[R] R을 활용한 미분과 적분 계산

1. Mathematical functions
2. Differentiation
3. Calculate derivatives using R function
- 3. 1. D function
- 3. 2. Deriv packages
4. Calculate integration using R function
- 4. 1. integrate() function
5. Exercise

이번 포스팅은 제가 학교에서 진행했던 수업자료의 일부로 R을 활용하여 미분과 적분을 하는 방법에 대해서 소개해드리겠습니다.

1. Mathematical functions

Anything to the power zero is 1 :
$x^{0} = 1$
One raised to any power is still 1 :
$1^{x} = 1$
Infinity plus 1 is infinity :
$\infty + 1 = \infty$
One over infinity (the reciprocal of infinity, $\infty^{-1}$ ) is zero :
$\frac{1}{\infty} = 0$
A fraction raised to the power infinity is zero :
$0.99^{\infty} = 0$
Negative power are reciprocals :
$x^{-b} = \frac{1}{x^{b}}$
Functional powers are roots :
$x^{1/3} = \sqrt[3]{x}$
The base of natural logarithms $e$ is 2.71828, so :
$e^{\infty} = \infty$

1. 1. Logarithmic functions, Exponential functions

$y = a \ln (bx)$

$y = a e^{bx}$

For example, $a=1$ and $b=1$

x = seq(from = 0, to = 10, by = 0.1)
y = log(x)
z = exp(x)

par(mfrow = c(1, 2))
plot(x, y, type = "l", main = expression(log(x)), lwd = 2)
grid()

plot(x, z, type = "l", main = expression(e^x), lwd = 2)
grid()

1. 2. Trigonometric function

Here are the sine, cosine and tangent functions of $x$ over the range $0$ to $2\pi$ .
Recall that the full circle is $2 \pi$ radians, so $1$ radian = $360/2\pi$ = $57.29578$ degrees.
$\sin$ function, $\cos$ function, $\tan$ function

x = seq(from = 0, to = 2*pi, length.out = 100)
y = sin(x)
z = cos(x)
w = tan(x)

par(mfrow = c(2, 2))

plot(x, y, type = "l", main = expression(sin(x)), lwd = 2)
abline(v = 0, h = 0, lty = 3)

plot(x, z, type = "l", main = expression(cos(x)), lwd = 2)
abline(v = 0, h = 0, lty = 3)

plot(x, w, type = "l", ylim = c(-3, 3), main = expression(tan(x)), lwd = 2)
abline(v = 0, h = 0, lty = 3)

1. 3. Polynomial functions

$f(x) = c_{0} + c_{1}x + c_{2}x^{2} + ... + c_{n}x^{n}$

$y = 2x^{2} + 5x + 2$

x = seq(from = -10, to = 10, by = 0.1)
y = 2*x^{2} + 5*x + 2

par(mfrow = c(1, 1))
plot(x, y, type = "l", main = expression(2*x^{2} + 5*x + 2), lwd = 2)
grid()

$y = -2x^{2} + 5x + 2$

x = seq(from = -10, to = 10, by = 0.1)
y = -2*x^{2} + 5*x + 2

plot(x, y, type = "l", main = expression(-2*x^{2} + 5*x + 2), lwd = 2)
grid()

$y = 0.04x^{3} - 0.6x^{2} + 4x + 2$

x = seq(from = 0, to = 10, by = 0.1)
y = 0.04*x^{3} - 0.6*x^{2} + 4*x + 2

plot(x, y, type = "l", main = expression(0.04*x^{3} - 0.6*x^{2} + 4*x + 2), lwd = 2)
grid()

$y = 0.04x^{4} - 0.6x^{3} + 2x^{2} + 4x + 2$

x = seq(from = 0, to = 10, by = 0.1)
y = 0.04*x^{4} - 0.6*x^{3} + 2*x^{2} + 4*x + 2

plot(x, y, type = "l", main = expression(0.04*x^{4} - 0.6*x^{3} + 2*x^{2} + 4*x + 2), lwd = 2)
grid()

1. 4. Basic mathematical operator in R

+ : addition
```
## [1] 2
```
```
1 + 1
```
- : subtraction
```
## [1] 1
```
```
10 - 9
```
* : multiplication
```
## [1] 0
```
```
0 * 1000
```
/ : division
```
## [1] 2
```
```
10 / 5
```

log() : logarithm

## [1] 0

log10(10)

## [1] 1

log2(2)

## [1] 1

log(10)

## [1] 2.302585

```
log(1)
```
exp() : exponent
```
## [1] 2.718282
```
```
log(exp(1))
```
```
## [1] 1
```
```
exp(1)
```

^ : power

## [1] 100

10^2+1

## [1] 101

10^{2+1}

## [1] 1000

```
10^2
```
sqrt() : square root
```
## [1] 1
```
```
sqrt(2)
```
```
## [1] 1.414214
```
```
sqrt(1)
```

sin(), cos(), tan() : trigonometrical function

## [1] 0

cos(0)

## [1] 1

tan(45)

## [1] 1.619775

```
sin(0)
```

2. Differentiation

$f^{'}= \frac{d}{dx} (f) = \frac{df}{dx} = \frac{dy}{dx}$

A differentiation is a type of transformation that takes a new function from original function.
A new function created by a differential, called derivative function, represents the slope of the original function.

