Is 0 a natural number, or does it stand apart from the counted set that begins with 1? This seemingly simple question invites a dive into the history of numbers, the evolution of mathematical thought, and the nuances of number theory. In this article, we will embark on a journey through the ages, exploring the […]
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“Applied Calculus” is a single-level course that covers the basics of several topics such as functions, derivatives and integrals. It is also known as “baby calculus” and discusses several topics which are also part of a calculus course. In this topic, we will discuss applied calculus, its similarities and differences with calculus, and its related […]
Delving into the realm where patterns, functions, and behaviors take the forefront, we explore how to find end behavior in mathematics. An intriguing notion is ‘end behavior,’ deeply ingrained in mathematical analysis and calculus. This term provides us with a window into the future trajectory of a function, depicting the path it will take as its […]
In the awe-inspiring universe of mathematics and geometry, concepts extend beyond the standard three dimensions we experience daily. One such captivating idea is that of a hypersphere, an object existing in four or more dimensions, transcending our usual comprehension of space. Known as a higher-dimensional analogue of a sphere, the hypersphere represents a quantum leap in our understanding of geometric […]
Within the expansive realm of calculus, the antiderivative, including the antiderivative of tan(x), assumes a pivotal role in solving numerous mathematical problems. When we delve into the intricacies of trigonometric functions, one of the most frequently encountered functions is the tangent function or tan(x). Therefore, understanding the antiderivative of tan(x) broadens our grasp of integral […]
The sigmoid function, often heralded as the quintessential activation function in the realm of neural networks and deep learning, boasts an elegant S-shaped curve that smoothly transitions between 0 and 1. We will delve deeply into the nuances of the derivative of the sigmoid function in this article, revealing its mathematical beauty and comprehending its […]
Rational root theorem, also known as rational zero theorem or rational root test, states that the rational roots of a single-variable polynomial with integer coefficients are such that the leading coefficient of the polynomial is divisible by the denominator of the root and the constant term of the polynomial is divisible by the numerator of […]
The law of cosines or cosine theorem is a rule that provides us with the relation between the sides and angles of a triangle. The relationship is described using the formula: $c^2 = a^2 + b^2 -2abcos (z)$ or $c = sqrt{a^2 + b^2 -2abcos (z)}$, where $a$, $b$ and $c$ are the three sides […]
Inverse function theorem gives a sufficient condition for the existence of the inverse of a function around a certain point and also tells us how to find the derivative of the inverse function at that point. To understand the inverse function theorem, let us first recall what is a function and what is the inverse […]
Perimeter of a rectangle is the total length of all of its sides. It is calculated with the help of the following formula: $textrm{Perimeter of a Rectangle} = 2 ( textrm{Length} + textrm{Width})$. Perimeter is defined as the boundary that surrounds a shape. It can also be defined as the length of a shape’s sides. […]
The perimeter of a parallelogram is the total length of its outer boundaries. A parallelogram, similar to a rectangle, is a quadrilateral with equal opposite sides. So if the length and width of a parallelogram are $a$ and $b$, such as in the figure above, we can calculate the perimeter as: Perimeter = $2(a + […]
The perimeter of a rhombus is the total length measured across its boundaries. All sides of a rhombus are equal to each other. If the length of any single side is equal to $x$, as shown in the figure above, then the perimeter is given as Perimeter $=4x$ We get the perimeter of a rhombus […]
The perimeter of a square is the total length measured across its boundaries. Let $x$ be the length of each side of the square, as shown in the figure below: The perimeter is calculated using the formula: $textrm{Perimeter} = 4x$ The word perimeter is the combination of two Greek words, “Peri” meaning surrounding or enclosing […]
In mathematics, you will come across relations and functions quite often, but one burning question that arises in many students’ minds is which relation is not a function. A relation that does not have the properties of a function is just a simple relation. Every function is a relation but every relation is not a […]
The perimeter of a triangle can be defined as the total length across all the boundaries of a triangle. Let the lengths of three sides of a triangle be given as $a$, $b$ and $c$, as shown in the figure above. With this information, the perimeter is calculated as: $Perimeter = a + b + […]
