=0,$ so ${a}_{2}=\frac{1}{2},{a}_{3}=\frac{1}{6},{a}_{4}=0,$ and ${a}_{5}=-\frac{1}{120}.$

Suppose that $\sum _{n=0}^{\infty }{a}_{n}{x}^{n}$ converges to a function $y$ such that $y\text{″}-{y}^{\prime }+y=0$ where $y\left(0\right)=0$ and ${y}^{\prime }\left(0\right)=1.$ Find a formula that relates ${a}_{n+2},{a}_{n+1},$ and ${a}_{n}$ and compute ${a}_{1},...,{a}_{5}.$

The error in approximating the integral ${\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.1em}{0ex}}dt$ by that of a Taylor approximation ${\int }_{a}^{b}{P}_{n}\left(t\right)\phantom{\rule{0.1em}{0ex}}dt$ is at most ${\int }_{a}^{b}{R}_{n}\left(t\right)\phantom{\rule{0.1em}{0ex}}dt.$ In the following exercises, the Taylor remainder estimate ${R}_{n}\le \frac{M}{\left(n+1\right)\text{!}}{|x-a|}^{n+1}$ guarantees that the integral of the Taylor polynomial of the given order approximates the integral of $f$ with an error less than $\frac{1}{10}.$

1. Evaluate the integral of the appropriate Taylor polynomial and verify that it approximates the CAS value with an error less than $\frac{1}{100}.$
2. Compare the accuracy of the polynomial integral estimate with the remainder estimate.

[T] ${\int }_{0}^{\pi }\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{t}dt;{P}_{s}=1-\frac{{x}^{2}}{3\text{!}}+\frac{{x}^{4}}{5\text{!}}-\frac{{x}^{6}}{7\text{!}}+\frac{{x}^{8}}{9\text{!}}$ (You may assume that the absolute value of the ninth derivative of $\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{t}$ is bounded by $0.1.\right)$

a. (Proof) b. We have ${R}_{s}\le \frac{0.1}{\left(9\right)\text{!}}{\pi }^{9}\approx 0.0082<0.01.$ We have ${\int }_{0}^{\pi }\left(1-\frac{{x}^{2}}{3\text{!}}+\frac{{x}^{4}}{5\text{!}}-\frac{{x}^{6}}{7\text{!}}+\frac{{x}^{8}}{9\text{!}}\right)\phantom{\rule{0.1em}{0ex}}dx=\pi -\frac{{\pi }^{3}}{3·3\text{!}}+\frac{{\pi }^{5}}{5·5\text{!}}-\frac{{\pi }^{7}}{7·7\text{!}}+\frac{{\pi }^{9}}{9·9\text{!}}=1.852...,$ whereas ${\int }_{0}^{\pi }\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{t}dt=1.85194...,$ so the actual error is approximately $0.00006.$

[T] ${\int }_{0}^{2}{e}^{\text{−}{x}^{2}}dx;{p}_{11}=1-{x}^{2}+\frac{{x}^{4}}{2}-\frac{{x}^{6}}{3\text{!}}+\text{⋯}-\frac{{x}^{22}}{11\text{!}}$ (You may assume that the absolute value of the $23\text{rd}$ derivative of ${e}^{\text{−}{x}^{2}}$ is less than $2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{14}.\right)$

The following exercises deal with Fresnel integrals .

The Fresnel integrals are defined by $C\left(x\right)={\int }_{0}^{x}\text{cos}\left({t}^{2}\right)\phantom{\rule{0.1em}{0ex}}dt$ and $S\left(x\right)={\int }_{0}^{x}\text{sin}\left({t}^{2}\right)\phantom{\rule{0.1em}{0ex}}dt.$ Compute the power series of $C\left(x\right)$ and $S\left(x\right)$ and plot the sums ${C}_{N}\left(x\right)$ and ${S}_{N}\left(x\right)$ of the first $N=50$ nonzero terms on $\left[0,2\pi \right].$

Since $\text{cos}\left({t}^{2}\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{4n}}{\left(2n\right)\text{!}}$ and $\text{sin}\left({t}^{2}\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{4n+2}}{\left(2n+1\right)\text{!}},$ one has $S\left(x\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{4n+3}}{\left(4n+3\right)\left(2n+1\right)\text{!}}$ and $C\left(x\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{4n+1}}{\left(4n+1\right)\left(2n\right)\text{!}}.$ The sums of the first $50$ nonzero terms are plotted below with ${C}_{50}\left(x\right)$ the solid curve and ${S}_{50}\left(x\right)$ the dashed curve.

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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?
Abdul