Exponential Models

class: center, middle, inverse, title-slide

.title[
# Exponential Models
]
.subtitle[
## Session 07
]
.author[
### Alejandro Ucan
]
.date[
### 2023-10-08
]

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# Goals:

* Describe cases where the growth/decay is exponential. 
* Describe the criterium to decide if a model is exponential. 
* Describe the process to find an exponential model. 
* Use the exponential regression to predict values.

---
# Exponential behavior

> Sometimes our data shows a behavior where it grows or decays everytime by the same factor. This growth/decay of our data cannot be controlled by us, it is just the nature of the data. When this happens we say that our data has an **exponential behavior**. In this case, we can use an **exponential model** to describe our data.

--
 
#### Example

Consider the following data: 
| x | y |
|:-----------:|:------------:|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | ? |

---
# The exponential model:

> __Definition:__ The exponential function is a function of the form `$f(x)=ab^x$`, where `$a$` and `$b$` are constants and `$b>0$`. The relation between the dependent and independent variables is that the dependent one grows/decays by a factor of `$b$` everytime the independent variable increases by one unit.

--
<img src="index_files/figure-html/unnamed-chunk-1-1.png" width="100%" />
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# Criterium

> Assume that we have some real world data, and we want to know if it has an exponential behavior. We can use the following criterium to decide if our data has an exponential behavior: 
If three observations (with a unit of distance), then the ratio of the dependent variables is the same.

---
#### Example

Consider the following data: 
| `$x$` | `$y$` |
|:-----------:|:------------:|
| 1 | 5 |
| 2 | 10 |
| 3 | 20 |
| 4 | 40 |
| `$n$` | ? |

---
# Finding the exponential model

> To find the exponential model we need to find the values of `$a$` and `$b$` in the function `$f(x)=ab^x$`. To do this we need to use the following system of equations: 
`$$\begin{cases} f(x_1)=ab^{x_1} \\ f(x_2)=ab^{x_2} \end{cases}$$`

---
#### Example

> The population of a city in 2010 was 100,000 people. If the rate of change of the population is 2.5% per year, find the exponential model that describes the population of the city.

--
 
#### Solution

We know that the population of the city in 2010 was 100,000 people, so we have the point `$(0,100,000)$`. We also know that the rate of change of the population is 2.5% per year, so we have the point `$(1,102,500)$`. Therefore, we have the following system of equations: 
`$$\begin{cases} f(0)=ab^0=100,000 \\ f(1)=ab^1=102,500 \end{cases}$$`

---
#### Example

> The INEGI reported that in the month June 2019. The INPC (National Consumer Price Index) registered a montly increase of 0.06%. Assuming that in the following months, this index follows the same behavior. Find the exponential model that describes the INPC if at June this index was 100.00.

--
 
#### Solution

We know that the INPC in June 2019 was 100.00, so we have the point `$(0,100.00)$`. We also know that the INPC in July 2019 was 100.06, so we have the point `$(1,100.06)$`. Therefore, we have the following system of equations:

`$$\begin{cases} f(0)=ab^0=100.00 \\ f(1)=ab^1=100.06 \end{cases}$$`

---
## Applications to business

> The compound interest is an example of an exponential model. The compound interest is the interest that is added to the principal of a deposit or loan so that the added interest also earns interest from then on. The formula to calculate the compound interest is: 
`$$A=P\left(1+\frac{r}{n}\right)^{nt}$$` where `$A$` is the amount of money accumulated after `$t$` years, `$P$` is the principal, `$r$` is the annual interest rate, and `$n$` is the number of times the interest is compounded per year.

---
#### Example

> A person deposits $\$1000$ in a bank account that pays 5% annual interest, compounded monthly. Find the amount of money accumulated after 10 years.

--
 
#### Solution

We know that the principal is $\$1000$, the annual interest rate is 5%, and the number of times the interest is compounded per year is 12. Therefore, we have the following information: 
`$$\begin{cases} P=1000 \\ r=0.05 \\ n=12 \\ t=10 \end{cases}$$`

---
#### Example

> A person deposits $\$1500$ in a bank account that pays 11.6% annual interest, compounded three-monthly. Find the amount of money accumulated after 5 years.

---
## Base e exponential model

> Suppose that you invest $\$1$ in a bank account that pays 100% annual interest, compounded `$n$` times a year. The amount of money accumulated after `$t$` years is given by the formula: `$$A=\left(1+\frac{1}{n}\right)^{nt}$$`

What happend if we increase the number of compounded times a year? Well, the number `$(1+\frac{1}{n})$` will be closer to a number `$e$`.

---
## Base e exponential model

> The number `$e$` is a number that is approximately equal to `$2.7182818284590452353602874713527$`. The number `$e$` is a very important number in mathematics, because it is the base of the natural logarithm. The number `$e$` is also the base of the exponential model that is used in many applications.