Refactor lecture notes: Update objective function notation and remove…

… duplicate content in Lecture0_notes.md

Lectures/0. Intro/Lecture0_notes.md

@@ -5,8 +5,6 @@
 - Deliverables  
 - Expectations and learning outcomes  
 
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 ## What Is Optimization?
 
 In the syllabus, I mentioned:
@@ -20,19 +18,17 @@ Let's unpack what that means. Suppose we have a real-world problem or a goal we
 
 2. **Define an Objective Function**  
    We then formulate a metric to measure how close we are to achieving our goal. If the goal is to minimize cost or error, we could write $\min_x f(x)$. For example, we might minimize 
-   
+
    $$
    f(x) = (x_1 - 100)^2 + (x_2 - 50)^2
-   $$
-
+   $$ 
    to keep the budgets near some target values.
 
 3. **Identify Relevant Constraints**  
    Finally, we specify any bounds or conditions that our choices must satisfy, such as $x_1 + x_2 \leq 150$ to keep total spending below 150, or $x_1, x_2 \ge 0$ to ensure the budgets are nonnegative.
 
 This process—called *optimization modeling*—often involves creativity and iteration. Real-world complexity requires you to rethink which objectives and constraints really matter, and you might consult domain experts to confirm whether your formulation makes sense.
 
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 ## Application Areas and Their Optimization Goals
 

rough_outline.md

@@ -44,7 +44,7 @@
 - Jacobians and the chain rule 
 - Formal gradients of nondifferentiable functions
 
-## You will never differentiate again
+## You will never differentiate by hand again
 ### Formal introduction to auto differentiation
 - Software: 
   - The details: Micrograd (https://github.com/karpathy/micrograd)