Definition

The Taylor series of a generic function \(f(x)\) in the point \(x = x_0\) is the following power series: \[ f(x_0) + \frac{f^{'}(x_0)}{1!} (x - x_0) + \frac{f^{''}(x_0)}{2!} (x - x_0)^2 + \frac{f^{'''}(x_0)}{3!} (x - x_0)^3 + \ldots \]

Taylor series for the square function

Let consider the function: \[f(x) = x^2\]

Suppose now to be interested in the Mac-Laurin approximation of order 0, 1, 2 and 3.

For our function \(f(x) = x^2\), only the first derivative is different from zero: - \(f^'(x) = 2\) The subsequent derivates are indeed all null: - \(f^{n}(x) = 0, n \ge 2\)

Let us define an R-function for each approximation:

#approximation of order 0
x2order0 <- function(x, x0=0) x0^2
#approximation of order 1
x2order1 <- function(x, x0=0) x0^2 + (x - x0)*2*x0
#approximation of order 2
x2order2 <- function(x, x0=0) 
  x0^2 + (x - x0)*2*x0 + (x - x0)^2 * 2 / factorial(2)
#since the derivates, starting from the order 3, are null
#we can avoid to define the higher order approximations

Note 1: in defining the components of the Taylor’s series, the default value 0 has been used for the point in which the function is approximated. This is the MacLaurin approximation, i.e. the Taylor series evaluated in the point \(x_0 = 0\).

Note 2: the following alternative solution exploits the lower order approximations in computing the higher order approximations:

x2order0 <- function(x, x0=0) x0^2
x2order1 <- function(x, x0=0) x2order0(x, x0) + (x - x0)*2*x0
x2order2 <- function(x, x0=0) 
  x2order0(x, x0) + x2order1(x, x0) + (x - x0)^2 * 2 / factorial(2)
#prova di chiamata dell'approssimazione di ordine 2 in due punti usando x0=1
x2order0(0)
[1] 0
x2order0(10)
[1] 0
#prova di chiamata della funzione per l'approssimazione di ordine 2 in due punti
x2order0(c(0, 10))
[1] 0
#per fare valutare la funzione in ciascun valore del vettore in input è necessario
#utilizzare la funzione Vectorize
x2order0vec <- Vectorize(x2order0, vectorize.args="x")
x2order0vec(c(0, 10))
[1] 0 0
#nell'approssimazione di ordine 0 è necessario usare la funzione Vectorize
#poichè la funzione è costante e non dipende da x
#non è necessario per le approssimazioni di ordine superiore
x2order1(c(0, 10))
[1] 0 0
x2order2(c(0, 10))
[1]   0 100
#preparo un vettore di interi sequenziali da usare come ascisse
ascisse <- -100:100
#rappresento la funzione da approssimare usando una linea nera
#il parametro main permette di impostare il titolo
#impostando il parametro type="l" ottengo una linea
#la funzione expression permette di utilizzare espressioni matematiche
plot(ascisse, ascisse^2, type="l", col="black", xlab="x", ylab="f(x)",
     main=expression(paste("f(x)=",x^2)))
#l'approssimazione di ordine 0 usando una linea rossa
lines(ascisse, x2order0vec(ascisse), col="red")
#l'approssimazione di ordine 1 usando una linea blue
lines(ascisse, x2order1(ascisse), col="blue")
#l'approssimazione di ordine 2 usando una linea verde tratteggiata (argomento 2)
#con uno spessore maggiore (argomento lwd)
lines(ascisse, x2order2(ascisse), col="green", lwd=5, lty=2)

#cambiando il punto di valutazione dell'approssimazione (argomento x0) si ottiene:
plot(ascisse, ascisse^2, type="l", col="black", xlab="x", ylab="f(x)",
     main=expression(paste("f(x)=",x^2)))     
lines(ascisse, x2order0vec(ascisse, x0=10), col="red")
lines(ascisse, x2order1(ascisse, x0=10), col="blue")
lines(ascisse, x2order2(ascisse, x0=10), col="green", lwd=5, lty=2)

Taylor series for the exponential function

Let consider now the exponential function: \[f(x) = exp(x)\] This is a particular function, since all the derivates are equal to the function itself: \[f^{(i)}(x) = exp(x), \forall i \ge 1\]

For such a function it is interesting to consider the Taylor series using a different number of components of such a series.

