.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/finite_differences/plot_02_fermat_adequality.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_calculus_finite_differences_plot_02_fermat_adequality.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_calculus_finite_differences_plot_02_fermat_adequality.py:


Fermat's adequality: maxima from a difference quotient
============================================================

Fermat found the maximum of a function by setting a difference quotient
(f(x+h) - f(x))/h equal to zero and then letting h vanish. This example
locates the maximum of Fermat's own problem, the largest product
x(a - x), and shows the O(h) error of the forward difference.

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.optimize import brentq

    from mathematicskit.calculus import forward_difference








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Fermat's problem: split a segment of length a to maximize x(a - x)
---------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-34

.. code-block:: Python


    a = 10.0


    def area(x):
        return x * (a - x)


    for h in (1.0, 0.1, 0.01):
        x_star = brentq(lambda x: forward_difference(area, x, h=h), 0.0, a)
        print(f"h = {h:5.2f}: difference quotient vanishes at x = {x_star:.4f}")
    print(f"letting h -> 0 gives x = a/2 = {a / 2}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    h =  1.00: difference quotient vanishes at x = 4.5000
    h =  0.10: difference quotient vanishes at x = 4.9500
    h =  0.01: difference quotient vanishes at x = 4.9950
    letting h -> 0 gives x = a/2 = 5.0




.. GENERATED FROM PYTHON SOURCE LINES 35-37

The forward difference is first-order accurate
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-47

.. code-block:: Python


    hs = np.logspace(-8, -1, 30)
    errors = [abs(forward_difference(np.sin, 1.0, h=h) - np.cos(1.0)) for h in hs]
    fig, ax = plt.subplots()
    ax.loglog(hs, errors, "o-", label="forward difference")
    ax.loglog(hs, 0.5 * np.sin(1.0) * hs, "--", label=r"$\frac{h}{2}|f''|$ truncation")
    ax.set_xlabel("step h")
    ax.set_ylabel("error in f'(1) for f = sin")
    ax.legend()
    ax.set_title("Truncation error falls as h, until rounding takes over")



.. image-sg:: /api/gallery/calculus/finite_differences/images/sphx_glr_plot_02_fermat_adequality_001.png
   :alt: Truncation error falls as h, until rounding takes over
   :srcset: /api/gallery/calculus/finite_differences/images/sphx_glr_plot_02_fermat_adequality_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'Truncation error falls as h, until rounding takes over')




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.085 seconds)


.. _sphx_glr_download_api_gallery_calculus_finite_differences_plot_02_fermat_adequality.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_fermat_adequality.ipynb <plot_02_fermat_adequality.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_fermat_adequality.py <plot_02_fermat_adequality.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_fermat_adequality.zip <plot_02_fermat_adequality.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_