Mathematics

Caculus – Exponential Functions

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Resource: The Exponential Function Suppose we have a quantity \(y\), whose rate of change over time is proportional to the amount present. We can describe this relationship using a differential equation: \[ \frac{dy}{dt} = ky \] If we define the initial state where \(y = y_0\) at time \(t = 0\), the solution to this ... Read More

Systems of Linear Equations and Matrices

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References: Linear Equation: At its simplest, a linear equation is an algebraic equation that creates a straight line when plotted on a graph. Every variable in the equation is raised to the first power (meaning no exponents like $x^2$ or $y^3$), and there are no variables multiplied by each other. The Standard Forms Depending on ... Read More

Calculus – Natural Logarithms

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Resources: What is Natural Logarithm? In calculus, the natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. Unlike base-10 logarithms (common logs), the natural logarithm, denoted as $ln(x)$, is preferred in calculus because it simplifies the differentiation and integration of exponential functions. 1. ... Read More

Euler’s Number $e$

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Resources: What is Euler’s Number? Euler’s number (eā‰ˆ2.71828e is approximately equal to 2.71828) was first discovered by Swiss mathematician in Jacob Bernoulli in1683 while studying compound interest. However, it is named after Leonhard Euler, who popularized the constant, discovered its connection to calculus, calculated its value to 23 decimal places, and used the letter š‘’ ... Read More

Calculus – Inverse Functions

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Resources: What is Inverse Function in Calculus? In calculus, an inverse function is a function that “undoes” the action of another function. If a function $f$ takes an input $x$ and gives an output y, its inverse $f^{āˆ’1}$ takes $y$ and brings you back to $x$. Mathematically: $f(x) = y \iff f^{-1}(y) = x$ 1. ... Read More

Calculus – Transcendental Functions

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References: Transcendental Functions In mathematics, a transcendental function is a function that does not satisfy a polynomial equation with polynomial coefficients. To put it simply: it is any function that is not algebraic. While algebraic functions can be constructed using a finite number of elementary operations (addition, subtraction, multiplication, division, and taking roots), transcendental functions ... Read More

Calculus – Application of Integrals (Area between Curves)

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References: Integrals can be used to calculate many things such as area between curves, volumes and surfaces of solids, length of curves etc. Areas between Curves: In calculus, we define the exact area under a curve not by measuring it directly, but by slicing it into rectangles, adding them up, and then making the slices ... Read More

Probability – Introduction

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Book Probability by Jim Pitman, Springer The main concepts of probability theory include: These three concepts form the fundamental framework of probability theory. They describe the structure of any random process, from a simple coin toss to complex risk modeling. Outcome Space (Sample Space) The outcome space (often denoted by the Greek letter Omega, Ī©, ... Read More

Linear Algebra – Gaussian Elimination

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Book: Introduction to Linear Algebra, Johnson et al. Third Edition Gaussian Elimination is a method in linear algebra used to solve systems of linear equations.It systematically transforms a system of equations into a simpler form so the solutions become easy to find. It works by manipulating the augmented matrix of the system using basic row ... Read More

Rank of Matrix

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The Rank of a Matrix is a fundamental concept in linear algebra that tells you how much “information” or “dimensionality” a matrix preserves. At its core, the rank represents the number of linearly independent rows or columns in a matrix. It is the dimension of the vector space generated by its columns (or rows). https://www.mathsisfun.com/algebra/matrix-rank.html ... Read More
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