This week we go over models which use the exponential and logarithms and then start in with trigonometry. The second test will cover the material through 3.5 (There is a typo in the weekly syllabus that I just corrected)
We now have the ability to manipulate equations involving exponential and logarithmic functions. Now lets put them to work. The reason we learn these functions is that they appear in nature. We saw this with compounded interest:
If we compound monthly we get the formula for the principal after $n$ months.
$$P_n = P_0(1 + \frac{r}{12})^{n}$$
Translating to years we get then
$$P_t = P_0(1+\frac{r}{12})^{12t}$$
This allows us to change the compounding period. If we do it 365 times per year the formula becomes
$$P_t = P_0(1+\frac{r}{365})^{365t}$$
If we do it a 1000 times per year we get
$$P_t = P_0(1+\frac{r}{1000})^{1000t}$$
And if we constantly compound, then this expression becomes simply
$$P_t = P_0 e^{rt}$$
The point being that money naturally has the exponential functions appearing. The reason is that the amount you earn in interest is proportional to the amount you have in the bank at a given time.
Now other models for nature use exponentials and logs. The examples in the book list
Just like interest only applied to things such as population. We get the population at time $t$ is given by
$$P_t = P_0 e^{rt}$$
where $r>0$ is a parameter that changes how fast things grow. If r is large they grow fast. If r small not very fast. (in some countries $r<0$ and we don’t grow but decay.)
Problems here are how to find $r$? This can often be figured out from a doubling rate. Suppose all you know is that it take 30 years for the population to double. This tells you $r$. How???? Let’s write down an equation and solve. Start with the original equation and put in all you know:
$$P_{30} = 2P_0 = P_0 e^{r30}$$
Now solve, first divide by $P_0$ – it appears on both sides to get
$$2 = e^{r30}$$
Then take the natural log and divide by 30:
$$\ln(2) = 30r$$
giving $r = \ln(2)/30 = .0231$. If it took 100 years you would get $r = \ln(2)/100$, etc.
This is the case when $r<0$. The graph declines. The formula looks the same:
$$P_t = P_0 e^{rt}$$
but when $r<0$ the graph decays. This models decaying population, decay of nuclear isotopes.
Now, there are refinements to these that are more realistic. For example, eventually a population stops growing so fast and saturates. This is handled with the logistic equation. As well, exponential decay doesn’t always decay to 0. For example, the coffee I am drinking while typing this up, won’t cool to 0 degrees but rather room temperature about 72 degrees. A model for this would look like
$$T_t = T_{room} + (T_0-T_{room})e^{-rt}$$
There $T_t$ is temperature at time $t$. $T_0$ initial temperature and $T_{room}$ the room temperature.
Unfortunately, these don’t appear in the homework.
After exponentials and logarithms, comes trigonometry. Believe it or not, this will be our main topic for the rest of the semester. What do we cover? New functions that are related to circles and oscillating behavior. Think about a ferris wheel spinning around. A person moves in a circular manner. However, what function describes their height as a function of time. Notice they move up and down and up and down. We don’t see this in the graphs of exponentials or logs which move just up or just down. We don’t see this in the polynomials we studied which eventually move off to $\infty$ or $-\infty$. We need new functions to describe this movement. These will be the $\sin(x)$ and $\cos(x)$ functions. They are related to circles and triangles.
First though, we begin with a discussion about angles. This is the material in section 4.1. It is very important that we understand what this section is about. Without it, the rest of the chapter is a complete loss.
Here are the questions we need to understand
As defined, we have two rays one called an initial side the other a terminal side. The angle between them is what we describe. Often we write the initial side as the positive $x$ axis, then the terminal side makes an angle. Angles can be both positive and negative, and in particular they picture can repeat. That is the angle $400^\circ$ will look like $40^\circ$
You are probably familiar with the concept of degrees. 90 of them is a right angle, 180 of them is a straight line, there are 360 in a circle. They are denoted with a degree symbol: $360^\circ$ for example.
In college classes, we also learn about radians. Why? They are related to distances unlike degrees and so have a physical meaning. (We can multiply by them and keep the proper units). The key figure is figure 4.6 in the book A circle of radius $r$ has an angle $\theta$ drawn on it. The angle in radians is the same as the arclegnth divided by $r$. So if you could measure the length of the curve piece you can tell me the angle measurement. (How do you find the degree measurement?)
They both say the same thing, we should be able to convert. The simple way to figure this out is that a circle has $360^\circ$ and $2\pi$ radians. So we need to multiply by this ratio. For example, $125^\circ$ is
$$125^\circ \frac{2\pi}{260^\circ} = \frac{125\pi}{180} = 2.1817$$
and $\pi/8$ radians is
$$\frac{\pi}{8} \frac{360^\circ}{2\pi} = \frac{360^\circ}{16} = 22.5^\circ$$
(Notice, I could have used any equivalent ratio. You may have learned to use
$$\frac{180^\circ}{\pi} or \frac{\pi}{180^\circ}.$$
They give you the same thing, don’t be confused about this.
If you have question with the homework, please ask them. This material is supposed to be basic. If it is not, you will be in big trouble this chapter.