A heptagon is a geometric figure that belongs to the broader category of polygons. Polygons are closed, two-dimensional shapes composed of straight lines. While it may seem like just another simple geometric shape, the heptagon presents fascinating complexities in its properties, historical uses, and real-world applications. Brief Overview of the Article This exploration into the […]
In the vast expanse of mathematical vectors, the ‘parametric vector form’ stands as a beacon, illuminating complex geometric and algebraic problems with precision and clarity. This form, often an unsung hero, enables mathematicians and physicists alike to define lines and curves in space more efficiently than traditional methods. Dive into this article as we unravel […]
In mathematics, few concepts possess the real number system‘s captivating power and profound significance. From the enchanting elegance of whole numbers to the mystifying infinitude of decimals, the real number system is the bedrock upon which mathematical analysis is built. While we may encounter real numbers daily, their true nature and remarkable properties often remain […]
The volume of a parallelepiped serves as an intriguing point of exploration, while embarking on a journey into the realm of three-dimensional space. As a polyhedron enveloped by six parallelograms, a parallelepiped is a geometric marvel that offers rich insights into the interplay of vectors and spatial dimensions. This article aims to unfold the intricacies […]
In the vast and interconnected realm of mathematical functions, quartic functions hold a position of unique interest and versatility. Characterized by a degree of four, these functions, defined by a fourth-degree polynomial, wield significant influence across numerous aspects of mathematical theory and its many practical applications. As the next step beyond linear, quadratic, and cubic […]
Within the world of calculus, we explore the derivative of x² through applications and examples that help us make sense of the myriad phenomena in science and engineering. The derivative is a tool that helps us understand rates of change and slopes of curves. A classic and instructive example is the derivative of x², a simple parabolic function. In this […]
In the vast expanse of mathematical series, the telescoping series stands as an intriguing and illuminating concept. With its unique properties, it collapses, or ‘telescopes,’ into a simpler and more manageable form, much like the sliding mechanism of a handheld telescope. In this article, we will gain insight into its behavior, its utility in mathematics, […]
This article dives deep into the mathematical and statistical implications of averaging averages, elucidating potential pitfalls, best practices, and the underlying principles that guide this practice. Defining Average of Averages The “average of averages” refers to a calculation where individual averages from separate subsets of data are combined to produce a single overall average. In […]
The concept of relative extrema, also known as local extrema, is a key player in calculus that offers valuable insights into the local behavior of functions. These extrema signify a function’s local maximum and minimum values, providing a nuanced comprehension of its shape, trend, and turning points. Understanding the peaks and valleys of functions is […]
The second derivative implicit differentiation is a powerful tool to differentiate implicitly defined functions concerning an independent variable not explicitly expressed. Exploring the intricacies of calculus often leads us to fascinating techniques that unveil the hidden properties of equations and functions. While implicit differentiation enables us to find the first derivative of such functions, delving […]
In the beautifully interconnected realm of mathematics, the transverse axis offers a compelling thread that weaves together multiple disciplines, from geometry to calculus. As we explore this crucial concept, its underpinning role in the integrals world cannot be overstated. In this article, we spotlight the transverse axis, dissecting its unique position in the mathematical landscape and, […]
The concept of vertical intercept and its application to real-world scenarios is fundamentally the fascinating realm of mathematics. It provides an essential reference point in the graphical representation of linear equations, functions, and data trends. This vital intersection point on the y-axis delivers invaluable insight into the inherent characteristics of the relationship described by the equation […]
The curvature formula an expressed equation gives us a way to quantitatively understand how shapes bend and reveal their inherent properties. In this article, we will dive into the intricacies of the curvature formula examine how it was derived, and highlight its importance in disciplines such, as architecture and theoretical physics. Definition of the Curvature […]
