[R] R을 활용한 미분과 적분 계산

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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

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

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