L2.Rmd

---
title: "L2"
author: "Xiaoting Chen"
date: "2023-08-08"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(dplyr)
library(ggplot2)
library(gridExtra)
library(scales)
```


## Exponential function

### compound interest
```{r}

```

practice:\
bacteria double in size every 4h. a culture starts with 1000, the population after 10h/24h?
```{r}
bac_growth <- function(initial_p, time) {
  p_i <- initial_p
  t <- time
  p_n <- p_i * 2^(t/4)
  p_n
}

bac_growth(1000, 10)
bac_growth(1000, 24)
```

### the number e

```{r}

```



## Logarithmic function

LSE
```{r}
LSE <- function(x) {
  c <- max(x)
  y <- c + log(sum(exp(x - c)))
  y
}

X <- c(1e3, 1e3 + 1, 1e3 + 2)
LSE(X)

```

softmax_log
```{r}
softmax_log <- function(x) {
  x_new <- x - LSE(x)
  x_new
}
softmax_log(X)
sum(exp(softmax_log(X)))
```


## derivatives

```{r}
n <- 100
t <- seq(0, pi, length = n)
x <- sin(t) * cos(t)
dxdt <- cos(t)^2 - sin(t)^2  # derived derivative

d <- tibble(t, x, dxdt)
p <- ggplot(d, aes(t, x))
p + geom_line() + geom_line(aes(y = dxdt), col = 'red')
```

approximate the derivative
```{r}
s <- pi / (n - 1)        # choose the same step s as the increment in the t sequence
all.equal(s, diff(t)[1]) # check that the above statement is true

appr_dxdt <- diff(x)/s   # derivative ≈ rise / run
d <- d %>%
  mutate(appr_dxdt = c(NA, appr_dxdt))
p <- ggplot(d, aes(t, x))
p + geom_line() + geom_line(aes(y = dxdt), col = 'red') +
  geom_line(aes(y = appr_dxdt), col = 'blue', size = 5, alpha = 1/5) 
# the approximated derivative value (blue band) cover the true value (red line)
```