2. 1. Basic differential formula

Constant :
$\frac{d}{dx}(c)=0$
Power :
$\frac{d}{dx}(x^{n}) = n x^{n-1}$
Log :
$\frac{d}{dx}(\log x)=\frac{1}{x}$
Exponent :
$\frac{d}{dx}(e^{x}) = e^{x}$
Linear combination : $\frac{d}{dx}(f_1 + f_2) = \frac{df_1}{dx} + \frac{df_2}{dx}$ ex) $y = 1 + 2x + 3x^{2} + 4e^{x} + 5\log(x)$
$\frac{dy}{dx} = 2 + 6x + 4e^{x} + \frac{5}{x}$
$\frac{d}{dx}(c_1 f_1 + c_2 f_2) = c_1 \frac{df_1}{dx} + c_2 \frac{df_2}{dx}$
$\frac{d}{dx}(cf) = c \frac{df}{dx}$

2. 2. Multiplication rule for differentiation

$\frac{d}{dx}(f \times g) = \frac{df}{dx} \times g + \frac{dg}{dx} \times f$

When the function is multiplied by two functions, derivative of the function is gained to use the derivative of each individual function.
ex) $f = x e^{x}$ , $\frac{df}{dx} = e^{x} + xe^{x}$

2. 3. Chain rule for differentiation

$f(x) = h(g(x))$

$\frac{df}{dx} = \frac{dh}{dg} \times \frac{dg}{dx}$

For example, the following rules can be obtained by applying chain rule to a logarithmic function :
$\frac{d}{dx} \log{f(x)} = \frac{f^{'}(x)}{f(x)}$

2. 4. Partial differentiation

$f(x, y) = x^{2} + xy + y^{2}$

$f_{x}(x, y) = \frac{\partial f}{\partial x} = 2x + y$

$f_{y}(x, y) = \frac{\partial f}{\partial y} = x + 2y$

3. Calculate derivatives using R function

In this chapter, we will learn two convenient ways of various methods by using R function.

3. 1. `D` function

We calculate the derivative function by using D() function.
STEP1) Define the function to be differentiate by using expression() function.
- $f(x) = 5x^{2} + 10x + 5$
```
f <- expression(5*x^2 + 10*x + 5)
```
STEP2) Use D() function
```
## 5 * (2 * x) + 10
```
- As a result, We obtain $f^{'}(x) = 5 \times (2 \times x) + 10 = 10x + 10$

D(f, "x")                         # 함수 f를 X로 미분

Some complex examples

$f(x) = e^{x} + 10x^{3} + 2x$
```
## exp(x) + 10 * (3 * x^2) + 2
```
- $f^{'}(x) = e^{x} + 30x^{2} + 2$

f <- expression(exp(x) + 10*x^3 + 2*x)
D(f, "x")

$f(x) = \log{x^{3}} + \log{x}$
```
## 3 * x^2/x^3 + 1/x
```
- $f^{'}(x) = \frac{3}{x} + \frac{1}{x}$

f <- expression(log(x^3) + log(x))
D(f, "x")

$f(x) = \log(x^{3} + x + 1) + e^{x^{2}}$
```
## (3 * x^2 + 1)/(x^3 + x + 1) + exp(x^2) * (2 * x)
```
- $f^{'}(x) = \frac{3x^{2} + 1}{x^{3} + x + 1} + 2xe^{x^{2}}$

f <- expression(log(x^3 + x + 1) + exp(x^2))
D(f, "x")

$f(x) = e^{e^{x+1}} + \log{x}$
```
## exp(exp(x + 2)) * exp(x + 2) + 1/x
```
- $f^{'}(x) = e^{e^{x+2}} e^{x+2} + \frac{1}{x}$

f <- expression(exp(exp(x+2)) + log(x))
D(f, "x")

$f(x, y) = e^{e^{y+1}} + \log{x}$
```
## 1/x
```
- $f_{x}^{'}(x, y) = \frac{1}{x}$

f <- expression(exp(exp(y+2)) + log(x))
D(f, "x")

3. 2. `Deriv` packages

This method is immediate and quick but to run Deriv() function, you must install Deriv package.
```
library(Deriv)
```
In Deriv() function, it is similar to D() function but does not need to specify a objective function.

Some examples

$f(x) = \sin(x) + \cos(x)$
```
## cos(x) - sin(x)
```
- $f^{'}(x) = \cos(x) - \sin(x)$
```
Deriv(sin(x) + cos(x), "x")
```
$f(x) = \sin(x) \times \cos(x)$
```
## cos(x)^2 - sin(x)^2
```
- $f^{'}(x) = \cos^{2}(x) - \sin^{2}(x)$
```
Deriv(sin(x)*cos(x), "x")
```
$f(x) = x^{3} + x^{-2} + 1$
```
## 3 * x^2 - 2/x^3
```
- $f^{'}(x) = 3x^{2} - \frac{2}{x^{3}}$
```
Deriv(x^(3) + x^(-2) + 1, "x")
```

4. Calculate integration using R function

The integral of the function is one of the most important computation and is to calculate the extent and volumne of the function.

4. 1. `integrate()` function

We define the formula(objective function) by using function() {expression} function to run integrate() function.
```
## function(x) {x^3 - 7*x^2 + 13*x + 12}
## <environment: 0x000000001606b770>
```

f <- function(x) {x^3 - 7*x^2 + 13*x + 12}
f

Then, use integrate() function. But it must be established lower bound and upper bound.
```
## 50.25 with absolute error < 5.6e-13
```
```
integrate(f, lower = 1, upper = 4)
```

5. Exercise

$f(x) = x^{3} - 1$
$f(x) = \log(x^{2} - 2)$
$f(x) = e^{2x^{3}}$
$f(x) = x^{4} + 3x^{3} - 2x^{2} + 4$

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