\(\(微分積分と近似計算

        1\ \ 極限値

        2\ \ 微分

        3\ \ 積分

        4\ \ ニュ\) ートン法\)\(5\ \ シンプソンの公式\)

\(1\ \ 極限値\)

Limit[(((x^3) - (a^3))/(x - a)), x -> a]

Limit[(((Sqrt[1 + x]) - 1)/x), x -> 0]

Limit[(((((x + 1)/(x - 1))))^x), x -> ∞]

\(2\ \ 微分\)

D[(x + 1) ((x^2) - 2  x + 3), x] // Expand

D[Log[x + (Sqrt[(x^2) + A])], x] // Simplify

D[Exp[-a x] Sin[b x], x] // Factor

\(3\ \ 積分\)

Integrate[Cos[x], x]

Integrate[(1/((1 - (x^2)) (Sqrt[1 + (x^2)]))), x] // Simplify

Integrate[Cos[x]^10, {x, 0, (π/2)}]

Integrate[Sin[x]/x, {x, 0, ∞}]

\(\(4\ \ ニュ\) ートン法\)\(\ \ \ \ \ \ \ 
    f \((x)\) = 
      \(0\ の解\ x = 
        \(c\ の近似値として\ x = \ 
          x\_0\ をとる\)\)\)      
    (Subscript[x, n + 1]) = 
      (Subscript[x, n]) - ((f ((Subscript[x, n])))/(f' ((Subscript[x, n]))))   
          (n = 0, 1, 2,  ... )\(\ \ \ \ 
    適当な条件の下で数列\ {x\_n}\ 
      は\ c\ に収束する
\)              
    (x^3) = 2                     cos (x) = x

\(f[x_] := x\^3 - 2; 
df[x_] := D[f[x], x]\)g[x_] = Simplify[x - f[x]/df[x]]\(x := 1.5; 
eps := 10^\((\(-10\))\); 

    While[Abs[g[x] - x] > eps, 
	\ \ \ x = g[x]]\)N[g[x], 10]

Plot[Cos[x] - x, {x, 0, 1}]

\(f[x_] := Cos[x] - x; 
df[x_] := D[f[x], x]\)g[x_] = Simplify[x - f[x]/df[x]]\(x := 0.7; 
eps := 10^\((\(-10\))\); 

    While[Abs[g[x] - x] > eps, 
	\ \ \ x = g[x]]\)N[g[x], 10]

\(5\ \ シンプソンの公式\)         
    (∫_0^1) (4/(1 + (x^2))) 
        \[DifferentialD]x
                                        
        (∫_0^1) (1/(Sqrt[2  π])) 
            (E^(-((x^2)/2))) \[DifferentialD]x

f[x_] := (4/(1 + (x^2)))\(a = 0; 
\ b = 1; 
\ n = 10; 
h = \(b - a\)\/\(2\ n\); 

    S = \(h\/3\) 
          \((f[a] + 4\ Sum[f[a + \((2\ i - 1)\) h], {i, 1, n}] + 
              2\ Sum[f[a + 2\ i\ h], {i, 1, n}] + f[b])\) // N\)

f[x_] := (1/(Sqrt[2  π])) (E^(-((x^2)/2)))\(a = 0; 
\ b = 1; 
\ n = 10; 
h = \(b - a\)\/\(2\ n\); 

    S = \(h\/3\) 
          \((f[a] + 4\ Sum[f[a + \((2\ i - 1)\) h], {i, 1, n}] + 
              2\ Sum[f[a + 2\ i\ h], {i, 1, n}] + f[b])\) // N\)