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https://github.com/davidthorn/learning-calculus


https://github.com/davidthorn/learning-calculus

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README 

          
# Learning Calculus



In order to learn calculus I decided to make this repo so as to save and document all of the work which I have done.



By no means should someone use this as a guide, its just how I learnt it.



# What is Calculus?



The difference between regular math and calculus: In a word, its the curve.



Calculus is that the mathematics of change.



# Real world examples calculus



You can calculate the area of the flat roof of a home with regular math.



With calculus you can compute the area of a complicated, nonspherical shape like the dome of a building.



# Diferentiation and Integration



## Diferentiation



Differentiation is the process of finding a derivative, and the derivative of a curve is just the fancy calculus term for the curve's slope of steepness; the slope of a curve is also a simple rate like miles per hour or profit per item.



## Integration



Integration is the second big idea in calculus, and is basically just fancy addition. Integration is the process of cutting up an area into tiny sections, figuring the areas of the small sections, and then adding up the little bits of area to get the whole area.



### Yeno's Paradoxes



It seems that it may be worth while reading through this document to understand things a bit further.



[Zeno's Paradoxes](https://plato.stanford.edu/entries/paradox-zeno/#ParPlu)



# Why Calculus Works



* Using limits to zoom in on curves

* Slope equals rise over run

* Area of a triangle equals one-half base times height

* The Pythagoreans theorem: a²+ b² = c²



# Pre-Algebra and Algebra Review



The deominator of a fraction can never equal zero.



```

0/5 equals to zero, but 5/0 is undefined

```

The ```reciprocal```of a number od expression is it multiplicative inverse -- whic is a fancy way of sayig that the prdocut of somethins is reciprocal is 1.



To get the reciprocal of a fraction, flip it upside down. Thus, the reciprocal of ```3/4 is 4/3```, the reciprocal of ```6```, which equals ```6/1```, is ```1/6,``` and the reciprocal of ```x - 2 ```is ```1/x-2```.



### Polynominal



In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.



Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.



Reference: [Wikipedia - Polynomial](https://en.wikipedia.org/wiki/Polynomial)



### Coefficient



In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables.



For example, in



> 7x2 - 3xy + 1.5 + y



See more ...



Reference: [Wikipedia - Coefficient](https://en.wikipedia.org/wiki/Coefficient)



### Power Rule



>x0 = 1



This it the rule regardless of what ```x``` equals -- a fraction, a negative, anything -- except for zero (zero raised to the zero power is ```undefined```)



>x0 = 1



>00 = undefined



#### Examples



>x-3 = 1⁄x3



>x-a = 1⁄xa



>4-2 = 1⁄42 =  1⁄16 



This part definitely needs more working on and possibly a seperate repo to learn more.



[Learning more about Power Rule in Algebra](https://github.com/davidthorn/power-rule)



### Square Roots



The mathematical expression "The (principal) square root of x"



For example, √25 = 5, since 25 = 5⋅5, or 52 (5 squared).

In mathematics, a square root of a number a is a number y such that y2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or y⋅y) is a.[1] For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16. Every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 32 = 3 • 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.



Every positive number a has two square roots: √a, which is positive, and −√a, which is negative. Together, these two roots are denoted as ±√a (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2.[2]



Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)



Reference: [Wikipedia Square Roots](https://en.wikipedia.org/wiki/Square_root)



### Logarithms



In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm to base 10" of 1000 is 3. The logarithm of x to base b is denoted as logb (x) (or, without parentheses, as logb x, or even without explicit base as log x, when no confusion is possible). More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:



> logb(x) = y      exactly if       by = x



Reference: [Wikipedia - Logarithm](https://en.wikipedia.org/wiki/Logarithm)