Let define an R-function for the first five subsequent terms of the Taylor series:

#Taylor approximation of order 0
expXorder0 <- function(x, x0=0) exp(x0)
#Taylor approximation of order 1
expXorder1 <- function(x, x0=0) exp(x0) + (x - x0)*exp(x0)
#Taylor approximation of order 2
expXorder2 <- function(x, x0=0) 
  exp(x0) + (x - x0)*exp(x0) + (x - x0)^2 * exp(x0) / factorial(2)
#Taylor approximation of order 3
expXorder3 <- function(x, x0=0) 
  exp(x0) + (x - x0)*exp(x0) + 
  (x - x0)^2 * exp(x0) / factorial(2) +
  (x - x0)^3 * exp(x0) / factorial(3)
#Taylor approximation of order 4
expXorder4 <- function(x, x0=0) 
  exp(x0) + (x - x0)*exp(x0) + 
  (x - x0)^2 * exp(x0) / factorial(2) +
  (x - x0)^3 * exp(x0) / factorial(3) +   
  (x - x0)^4 * exp(x0) / factorial(4)
#Taylor approximation of order 5
expXorder5 <- function(x, x0=0) 
  exp(x0) + (x - x0)*exp(x0) + 
  (x - x0)^2 * exp(x0) / factorial(2) +
  (x - x0)^3 * exp(x0) / factorial(3) + 
  (x - x0)^4 * exp(x0) / factorial(4) + 
  (x - x0)^5 * exp(x0) / factorial(5)

Since the first component is constant, we need to exploit the Vectorize function so to have a return value for each component of input vector:

expXorder0vec <- Vectorize(expXorder0, vectorize.args="x")

In order to plot the original function as well as its subsequent approximations, it is useful to define a sequence of integers:

abscissa <- 0:5

The following plot depicts the original function (solid black line) along with the first 5 approximation of the Taylor series (lines of different colors):

#rappresento la funzione da approssimare usando una linea nera
#NOTA: la funzione expressione permette di utilizzare formule sulle
#      rappresentazioni grafiche
#      Per una carrellata delle opzioni disponibili consultare
#      demo(plotmath) e example(plotmath)
plot(abscissa, exp(abscissa), 
     type="l", col="black", xlab="x", ylab="f(x)",
     main=expression(paste("f(x)=",e^x)))
#l'approssimazione di ordine 0 usando una linea rossa
lines(abscissa, expXorder0vec(abscissa), col="red")
#l'approssimazione di ordine 1 usando una linea blue
lines(abscissa, expXorder1(abscissa), col="blue")
#l'approssimazione di ordine 2 usando una linea verde
lines(abscissa, expXorder2(abscissa), col="green")
#l'approssimazione di ordine 3 usando una linea magenta
lines(abscissa, expXorder3(abscissa), col="magenta")
#l'approssimazione di ordine 4 usando una linea marrone
lines(abscissa, expXorder4(abscissa), col="brown")
#l'approssimazione di ordine 5 usando una linea arancione
lines(abscissa, expXorder5(abscissa), col="orange")

What happens if we change the value \(x_0\) in which the function is approximated? Here is the answer for the case \(x_0 = 10\):

#cambiando il punto di valutazione dell'approssimazione (argomento x0) si ottiene:
plot(abscissa, exp(abscissa), 
     type="l", col="black", xlab="x", ylab="f(x)",
     main=expression(paste("f(x)=",e^x)))
lines(abscissa, expXorder0vec(abscissa, x0=3), col="red")
lines(abscissa, expXorder1(abscissa, x0=3), col="blue")
lines(abscissa, expXorder2(abscissa, x0=3), col="green")
lines(abscissa, expXorder3(abscissa, x0=3), col="magenta")
lines(abscissa, expXorder4(abscissa, x0=3), col="brown")
lines(abscissa, expXorder5(abscissa, x0=3), col="orange")

