Numworks/Casio MicroPython/Python: Calculus
Introduction
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 *
# 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
Eddie
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
Introduction
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 *
# 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
Eddie
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
