# 2.2 Completing the basics  (Page 6/6)

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The second output line illustrated in Table 3.4 displays the same type of items as the first line, except that the approximation to ex requires the two of two terms of the approximating polynomial. Notice also that the first item on the second line, the value obtained by the exp() function, is the same as the first item on the first line. This means that this item does not have to recalculated; the value calculated for the first line can simply be displayed a second line. Once the data for the second line have been calculated, a single cout statement can again be used to display the required values.

Finally, only the second and third items on the last two output lines shown in Figure 1 need to be recalculated because the first item on these lines is the same as previously calculated for the first line. Thus, for this problem, the complete algorithm described in pseudocode is:

`Display a prompt for the input value of x.`

`Read the input value.`

`Display the heading lines.`

`Calculate the exponential value of x using the exp() function.`

`Calculate the first approximation.`

`Calculate the first difference.`

`Print the first output line.`

`Calculate the second approximation.`

`Calculate the second difference.`

`Print the second output line.`

`Calculate the third approximation.`

`Calculate the third difference.`

`Print the third output line.`

`Calculate the fourth approximation.`

`Calculate the fourth difference.`

`Print the fourth output line.`

To ensure that we understand the processing used in the algorithm, we will do a hand calculation. The result of this calculation can then be used to verify the result produced by the program that we write. For test purposes, we use a value of 2 for x, which causes the following approximations:

Using the first term of the polynomial, the approximation is

e^2 = 1

Using the first two terms of the polynomial, the approximation is

e^2 = 1 + 2/1 = 3

Using the first three terms of the polynomial, the approximation is

e^2 = 3 + 2^2/2 = 5

Using the first four terms of the polynomial, the approximation is

e^2 = 5 + 2^3/6 = 6.3333

Notice that the first four terms of the polynomial, it was not necessary to recalculate the value of the first three terms; instead, we used the previously calculated value.

## Step 3: code the algorithm

The following program represents a description of the selected algorithm in C++.

`// This program approximates the function e raised to the x power`

`// using one, two, three, and four terms of an approximating polynomial.`

`#include<iostream.h>`

`#include<iomanip.h>`

`#include<math.h>`

`int main()`

`{`

`double x, funcValue, approx, difference;`

`cout<<“\n Enter a value of x: “;`

`cin>>x;`

`// print two title lines`

`cout<<“ e to the x Approximation Difference\n”`

`cout<<“------------ --------------------- --------------\n”;`

`funcValue = exp(x);`

`// calculate the first approximation`

`approx = 1;`

`difference = abs(funcValue – approx);`

`cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue`

`<<setw(18)<<approx`

`<<setw(18)<<difference<<endl;`

`// calculate the first approximation`

`approx = 1;`

`difference = abs(funcValue – approx);`

`cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue`

`<<setw(18)<<approx`

`<<setw(18)<<difference<<endl;`

`// calculate the second approximation`

`approx = approx + x;`

`difference = abs(funcValue – approx);`

`cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue`

`<<setw(18)<<approx`

`<<setw(18)<<difference<<endl;`

`// calculate the third approximation`

`approx = approx + pow(x,2)/2.0;`

`difference = abs(funcValue – approx);`

`cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue`

`<<setw(18)<<approx`

`<<setw(18)<<difference<<endl;`

`// calculate the fourth approximation`

`approx = approx + pow(x,3)/6.0;`

`difference = abs(funcValue – approx);`

`cout<<setw(10)<<setiosflags(iso::showpoint)<<funcValue`

`<<setw(18)<<approx`

`<<setw(18)<<difference<<endl;`

`return 0;`

`}`

In reviewing the program, notice that the input value of x is obtained first. The two title lines are then printed prior to any calculations being made. The value of ex is then computed using the exp() library function and assigned to the variable funcValue. This assignment permits this value to be used in the four difference calculations and displayed four times without the need for recalculation.

Since the approximation to the ex is “built up” using more and more terms of the approximating polynomial, only the new term for each approximation is calculated and added to the previous approximation. Finally, to permit the same variables to be reused, the values in them are immediately printed before the next approximation is made.

## Step 4: test and correct the program

The following is the sample run produced by the above program is:

The first two columns of output data produced by the sample run agree with our hand calculation. A hand check of the last column verifies that it also correctly contains the difference in values between the first two columns.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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