---
title: "Taylor series"
output: html_notebook
---

# Definition
The Taylor series of a generic function $f(x)$ in the point $x = x_0$ is the following power series:
$$
f(x_0) + \frac{f^{'}(x_0)}{1!} (x - x_0) + \frac{f^{''}(x_0)}{2!} (x - x_0)^2 + \frac{f^{'''}(x_0)}{3!} (x - x_0)^3 + \ldots
$$

# Taylor series for the square function

Let consider the function:
$$f(x) = x^2$$

Suppose now to be interested in the Mac-Laurin approximation of order 0, 1, 2 and 3.

For our function $f(x) = x^2$, only the first derivative is different from zero:
- $f^'(x) = 2$
The subsequent derivates are indeed all null:
- $f^{n}(x) = 0, n \ge 2$

Let us define an R-function for each approximation:
```{r}
#approximation of order 0
x2order0 <- function(x, x0=0) x0^2
#approximation of order 1
x2order1 <- function(x, x0=0) x0^2 + (x - x0)*2*x0
#approximation of order 2
x2order2 <- function(x, x0=0) 
  x0^2 + (x - x0)*2*x0 + (x - x0)^2 * 2 / factorial(2)
#since the derivates, starting from the order 3, are null
#we can avoid to define the higher order approximations
```
Note 1: in defining the components of the Taylor's series, the default value 0 has been used for the point in which the function is approximated. This is the MacLaurin approximation, i.e. the Taylor series evaluated in the point $x_0 = 0$.

Note 2: the following alternative solution exploits the lower order
approximations in computing the higher order approximations:
```{r}
x2order0 <- function(x, x0=0) x0^2
x2order1 <- function(x, x0=0) x2order0(x, x0) + (x - x0)*2*x0
x2order2 <- function(x, x0=0) 
  x2order0(x, x0) + x2order1(x, x0) + (x - x0)^2 * 2 / factorial(2)
```

```{r}
#function call in two distinct points
x2order0(0)
x2order0(10)
#function call passing the two points in a unique call
x2order0(c(0, 10))
```

```{r}
#the Vectorize function is needed in order to evaluate the function
#separately for each element of the input vector
x2order0vec <- Vectorize(x2order0, vectorize.args="x")
x2order0vec(c(0, 10))
```

```{r}
#the use of the Vectorize function is needed only for the order 0 
#approximation, since the function is costant and does not depend on x
#while it is not needed for higher order approximations
x2order1(c(0, 10))
x2order2(c(0, 10))
```

```{r}
#vector of points in which the function is evaluated
abscissa <- -100:100
```

```{r}
#the square function is plotted using a solid black line
#the main argument is used to set the plot title
#the argument type="l" is used to obtain a line
#the expression function is used to label the plot using formulas
plot(abscissa, abscissa^2, type="l", col="black", xlab="x", ylab="f(x)",
     main=expression(paste("f(x)=",x^2)))
#the order 0 approximation is plotted using a red line
lines(abscissa, x2order0vec(abscissa), col="red")
#the order 1 approximation is plotted using a blue line
lines(abscissa, x2order1(abscissa), col="blue")
#the order 2 approximation is plotted using a dashed green line
#the argument lwd is used to obtain a thicker line
lines(abscissa, x2order2(abscissa), col="green", lwd=5, lty=2)
```

```{r}
#the same plot is now obtained changig the point x0 in which the
#function is approximated
plot(abscissa, abscissa^2, type="l", col="black", xlab="x", ylab="f(x)",
     main=expression(paste("f(x)=",x^2)))     
lines(ascisse, x2order0vec(abscissa, x0=10), col="red")
lines(ascisse, x2order1(abscissa, x0=10), col="blue")
lines(ascisse, x2order2(abscissa, x0=10), col="green", lwd=5, lty=2)
```

## Taylor series for the exponential function
Let consider now the exponential function:
$$f(x) = exp(x)$$
This is a particular function, since all the derivates are equal to the function itself:
$$f^{(i)}(x) = exp(x), \forall i \ge 1$$ 

For such a function it is interesting to consider the Taylor series using a different number of components of such a series.

