Added heuns method

1 files changed, 20 insertions, 2 deletions

diff --git a/tutorials/module_3/1_numerical_differentiation.md b/tutorials/module_3/1_numerical_differentiation.md
index 0926851..441f838 100644
--- a/tutorials/module_3/1_numerical_differentiation.md
+++ b/tutorials/module_3/1_numerical_differentiation.md
@@ -45,7 +45,7 @@ plt.show()
 
 
 ### Backwards Difference
-Uses the point at which we want to find and the previous point in the array.
+Uses the point at which we want to find the derivative and the previous point in the array.
 $$
 f'(x_i) = \frac{f(x_{i})-f(x_{i-1})}{x_i - x_{i-1}}
 $$
@@ -68,10 +68,28 @@ plt.show()
 Try plotting both forward and backwards
 
 ### Central Difference
-
 $$
 f'(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}}
 $$
+### Problem 1
+Use the forward difference formula to approximate the derivative of $f(x)$ at $x = 1$ using step sizes: $h=0.5$ and $h=0.1$ for the following function.
+$$
+f(x) = \ln(x^2 + 1)
+$$
+Compare your results with the analytical solution at $x=1$. Comment on how the choice of $h$ affects the accuracy.
+```python
+
+```
+### Problem 2
+Use the central difference formula to approximate the derivative of $f(x)$ at $x = 1.2$ using step sizes: $h=0.5$ and $h=0.1$ for the following function.
+$$
+f(x) = e^{-x^2}
+$$
+Compare your results with the analytical solution at $x=1.2$. Comment on how the choice of $h$ affects the accuracy.
+```python
+
+```
+
 
 ---
 # Advanced Derivatives