指数增长

(Video) Exponential Growth

by mathman1024

Here's a video by mathman1024 talking about growth and decay.

$\large{\text{Summary}}$

Both growth and decay can be represented with an exponential function: $$y=y_0e^{kt}$$ Where,

• $y_0=$ the initial value or amount
• $k=$ the rate (constant)
• $t=$ time

You might have noticed that this formula looks similar to the formula for continuous compounding interest, $A=Pe^{rt}$. Well that's because they're pretty much the same. Continuous compounding interest is a type of exponential growth.

Let's see if you are understanding exponential growth.

How long would it take to double an investment at $8\%$ interest rate compounded continuously? No starting amount has been given though but we know that at the end we want our final value to be double the starting value. So if we use our continuous compounding interest formula we get

• $A=2P$ since we want it to be doubled
• $P$ is unknown
• $r=0.08$
• $t$ is what we're solving for $$\begin{align} &A=Pe^{rt}\\ &\color{#8925AE}{2P}=Pe^{\color{#8925AE}{0.08}t}\\ &\frac{2P}{\color{#8925AE}{P}}=\frac{Pe^{0.08t}}{\color{#8925AE}{P}}\\ &2=e^{0.08t}\\ &\color{#8925AE}{\ln}2=\color{#8925AE}{\ln}e^{0.08t}\\ &\ln2=0.08t\\ &\frac{\ln2}{\color{#8925AE}{0.08}}=\frac{0.08t}{\color{#8925AE}{0.08}}\\ &\frac{\ln2}{0.08}=t\\ &t\approx8.664\text{ years} \end{align}$$

You go to the beach and start collecting shells on the first day. The first day, you only collect four shells. After the first day, it gets quadrupled, so you have 16 shells. On the second day you get 4 dollars, and so on. If we make a table we see: $$\begin{array}{c|c} \text{shells}&\text{day}\\ \hline 4&1\\ 16&2\\ 64&3\\ 256&4 \end{array}$$ You might notice that pattern to be $4^n$.

(Video) Exponential Growth and Decay

by ThinkwellVids

Here's a great video by ThinkwellVids showing you how to solve a problem involving exponential growth.

$\large{\text{Summary}}$

The equation for exponential growth is $y=ax^{bx}$. You're usually given two different sets of points. With those you make two equations and use them to find any unknowns. Then once you have a formula you can plug in any value you want and find its solution.

(Video) Exponential Growth Application

by GVSUmath

In this video by GVSUmath the concept of exponential growth is explained.

Exponential Growth

Bacteria are one of the most common ways to look at exponential growth. If your population is larger, then the rate of increase or growth should be increasingly larger. This concept is modeled by the function $$y=ae^{bx} \text{ with } b>0$$

So, let's look at an example.

A student measures bacteria at two different times. After $2$ hours, there are $100$ bacteria. After $4$ hours, there are $400$ bacteria. How many bacteria are present at $5$ hours?

We have the following information:

• $(2, 100)$

• $(4, 400)$

When $x=2$ and $y=100$ the equation is correct. So, $100=ae^{(2x)}$. This means, $a=\frac{100}{e^{2b}}$.

Now, we see when $x=4$ and $y=400$ the equation is correct. So, $400=ae^{(4b)}$. We can plug in $a=\frac{100}{e^{2b}}$. We see,

$$\begin{align} 400&=ae^{(4b)}\\ 400&=\frac{100\times{e^{(4b)}}}{e^{2b}}\\ 400&=100\times{e^{(2b)}}\\ 4&=e^{(2b)}\\ \ln{4}&=\ln{e^{2b}}\\ \ln{4}&=2b\\ b&=\frac{1}{2}\ln{4} \end{align}$$

We can now plug $b$ in to the other equation to find $a$. We see, $$\begin{align} 400&=ae^{(4b)}\\ 400&=ae^{4\times{\frac{1}{2}\ln{4}}}\\ 400&=ae^{\ln{4}^2}\\ 400&=16a\\ a&=25 \end{align}$$

Finally, we can plug in $a$ and $b$. $$y=25e^{\frac{x}{2}\ln{4}}$$

Now, we can find any size of the bacteria population given any time!

Exponential Growth

We see exponential growth in a lot of different industries, including biology, economics, and math!

We use the following formula to calculate exponential growth:

$$y=a(1+r)^x$$

, where $a$ represents the initial amount, $r$ represents the growth rate, and $x$ represents the time.