Let define an R-function for the first five subsequent terms of the Taylor series:

```{r}
#Taylor approximation of order 0
expXorder0 <- function(x, x0=0) exp(x0)
#Taylor approximation of order 1
expXorder1 <- function(x, x0=0) exp(x0) + (x - x0)*exp(x0)
#Taylor approximation of order 2
expXorder2 <- function(x, x0=0) 
  exp(x0) + (x - x0)*exp(x0) + (x - x0)^2 * exp(x0) / factorial(2)
#Taylor approximation of order 3
expXorder3 <- function(x, x0=0) 
  exp(x0) + (x - x0)*exp(x0) + 
  (x - x0)^2 * exp(x0) / factorial(2) +
  (x - x0)^3 * exp(x0) / factorial(3)
#Taylor approximation of order 4
expXorder4 <- function(x, x0=0) 
  exp(x0) + (x - x0)*exp(x0) + 
  (x - x0)^2 * exp(x0) / factorial(2) +
  (x - x0)^3 * exp(x0) / factorial(3) +   
  (x - x0)^4 * exp(x0) / factorial(4)
#Taylor approximation of order 5
expXorder5 <- function(x, x0=0) 
  exp(x0) + (x - x0)*exp(x0) + 
  (x - x0)^2 * exp(x0) / factorial(2) +
  (x - x0)^3 * exp(x0) / factorial(3) + 
  (x - x0)^4 * exp(x0) / factorial(4) + 
  (x - x0)^5 * exp(x0) / factorial(5)
```

Since the first component is constant, we need to exploit the Vectorize function so to have a return value for each component of
input vector:
```{r}
expXorder0vec <- Vectorize(expXorder0, vectorize.args="x")
```

In order to plot the original function as well as its subsequent approximations, it is useful to define a sequence of integers:
```{r}
abscissa <- 0:5
```

The following plot depicts the original function (solid black line) along with the first 5 approximation of the Taylor series (lines of different colors):
```{r}
#solid black line for the function
#NOTE: the expression function allows to use formulas to label
#      the plot
#      See demo(plotmath) and example(plotmath) for a complete gallery
plot(abscissa, exp(abscissa), 
     type="l", col="black", xlab="x", ylab="f(x)",
     main=expression(paste("f(x)=",e^x)))
#red line for the approximation of order 0
lines(abscissa, expXorder0vec(abscissa), col="red")
#blue line for the approximation of order 1
lines(abscissa, expXorder1(abscissa), col="blue")
#green line for the approximation of order 2
lines(abscissa, expXorder2(abscissa), col="green")
#magenta line for the approximation of order 3
lines(abscissa, expXorder3(abscissa), col="magenta")
#brown line for the approximation of order 4
lines(abscissa, expXorder4(abscissa), col="brown")
#orange line for the approximation of order 5
lines(abscissa, expXorder5(abscissa), col="orange")
```

What happens if we change the value $x_0$ in which the function is approximated? Here is the answer for the case $x_0 = 10$:
```{r}
plot(abscissa, exp(abscissa), 
     type="l", col="black", xlab="x", ylab="f(x)",
     main=expression(paste("f(x)=",e^x)))
lines(abscissa, expXorder0vec(abscissa, x0=3), col="red")
lines(abscissa, expXorder1(abscissa, x0=3), col="blue")
lines(abscissa, expXorder2(abscissa, x0=3), col="green")
lines(abscissa, expXorder3(abscissa, x0=3), col="magenta")
lines(abscissa, expXorder4(abscissa, x0=3), col="brown")
lines(abscissa, expXorder5(abscissa, x0=3), col="orange")
```