From 18eb6dbfa545cdbb69103e312ccb0992d1b20b00 Mon Sep 17 00:00:00 2001
From: Christian Kolset <christian.kolset@gmail.com>
Date: Fri, 12 Sep 2025 17:24:40 -0600
Subject: Worked on module 3

---
 tutorials/module_3/1_numerical_differentiation.md | 41 ++++++++++++++++-------
 1 file changed, 29 insertions(+), 12 deletions(-)

(limited to 'tutorials/module_3/1_numerical_differentiation.md')

diff --git a/tutorials/module_3/1_numerical_differentiation.md b/tutorials/module_3/1_numerical_differentiation.md
index aa45c3a..68e0953 100644
--- a/tutorials/module_3/1_numerical_differentiation.md
+++ b/tutorials/module_3/1_numerical_differentiation.md
@@ -11,22 +11,30 @@ s = 34 * np.exp(3 * t)
 
 Then we can use the following methods to approximate the definitive derivatives as follows.
 
-## Forward Difference
+## Finite Difference
+### Forward Difference
 Uses the point at which we want to find the derivative and a point forwards in the array.
 $$
 f'(x_i) = \frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i}
 $$
 *Note: If we apply this to an array, consider what happens at the last point.*
 
+```python
+for s in t:
+	ds_dt = f(x+)
+```
+
+
+
 ```python
 # Forward difference using python arrays
-dsdt =  (y[1:] - y[:-1]) / (x[1:] - x[:-1])
+dsdt =  (s[1:] - s[:-1]) / (t[1:] - t[:-1])
 
 import matplotlib.pyplot as plt
 
 # Plot the function
-plt.plot(x, s, label=r'$y(x)$')
-plt.plot(x, dsdt, label=b'$/frac{ds}{dt}$')
+plt.plot(t, s, label=r'$s(t)$')
+plt.plot(t, dsdt, label=b'$/frac{ds}{dt}$')
 plt.xlabel('Time (t)')
 plt.ylabel('Displacement (s)')
 plt.title('Plot of $34e^{3t}$')
@@ -36,7 +44,7 @@ plt.show()
 ```
 
 
-## Backwards Difference
+### Backwards Difference
 Uses the point at which we want to find and the previous point in the array.
 $$
 f'(x_i) = \frac{f(x_{i})-f(x_{i-1})}{x_i - x_{i-1}}
@@ -44,23 +52,32 @@ $$
 
 
 ```python
-dsdt =  (y[1:] - y[:-1]) / (x[1:] - x[:-1])
+dsdt =  (s[1:] - s[:-1]) / (t[1:] - t[:-1])
 
 # Plot the function
-plt.plot(x, y, label=r'$y(x)$')
-plt.plot(x, dydx, label=b'$/frac{ds}{dt}$')
-plt.xlabel('x')
-plt.ylabel('y')
-plt.title('Plot of $34e^{3x}$')
+plt.plot(t, s, label=r'$s(t)$')
+plt.plot(t, dydx, label=b'$/frac{ds}{dt}$')
+
+plt.xlabel('Time (t)')
+plt.ylabel('Displacement (s)')
+plt.title('Plot of $34e^{3t}$')
 plt.grid(True)
 plt.legend()
 plt.show()
 ```
-## Central Difference
+Try plotting both forward and backwards
+
+### Central Difference
 
 $$
 f'(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}}
 $$
 
+---
+# Advanced Derivatives
 
+## High-Accuracy Differentiation Formulas
+## Richardson Extrapolation
+## Derivative of Unequally Spaced Data
+## Partial Derivatives
 
-- 
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