# 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.

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s.
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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s.
Graphene has a hexagonal structure
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Cied
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Azam
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Prasenjit
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Azam
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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