# 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 $x=-6$ .

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
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Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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