5.3 Exponential curves  (Page 2/2)

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Once again, this is predictable from the rules of exponents: $2\cdot {2}^{x}={2}^{1}\cdot {2}^{x}={2}^{x+1}$

Using exponential functions to model behavior

In the first chapter, we talked about linear functions as functions that add the same amount every time . For instance, $y=3x+4$ models a function that starts at 4; every time you increase $x$ by 1, you add 3 to $y$ .

Exponential functions are conceptually very analogous: they multiply by the same amount every time . For instance, $y=4×{3}^{x}$ models a function that starts at 4; every time you increase $x$ by 1, you multiply $y$ by 3.

Linear functions can go down, as well as up, by having negative slopes : $y=-3x+4$ starts at 4 and subtracts 3 every time. Exponential functions can go down, as well as up, by having fractional bases : $y=4×\left(\frac{1}{3}{\right)}^{x}$ starts at 4 and divides by 3 every time.

Exponential functions often defy intuition, because they grow much faster than people expect.

Modeling exponential functions

Your father’s house was worth $100,000 when he bought it in 1981. Assuming that it increases in value by $8%\text{}$ every year, what was the house worth in the year 2001? (*Before you work through the math, you may want to make an intuitive guess as to what you think the house is worth. Then, after we crunch the numbers, you can check to see how close you got.) Often, the best way to approach this kind of problem is to begin by making a chart, to get a sense of the growth pattern. Year Increase in Value Value 1981 N/A 100,000 1982 $8%\text{}$ of 100,000 = 8,000 108,000 1983 $8%\text{}$ of 108,000 = 8,640 116,640 1984 $8%\text{}$ of 116,640 = 9,331 125,971 Before you go farther, make sure you understand where the numbers on that chart come from . It’s OK to use a calculator. But if you blindly follow the numbers without understanding the calculations, the whole rest of this section will be lost on you. In order to find the pattern, look at the “Value” column and ask: what is happening to these numbers every time? Of course, we are adding $8%\text{}$ each time, but what does that really mean? With a little thought—or by looking at the numbers—you should be able to convince yourself that the numbers are multiplying by 1.08 each time . That’s why this is an exponential function: the value of the house multiplies by 1.08 every year. So let’s make that chart again, in light of this new insight. Note that I can now skip the middle column and go straight to the answer we want. More importantly, note that I am not going to use my calculator this time—I don’t want to multiply all those 1.08s, I just want to note each time that the answer is 1.08 times the previous answer . Year House Value 1981 100,000 1982 $\text{100},\text{000}×1\text{.}\text{08}$ 1983 $\text{100},\text{000}×1\text{.}{\text{08}}^{2}$ 1984 $\text{100},\text{000}×1\text{.}{\text{08}}^{3}$ 1985 $\text{100},\text{000}×1\text{.}{\text{08}}^{4}$ $y$ $\text{100},\text{000}×1\text{.}{\text{08}}^{\text{something}}$ If you are not clear where those numbers came from, think again about the conclusion we reached earlier: each year, the value multiplies by 1.08. So if the house is worth $\text{100},\text{000}×1\text{.}{\text{08}}^{2}$ in 1983, then its value in 1984 is $\left(\text{100},\text{000}×1\text{.}{\text{08}}^{2}\right)×1\text{.}\text{08}$ , which is $\text{100},\text{000}×1\text{.}{\text{08}}^{3}$ . Once we write it this way, the pattern is clear. I have expressed that pattern by adding the last row, the value of the house in any year $y$ . And what is the mystery exponent? We see that the exponent is 1 in 1982, 2 in 1983, 3 in 1984, and so on. In the year $y$ , the exponent is $y-\text{1981}$ . So we have our house value function: $v\left(y\right)=\text{100},\text{000}×1\text{.}{\text{08}}^{y-\text{1981}}$ That is the pattern we needed in order to answer the question. So in the year 2001, the value of the house is $\text{100},\text{000}×1\text{.}{\text{08}}^{\text{20}}$ . Bringing the calculator back, we find that the value of the house is now$466,095 and change.

Wow! The house is over four times its original value! That’s what I mean about exponential functions growing faster than you expect: they start out slow, but given time, they explode. This is also a practical life lesson about the importance of saving money early in life—a lesson that many people don’t realize until it’s too late.

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
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
Anassong
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
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