# Introduction, limits, average gradient  (Page 2/3)

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## Interesting fact

Zeno (circa 490 BC - circa 430 BC) was a pre-Socratic Greek philosopher of southern Italy who is famous for his paradoxes.

One of Zeno's paradoxes can be summarised by:

Achilles and a tortoise agree to a race, but the tortoise is unhappy because Achilles is very fast. So, the tortoise asks Achilles for a head-start. Achilles agrees to give the tortoise a 1 000 m head start. Does Achilles overtake the tortoise?

We know how to solve this problem. We start by writing:

$\begin{array}{ccc}\hfill {x}_{A}& =& {v}_{A}t\hfill \\ \hfill {x}_{t}& =& 1000\mathrm{m}+{\mathrm{v}}_{\mathrm{t}}\mathrm{t}\hfill \end{array}$

where

 ${x}_{A}$ distance covered by Achilles ${v}_{A}$ Achilles' speed $t$ time taken by Achilles to overtake tortoise ${x}_{t}$ distance covered by the tortoise ${v}_{t}$ the tortoise's speed

If we assume that Achilles runs at 2 m $·$ s ${}^{-1}$ and the tortoise runs at 0,25 m $·$ s ${}^{-1}$ then Achilles will overtake the tortoise when both of them have covered the same distance. This means that Achilles overtakes the tortoise at a time calculated as:

$\begin{array}{ccc}\hfill {x}_{A}& =& {x}_{t}\hfill \\ \hfill {v}_{A}t& =& 1000+{v}_{t}t\hfill \\ \hfill \left(2\mathrm{m}·{\mathrm{s}}^{-1}\right)\mathrm{t}& =& 1000\mathrm{m}+\left(0,25\mathrm{m}·{\mathrm{s}}^{-1}\right)\mathrm{t}\hfill \\ \hfill \left(2\mathrm{m}·{\mathrm{s}}^{-1}-0,25\mathrm{m}·{\mathrm{s}}^{-1}\right)\mathrm{t}& =& 1000\mathrm{m}\hfill \\ \hfill t& =& \frac{1000\mathrm{m}}{1\frac{3}{4}\mathrm{m}·{\mathrm{s}}^{-1}}\hfill \\ & =& \frac{1000\mathrm{m}}{\frac{7}{4}\mathrm{m}·{\mathrm{s}}^{-1}}\hfill \\ & =& \frac{\left(4\right)\left(1000\right)}{7}\mathrm{s}\hfill \\ & =& \frac{4000}{7}\mathrm{s}\hfill \\ & =& 571\frac{3}{7}\mathrm{s}\hfill \end{array}$

However, Zeno (the Greek philosopher who thought up this problem) looked at it as follows: Achilles takes $t=\frac{1000}{2}=500\mathrm{s}$ to travel the 1 000 m head start that the tortoise had. However, in this 500 s, the tortoise has travelled a further $x=\left(500\right)\left(0,25\right)=125\mathrm{m}.$ Achilles then takes another $t=\frac{125}{2}=62,5\mathrm{s}$ to travel the 125 m. In this 62,5 s, the tortoise travels a further $x=\left(62,5\right)\left(0,25\right)=15,625\mathrm{m}.$ Zeno saw that Achilles would always get closer but wouldn't actually overtake the tortoise.

## Sequences, series and functions

So what does Zeno, Achilles and the tortoise have to do with calculus?

Well, in Grades 10 and 11 you studied sequences. For the sequence $0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},...$ which is defined by the expression ${a}_{n}=1-\frac{1}{n}$ the terms get closer to 1 as $n$ gets larger. Similarly, for the sequence $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},...$ which is defined by the expression ${a}_{n}=\frac{1}{n}$ the terms get closer to 0 as $n$ gets larger. We have also seen that the infinite geometric series has a finite total. The infinite geometric series is

$\begin{array}{c}\hfill {S}_{\infty }=\sum _{i=1}^{\infty }{a}_{1}.{r}^{i-1}=\frac{{a}_{1}}{1-r}\phantom{\rule{2.em}{0ex}}\mathrm{for}\phantom{\rule{2.em}{0ex}}r\ne -1\end{array}$

where ${a}_{1}$ is the first term of the series and $r$ is the common ratio.

We see that there are some functions where the value of the function gets close to or approaches a certain value.

Similarly, for the function: $y=\frac{{x}^{2}+4x-12}{x+6}$ The numerator of the function can be factorised as: $y=\frac{\left(x+6\right)\left(x-2\right)}{x+6}.$ Then we can cancel the $x-6$ from numerator and denominator and we are left with: $y=x-2.$ However, we are only able to cancel the $x+6$ term if $x\ne -6$ . If $x=-6$ , then the denominator becomes 0 and the function is not defined. This means that the domain of the function does not include $x=-6$ . But we can examine what happens to the values for $y$ as $x$ gets close to -6. These values are listed in [link] which shows that as $x$ gets closer to -6, $y$ gets close to 8.

 $x$ $y=\frac{\left(x+6\right)\left(x-2\right)}{x+6}$ -9 -11 -8 -10 -7 -9 -6.5 -8.5 -6.4 -8.4 -6.3 -8.3 -6.2 -8.2 -6.1 -8.1 -6.09 -8.09 -6.08 -8.08 -6.01 -8.01 -5.9 -7.9 -5.8 -7.8 -5.7 -7.7 -5.6 -7.6 -5.5 -7.5 -5 -7 -4 -6 -3 -5

The graph of this function is shown in [link] . The graph is a straight line with slope 1 and intercept -2, but with a missing section at $x=-6$ .

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
how did you get the value of 2000N.What calculations are needed to arrive at it
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