Anisha/calc

calculus_for_ML/README.md

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+# Learn Calculus for Machine Learning
+
+_🚧 This collection is a work in progress. Please help us add notebooks!_
+
+This collection of marimo notebooks is designed to teach you the basics of calculus for machine learning.
+
+**Help us build this course! ⚒️**
+
+We're seeking contributors to help us build these notebooks. Every contributor will be acknowledged as an author in this README and in their contributed notebooks. Head over to the [tracking issue](https://github.com/marimo-team/learn/issues/58) to sign up for a planned notebook or propose your own.
+
+**Running notebooks.** To run a notebook locally, use
+
+```bash
+uvx marimo edit <file_url>
+```
+
+You can also open notebooks in our online playground by appending marimo.app/ to a notebook's URL.
+
+**Thanks to all our notebook authors!**
+
+[Anisha Kalgi](https://github.com/k-anisha)

calculus_for_ML/RatesOfChange.py

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+import marimo
+
+__generated_with = "0.13.1"
+app = marimo.App(width="medium")
+
+
+@app.cell
+def _():
+    import marimo as mo
+    import numpy as np
+    import matplotlib.pyplot as plt
+    from sympy import symbols
+    from sympy import diff
+    return diff, mo, np, plt, symbols
+
+
+@app.cell
+def _(mo):
+    mo.md(
+        r"""
+        In this notebook, we'll learn about rates of change in Machine Learning. 
+
+        In simple terms, the rate of change measures how one quantity changes as we adjust its inputs or parameters. The rate of change can represent how a model's output changes as we adjust the input or parameters. 
+
+        In calculus, the rate of change of a function is given by calculating the derivative at a certain point. So, if f(x) is a function, then f'(x) is the rate of change. 
+
+        In the context of Machine Learning, rates of change are especially useful when working with optimization problems - for example, minimizing a loss function, or using gradient descent to try and find optimal parameters and minimize error.
+
+        Let's get into rates of change!
+        """
+    )
+    return
+
+
+@app.cell
+def _(np, plt):
+    # First, let's start by definiting a simple function. 
+
+    def f(x):
+        return x ** 2
+
+    # Now, we can plot the function.abs
+
+    x = np.linspace(-10, 10, 500)
+    y = f(x)
+
+    plt.plot(x, y, label=r'$f(x) = x^2$')
+    plt.title('Graph of f(x) = x^2')
+    plt.xlabel('x')
+    plt.ylabel('f(x)')
+    plt.axhline(0, color='black',linewidth=0.5)
+    plt.axvline(0, color='black',linewidth=0.5)
+    plt.legend()
+    plt.grid(True)
+    plt.show()
+    return f, x, y
+
+
+@app.cell
+def _(mo):
+    mo.md(r"""Our graph is a parabola. Now, let's find the derivative.""")
+    return
+
+
+@app.cell
+def _(plt, x):
+    def deriv_f(x):
+        return 2 * x
+
+    deriv_y = deriv_f(x)
+
+    plt.plot(x, deriv_y, label=r"$f'(x) = 2x$", color='red')
+    plt.title('Derivative of f(x) = x^2')
+    plt.xlabel('x')
+    plt.ylabel("f'(x)")
+    plt.axhline(0, color='black',linewidth=0.5)
+    plt.axvline(0, color='black',linewidth=0.5)
+    plt.legend()
+    plt.grid(True)
+    plt.show()
+    return deriv_f, deriv_y
+
+
+@app.cell
+def _(mo):
+    mo.md(r"""We can see that the rate of change, or the slope, is positive when x is also positive. This should all be review from high school math! :)""")
+    return
+
+
+@app.cell
+def _(diff, symbols):
+    # Now, we'll use SymPy, a library for symbolic math functions in Python.
+    # Here we've created a function and then found the derivative of it.
+
+    x_sym = symbols('x')
+    f_sym = x_sym ** 3 + 4 * x_sym + 5
+
+    f_deriv_sym = diff(f_sym, x_sym)
