Casio fx-CG50 and Python: Directional Derivative for f(x,y)
Introduction
Problem:
Given the following:
* A function of two variables, f(x, y)
* An evaluation point, (x0, y0)
* A directional vector v = < v1, v2 >
Find the directional derivative at the point (x0, y0).
Steps:
1. Determine the gradient of f, ∇f. (∇ is the nabla symbol.) The gradient is a vector of partial derivatives, ∇f = < ∂f / ∂x, ∂f / ∂y >.
2. Evaluate the gradient at the point (x0, y0).
3. Determine the unit vector of the directional vector. Let the norm of | v | = √(v1^2 + v2^2), and the unit vector is determined as u = < v1 / | v |, v2 / | v | > = < u1, u2 >.
4. Take the dot product of the gradient and unit vector: ∇f ⋅ u
The Casio basic code allows for more than one evaluation point.
Casio fx-CG 50 Program: DIRDERIV
Program DIRDERIV
304 bytes
“DIRECTION DERIV.”
Rad
1×10^-4 → H
“NO QUOTES NEEDED”
“F(X,Y)”? → fn1
“DIR VECTOR”? → Vct V
UnitV(Vct V) → Vct U
Lbl 1
“X0”? → R
“Y0”? → S
R + H → X
S → Y
fn1 → A
R – H → X
(A – fn1) ÷ (2 × H) → A
R → X
S + H → Y
(B – fn1) ÷ (2 × H) → B
DotP( [ [ A, B ] ] , Vct U) → D
“[ D _| DX, D _| DIR ] = “ ◢
[ [ A, B, D ] ]
Menu “AGAIN?”, “YES”, 1, “NO” , 0
Lbl 0
“THANK YOU.”
Function memory variables is found through the [ OPTN ] menu. In Run-Matrix Mode, function memory can only be accessed when the Input/Output setup is set at Linear.
Get the _|, the fraction category, by pressing the fraction key, [ []/[] ].
Python Script: dirderv.py
Programmed on the Casio fx-CG50. Only module that is used is the math module. The function f(x,y) is defined in the subroutine in the script.
Code:
from math import *
# define f(x,y) here
def f(x,y):
return x*sin(y) (or any function you want)
h = 0.0001
print(“Directional Deriv.”)
print(“Directional Vector: “)
vx = eval(input(“vx? “))
vy = eval(input(“vy? “))
n = sqrt(vx**2+vy**2)
ux = vx/n
uy = vy/n
print(“Point (x0,y0)”)
x0 = eval(input(“x0? “))
y0 = eval(input(“y0? “))
a = (f(x0+h,y0) – f(x0-h,y0))/(2 * h)
b = (f(x0,y0+h) – f(x0,y0-h))/(2 * h)
d = a*ux + b*uy
print(“dx=”,a)
print(“dy=”,b)
print(“dir.=”,d)
Examples
Example 1:
f(x,y) = x^3 + y^2
at (x0, y0) = (3, 2)
Directional Vector = < 1, 1 >
Norm: √(1^2 + 1^2) = √2
The unit vector: < 1/√2, 1/√2 >
Partial Derivatives:
∂f / ∂x = 3 * x^2, Value = 1
∂f / ∂y = 2 * y, Value = 1
Directional Derivative:
∇f * u ≈ 21.9203122
Example 2:
f(x,y) = x * sin y
at (x0, y0) = (0.5, -0.2)
Directional Vector = < -π, 0 >
Norm: √((-π)^2 + 0^2) = π
The unit vector: < -1, 0 >
Partial Derivatives:
∂f / ∂x = sin(y), Value ≈ -0.1986693308
∂f / ∂y = x * cos(y), Value ≈ 0.4900332881
Directional Derivative:
∇f * u ≈ 0.1986693308
Source
Dawkins, Paul “Section 13.7: Directional Derivatives”. Paul’s Online Notes. Last Modified November 16, 2022. Accessed July 4, 2024. https://tutorial.math.lamar.edu/classes/calciii/directionalderiv.aspx
Eddie
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