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[T] The Fresnel integrals are used in design applications for roadways and railways and other applications because of the curvature properties of the curve with coordinates ( C ( t ) , S ( t ) ) . Plot the curve ( C 50 , S 50 ) for 0 t 2 π , the coordinates of which were computed in the previous exercise.

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Estimate 0 1 / 4 x x 2 d x by approximating 1 x using the binomial approximation 1 x 2 x 2 8 x 3 16 5 x 4 2128 7 x 5 256 .

0 1 / 4 x ( 1 x 2 x 2 8 x 3 16 5 x 4 128 7 x 5 256 ) d x

= 2 3 2 −3 1 2 2 5 2 −5 1 8 2 7 2 −7 1 16 2 9 2 −9 5 128 2 11 2 −11 7 256 2 13 2 −13 = 0.0767732 ...

whereas 0 1 / 4 x x 2 d x = 0.076773 .

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[T] Use Newton’s approximation of the binomial 1 x 2 to approximate π as follows. The circle centered at ( 1 2 , 0 ) with radius 1 2 has upper semicircle y = x 1 x . The sector of this circle bounded by the x -axis between x = 0 and x = 1 2 and by the line joining ( 1 4 , 3 4 ) corresponds to 1 6 of the circle and has area π 24 . This sector is the union of a right triangle with height 3 4 and base 1 4 and the region below the graph between x = 0 and x = 1 4 . To find the area of this region you can write y = x 1 x = x × ( binomial expansion of 1 x ) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate π .

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Use the approximation T 2 π L g ( 1 + k 2 4 ) to approximate the period of a pendulum having length 10 meters and maximum angle θ max = π 6 where k = sin ( θ max 2 ) . Compare this with the small angle estimate T 2 π L g .

T 2 π 10 9.8 ( 1 + sin 2 ( θ / 12 ) 4 ) 6.453 seconds. The small angle estimate is T 2 π 10 9.8 6.347 . The relative error is around 2 percent.

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Suppose that a pendulum is to have a period of 2 seconds and a maximum angle of θ max = π 6 . Use T 2 π L g ( 1 + k 2 4 ) to approximate the desired length of the pendulum. What length is predicted by the small angle estimate T 2 π L g ?

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Evaluate 0 π / 2 sin 4 θ d θ in the approximation T = 4 L g 0 π / 2 ( 1 + 1 2 k 2 sin 2 θ + 3 8 k 4 sin 4 θ + ) d θ to obtain an improved estimate for T .

0 π / 2 sin 4 θ d θ = 3 π 16 . Hence T 2 π L g ( 1 + k 2 4 + 9 256 k 4 ) .

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[T] An equivalent formula for the period of a pendulum with amplitude θ max is T ( θ max ) = 2 2 L g 0 θ max d θ cos θ cos ( θ max ) where L is the pendulum length and g is the gravitational acceleration constant. When θ max = π 3 we get 1 cos t 1 / 2 2 ( 1 + t 2 2 + t 4 3 + 181 t 6 720 ) . Integrate this approximation to estimate T ( π 3 ) in terms of L and g . Assuming g = 9.806 meters per second squared, find an approximate length L such that T ( π 3 ) = 2 seconds.

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Chapter review exercises

True or False? In the following exercises, justify your answer with a proof or a counterexample.

If the radius of convergence for a power series n = 0 a n x n is 5 , then the radius of convergence for the series n = 1 n a n x n 1 is also 5 .

True

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Power series can be used to show that the derivative of e x is e x . ( Hint: Recall that e x = n = 0 1 n ! x n . )

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For small values of x , sin x x .

True

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The radius of convergence for the Maclaurin series of f ( x ) = 3 x is 3 .

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In the following exercises, find the radius of convergence and the interval of convergence for the given series.

n = 0 n 2 ( x 1 ) n

ROC: 1 ; IOC: ( 0 , 2 )

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n = 0 3 n x n 12 n

ROC: 12 ; IOC: ( −16 , 8 )

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n = 0 2 n e n ( x e ) n

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In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

f ( x ) = x 2 x + 3

n = 0 ( −1 ) n 3 n + 1 x n ; ROC: 3 ; IOC: ( −3 , 3 )

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f ( x ) = 8 x + 2 2 x 2 3 x + 1

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In the following exercises, find the power series for the given function using term-by-term differentiation or integration.

f ( x ) = tan −1 ( 2 x )

integration: n = 0 ( −1 ) n 2 n + 1 ( 2 x ) 2 n + 1

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f ( x ) = x ( 2 + x 2 ) 2

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In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

f ( x ) = x 3 2 x 2 + 4 , a = −3

p 4 ( x ) = ( x + 3 ) 3 11 ( x + 3 ) 2 + 39 ( x + 3 ) 41 ; exact

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f ( x ) = e 1 / ( 4 x ) , a = 4

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In the following exercises, find the Maclaurin series for the given function.

f ( x ) = cos ( 3 x )

n = 0 ( −1 ) n ( 3 x ) 2 n 2 n !

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In the following exercises, find the Taylor series at the given value.

f ( x ) = sin x , a = π 2

n = 0 ( −1 ) n ( 2 n ) ! ( x π 2 ) 2 n

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In the following exercises, find the Maclaurin series for the given function.

f ( x ) = e x 2 1

n = 1 ( −1 ) n n ! x 2 n

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f ( x ) = cos x x sin x

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In the following exercises, find the Maclaurin series for F ( x ) = 0 x f ( t ) d t by integrating the Maclaurin series of f ( x ) term by term.

f ( x ) = sin x x

F ( x ) = n = 0 ( −1 ) n ( 2 n + 1 ) ( 2 n + 1 ) ! x 2 n + 1

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Use power series to prove Euler’s formula : e i x = cos x + i sin x

Answers may vary.

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The following exercises consider problems of annuity payments .

For annuities with a present value of $ 1 million, calculate the annual payouts given over 25 years assuming interest rates of 1 % , 5 % , and 10 % .

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A lottery winner has an annuity that has a present value of $ 10 million. What interest rate would they need to live on perpetual annual payments of $ 250,000 ?

2.5 %

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Calculate the necessary present value of an annuity in order to support annual payouts of $ 15,000 given over 25 years assuming interest rates of 1 % , 5 % , and 10 % .

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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.
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Brian Reply
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?
LITNING Reply
what is a peer
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What is meant by 'nano scale'?
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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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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?
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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?
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Practice Key Terms 2

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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