# 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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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