(Casio Micropython/Numworks Python): Calculus

[Eddie W. Shore]

The following scripts creates the user functions for calculus:

f(x): define your function in terms of x here. This needs to be loaded into the script before running it. Each of the functions that follow will use f(x). You can call f(x) to evaluate the function at any value.

deriv(x): The approximate derivative at point x. The Five Stencil approximation is used.

sigma(a,b): Calculate the sum (Σ f(x)) from x = a to x = b.

integral(a,b,n): Calculates the definite integral ( ∫ f(x) dx) from x = a to x = b. The Simpson's rule is used with n divisions (n needs to be even)

solve(x0): Uses Newton's Rule to find roots for f(x).

Example

f(x) = -2x*^2 + 3x + 5
In Python: -2*x**2+3*x+5

f(0): 5
f(10): -165
f(-10): -225

deriv(10): -37.00004450971999

Σ f(x): x = 1 to 25: sigma(1,25): -8780

∫ f(x) dx: x = -3 to 1, n = 20: integral(-3,1,20): -10.666666666667

Solve f(x)=0, initial condition x0 = 2.5: solve(5): 2.5

Python Script: calculus.py

from math import *

Code:

# 2020-04-15 EWS

# define f(x) here
def f(x):
  return -2*x**2+3*x+5
  
# derivative
def deriv(x):
  # uses f(x), 5 stencil
  # h is tolerance
  h=1e-10
  d=(12*h)**-1*(f(x-2*h)-8*f(x-h)+8*f(x+h)-f(x+2*h))
  return d

# sum/sigma
def sigma(a,b):
  t=0
  n=b-a
  for i in range(n):
    t=t+f(i+1)
  return t

# integral by simpsons rule
def integral(a,b,n):
  t=f(a)+f(b)
  h=(b-a)/n
  for i in range(n-1):
    w=(i+1)/2
    if (w-int(w))==0:
      t=t+2*f(a+(i+1)*h)
    else:
      t=t+4*f(a+(i+1)*h)
  t=t*h/3
  return t

# solver
def solve(x0):
  tol=1e-14
  x1=x0-f(x0)/deriv(x0)
  while abs(x1-x0)>tol:
    x0=x1
    x1=x0-f(x0)/deriv(x0)
  return x1