Calculus

Data Science- Calculus

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Introduction to Calculus
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What do we need to know for backpropagation

Any new insights in visually understanding calculus

Intoduction

Chapter 1 | Intro

Indentation

Python Indentations (Exc 2)
Where in other programming languages the indentation in code is for readability only, in Python the indentation is very important.
Python uses indentation to indicate a block of code.

Chapter 1 | Python, class intro

Indentation

Python Indentations (Exc 2)
Where in other programming languages the indentation in code is for readability only, in Python the indentation is very important.
Python uses indentation to indicate a block of code.

Chapter 1 | Python, class intro

Intro to Functions

Chapter 1 | Intro to Functions

Derivative Visual

Chapter 3 | Derivative Visual

Derivative equation

Chapter 3 | Derivative

Derivative equation

Chapter 3 | Derivative

Derivative equation

Chapter 3 | Derivative

    Parameters
    ----------
    f : function
        Vectorized function of one variable
    a : number
        Compute derivative at x = a
    method : string
        Difference formula: 'forward'
    h : number
        Step size in difference formula

    Returns
    -------
    float
        Difference formula:
            forward: f(a+h) - f(a))/h
    '''

Integration

Chapter 4 | What is Integration

Greek astronomer Eudoxus (370 BC). He wanted to find a volume or area of a large object

Integration

Chapter 4 | What is Integration

That by breaking up the object into a small measurable piece, you could then add all those atomic pieces together to find the area.

The smaller this piece was, the more accurate this calculation was. It was not until the 17th century that we were able to refine this process into calculus.

Integration-Circle

Chapter 4 | Integration

Integration-Circle

Chapter 4 | Integration

The numerical integration of a circle is based on the circumference formula, why is that?

Integration

Chapter 4 | Numerical Integration

Integration

Chapter 4 | Numerical Integration

#f stands for function argument
#a stands for starting point
#b stands for ending point
#dx means the tiny piece

def integrate(f, a, b, dx=1):
    i = a
    s = 0
    while i <= b:
        s += f(i)*dx
        i += dx
    return s

How does this give us numerical integ?

Chain Rule

Chapter 5 | Chain Rule

What does the chain rule do?

Chain Rule

Chapter 5 | Chain Rule

What does the chain rule do?

Chain Rule

Chapter 5 | Chain Rule

The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x).
In other words, it helps us differentiate *composite functions*.
sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x². Using the chain rule and the derivatives of sin(x) and x², we can then find the derivative of sin(x²).

Chain Rule

Chapter 5 | Chain Rule

import numpy as np

x = 1.5

line =x
    
xSquared =x**2
    
sin = np.sin(x)


dline_dx = 1
    
d_xSquared = 2*x
    
d_sin = np.cos(xSquared)
    
dSquared_dx = d_xSquared * dline_dx

dsin_dx = d_sin * d_xSquared

Chain Rule Project

Chapter 5 | Chain Rule

Create a Class

To create a class, use the keyword class:

Example

Create a class named Snake, with a property named name

Chapter 1 | Python, class intro

The init() Function

Use the __init__() function to assign values to object properties, or other operations that are necessary to do when the object is being created.

Chapter 2 | More on classes