Newton's Method

This rule will use Newton's Method to find a root of a function.

Clear[Newt]
Newt[x_, f_, fpr_, 1] := N[x - f[x]/fpr[x]]
Newt[x_, f_, fpr_, n_Integer] := 
      Newt[x - f[x]/fpr[x], f, fpr, n - 1]

The format is Newt[init, f, fpr, n], where init is the initial guess, f is the function whose root we are seeking, fpr is the derivative of that function, and n is the number of iterations desired.

Left Endpoint Rule

This rule will numerically integrate a function using the Left Endpoint Rule.

Clear[Lft]
Lft[fctn_, a_, b_, 1] := (b - a)*fctn[a]
Lft[fctn_, a_, b_, n_Integer] := ((b - a)/n)*fctn[a] + 
        N[Lft[fctn, a + (b - a)/n, b, n - 1]]

Euler's Method

Clear[Eul]
Eul[fctn_, y_, {tb_, tf_}, d_] := {} /; 
        tb ≥ tf
Eul[fctn_, y_, {tb_, tf_}, d_] := 
      Prepend[Eul[fctn, y + d*fctn[tb, y], {tb + d, tf}, d],  
          y + d*fctn[tb, y]] /; tb < tf

ListPlot Helper

Clear[h]
h[{x_}, {y_}] := {{x, y}}
h[{x__, u_}, {y__, v_}] := Append[h[{x}, {y}], {u, v}]