3D System — Extreme Points z=f(x)

Finding Extreme Points, again:)— #PySeries#Episode 25

Hi, we are going to solve some exercises in Python, involving differential and integral to one variable.

We are going to use Python and Geogebra.

Welcome!

01 — Given the function z to respect to x, determine the extreme points:

Let’s get down to work! Take the Diferentiation of z to the respect of x:

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import sympy as sy
x= sy.symbols('x')
Z = x**3 - 2*x**2 - x + 2
Zx = sy.diff(Z, x)
print(f'Zx = {Zx}')
Zx = 3*x**2 - 4*x - 1

Equations

Z=x^3−2x^2−x+2

Zx=3x^2−4x−1=0

Roots of a Function

import numpy as np
c=[3,-4,-1]
np.roots(c)
array([ 1.54858377, -0.21525044])

So the roots are x=1.55;y=−0.21

Here are The 2 Equations (y1, y2):

x, y = sy.symbols('x y')
y1 = x**3 - 2*x**2 - x + 2
y2 = 3*x**2 - 4*x -  1

The Extreme points is (1.55, — 0.21) represent the Minimum

Plotting a polynomial in Geogebra

Equations:

Z=x^3−2x^2−x+2

Zx=3x^2−4x−1=0

Points:

A=(1.54858377,−0.21525044)

C=(1.54858377,−0.21525044,−0.44)

See the above Graph in Geogebra and Youtube Video; Please, Watch the video to understand the three-dimensional figure.

You notice that the point found was on the xy plane. If you project the point (A) along the z-axis, you will see that the location of the point fits perfectly in 3D figure (C).

02 — Given the function z to respect to x, determine the extreme points:

Let’s get down to work! Take the Diferentiation of z to the respect of x

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import sympy as sy
x= sy.symbols('x')
Z = x**3 - (27/2)*(x**2)
Zx = sy.diff(Z, x)
print(f'Zx = {Zx}')
Zx = 3*x**2 - 27.0*x

Equations

Z=x^3−272x^2

Zx=3x^2−27x

Roots of a Function

from sympy import solve
solve(3*x**2 - 27.0*x)
[0.0, 9.00000000000000]

Interpretation — Roots:

Ifx=0↦f(0)=0↦A(0,0)

Ifx=9↦f(9)=93−272∗92=−729/2↦A(9,−729/2)

To classify the critical points, we need to calculate the second order derivative of the function:

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import sympy as sy
x, y = sy.symbols('x y')
Z = Z = x**3 - (27/2)*(x**2)
Zx = sy.diff(Z, x)
Zxx = sy.diff(Zx, x)
print(f'Zx = {Zx}\nZxx = {Zxx}')
Zx = 3*x**2 - 27.0*x Zxx = 6*x - 27.0

Interpretation — Extreme Points:

ForA(0,0)↦Zxx=6∗0−27↦Zxx<0 Them MAX

ForA(9,−364.5)↦Zxx=6∗9−27↦Zxx>0 Them MIN

See these Graph in Geogebra and Youtube Video; Please, Watch the video to understand the three-dimensional figure

You notice that the point found was on the xy plane. If you project the point (A) along the z-axis, you will see that the location of the point fits perfectly in 3D figure (C).

The lecture is over here. Python is Powerful!

Thank you very much!

print("There you have it! Thank you very much! Python is awesome, doesn't it?")
There you have it! Thank you very much! Python is awesome, doesn't it?

That’s All for this lecture!

See you in the next Python Episode!

See answer below!

Bye!!!!

Colab File link:)

Google Drive link:)

Geogebra Solution link and link:)

Video link:)

Credits & References

Geogebra Solution by J3

Youtube vid by J3

Decimal to Fraction Calculator by calculatorsoup.com

Exercícios Resolvidos Assunto: Integral Dupla (pdf) by Universidade Federal Fluminense (Brazil — Niterói — RJ)

Welcome to Calculus with Python’s documentation! — Calculus with Python by calc-again.readthedocs.io

LaTeX/Mathematics by wikibooks.org

Eddie Woo Vids by Eddie Woo

Taking Derivatives in Python by Dario Radečić

Plotting a polynomial in Python by stackoverflow.com

Three-Dimensional Plotting in Matplotlib by jakevdp.github.io

Finding Extreme Points by https://www.dataquest.io/m/159-finding-extreme-points/

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