+    f_deriv_sym
+    return f_deriv_sym, f_sym, x_sym
+
+
+@app.cell
+def _(mo):
+    mo.md(r"""Now, let's get into gradient descent. Below is an example on a quadratic function.""")
+    return
+
+
+@app.cell
+def _(deriv_f, f, np, plt):
+    # Our parameters below
+
+    learning_rate = 0.1
+    initial = 8
+    iterations = 20
+
+    current = initial
+    history = [current]
+
+    for i in range(iterations):
+        grad = deriv_f(current)
+        current = current - learning_rate * grad
+        history.append(current)
+
+    x_vals = np.linspace(-10, 10, 400)
+    y_vals = f(x_vals)
+
+    plt.plot(x_vals, y_vals, label=r'$f(x) = x^2$')
+    plt.scatter(history, f(np.array(history)), color='red', label='Gradient Descent')
+    plt.title('Gradient Descent on f(x) = x^2')
+    plt.xlabel('x')
+    plt.ylabel('f(x)')
+    plt.legend()
+    plt.grid(True)
+    plt.show()
+    return (
+        current,
+        grad,
+        history,
+        i,
+        initial,
+        iterations,
+        learning_rate,
+        x_vals,
+        y_vals,
+    )
+
+
+@app.cell
+def _(mo):
+    mo.md(
+        r"""
+        What's happening here?
+
+        At a high level, gradient descent is essentially taking the partial derivative at each point in the function, until it converges to a minimum. Because our f(x) is a 'nice' function (i.e., there are no annoying curves, saddle points, or other minima/maxima), we are guaranteed that our gradient descent can find the minimum. 
+
+        What is the minimum in this case? It's the **point on the function where the rate of change is closest to 0**.
+        """
+    )
+    return
+
+
+@app.cell
+def _(mo):
+    mo.md(
+        r"""
+        Now, we can get into some applications of rates of change to fully understand the concept.
+
+        First, let's look at kinematics. We know that position, velocity, and acceleration are related, but how? Interestingly, **the derivative of position is velocity, and the derivative of velocity is acceleration!**
+        """
+    )
+    return
+
+
+@app.cell
+def _(np, plt):
+    # Let's say we have a position function of an object that is 5t^2, where t is time.
+
+    time = np.linspace(0, 10, 100)
+
+    position = 5 * time**2
+
+    velocity = 10 * time
+
+    acceleration = np.full_like(time, 10)
+
+    plt.plot(time, position, label='Position (s(t) = 5t^2)')
+    plt.plot(time, velocity, label='Velocity (v(t) = 10t)', linestyle='--')
+    plt.plot(time, acceleration, label='Acceleration (a(t) = 10)', linestyle=':')
+    plt.title('Position, Velocity, and Acceleration')
+    plt.xlabel('Time (t)')
+    plt.ylabel('Position / Velocity / Acceleration')
+    plt.legend()
+    plt.grid(True)
+    plt.show()
+    return acceleration, position, time, velocity
+
+
+@app.cell
+def _(mo):
+    mo.md(r"""Here, we can see that the orange line, velocity, represents the rate of change of position at the exact time t = 2 seconds. When we look at the green line, it tell us the rate of change of velocity  at that exact same point in time!""")
+    return
+
+
+@app.cell
+def _(mo):
+    mo.md(
+        r"""
+        We can use another example in economics. In economics, there is a term known as 'Price Sensitivity,' wjich essentially measures the rate of change in the quantity demanded or supplied of a good. The price elasticity is given by the formula:
+
+        Elasticity = % change in qty. demanded / % change in price
+
+        Using derivatives, we would say:
+
+        Elasticity = the rate of change of demand with respect to price * (price / quantity)
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    return
+
+
+@app.cell
+def _():
+    return
+
+
+if __name__ == "__main__":
+    app.run()