In the intricate field of mathematics, few topics can be as fascinating, yet enigmatic, as orthogonal trajectories. Bridging the world of calculus, geometry, and differential equations, these intriguing curves serve as a testament to the harmonious symphony of mathematics. This article endeavors to explore the concept of orthogonal trajectories, paths that intersect other curves at […]
In this article, we delve into the definition, calculation, and interpretation of the variance of xy, highlighting its significance in statistical analysis, correlation studies, and predictive modeling. Defining Variance of XY The variance of the product of two variables, denoted as Var(xy), represents the measure of the variability or dispersion of the outcomes obtained by […]
The right triangle, a cornerstone of geometry, is a geometric shape that embodies both simplicity and mathematical significance. With its distinctive form defined by one right angle, the right triangle holds a special place in various fields, including mathematics, physics, engineering, and architecture. Its elegant proportions and inherent properties make it a fundamental building block […]
Calculus, the mathematical study of change and motion, often brings two closely related but distinct concepts to the fore: antiderivatives and integrals. While these terms are occasionally used interchangeably in casual conversations, they address different aspects of calculus. Let’s embark on a journey to understand the nuances between them and explore their significance in the […]
Welcome to an engaging exploration of the Wronskian, an indispensable mathematical tool with profound applications. In this article, we embark on a journey to understand the intricacies and significance of the Wronskian. Defined as a determinant formed from a set of functions, the Wronskian serves as a powerful tool for analyzing relationships, testing linear dependence, […]
We explore the geometric series test, a cornerstone concept in mathematical sequences and series. This article will delve into the theory, proofs, and applications of this influential test. The geometric series test offers a gateway to understanding whether an infinite geometric series converges or diverges, providing a solid foundation for subsequent mathematical theories. Whether you’re […]
A semicircle, as the name suggests, is one-half of a circle. While circles have been central to human civilization for millennia, from the wheels of carts to the intricate designs in cathedrals, semicircles too have their unique charm and utility. Understanding the equation of a semicircle offers a glimpse into a world where mathematics marries […]
Delving into the depths of linear algebra, one encounters the powerful Gram-Schmidt Process, a mathematical algorithm that transforms a set of vectors into an orthogonal or orthonormal basis. It’s a fascinating process, fundamental to numerous areas in mathematics and physics, including machine learning, data compression, and quantum mechanics. This process simplifies computations and provides geometric […]
In the vast landscape of mathematical symbols, the “greater than or equal to” sign (≥) stands as a powerful tool to denote inclusivity and range. When transposed onto a number line, this seemingly simple concept takes on a visual dimension, making abstract comparisons tangible and easy to comprehend. In this article, we will embark on […]
To name a plane, three non-collinear points should be present on a two-dimensional flat surface. In geometry, a plane is considered a two-dimensional surface with no boundaries. If points A, B and C are present in two-dimensional flat surfaces, then we can call the plane ABC or the whole surface” P”. Hence, a plane is […]
Embarking on a journey through space and dimensions, vector translation is a fundamental tool for manipulating and transforming geometric objects with precision and accuracy. This transformative technology enables us to displace and relocate vectors while preserving their direction and magnitude. It is key to unlocking many practical applications in various fields, from computer graphics and […]
Geometry, with its intricate shapes and precise calculations, has always been at the heart of architectural marvels, spacecraft trajectories, and even simple everyday tools. What if you’re armed with the knowledge of just one side and an angle? How to find hypotenuse with angle and side? This article guides you through the process of finding […]
Euler’s Method is a cornerstone in numerical approximation, offering a simple yet powerful approach to solving differential equations. Named after the esteemed mathematician Leonhard Euler, this technique has revolutionized scientific and engineering disciplines by enabling researchers and practitioners to tackle complex mathematical problems that defy analytical solutions. Euler’s Method allows for approximating solutions to differential […]
In this article, we will unravel the intricacies of the partial sum formula, demonstrating its immense utility and showcasing the beauty of mathematical abstraction and simplification. Definition of Partial Sum Formula The partial sum formula pertains to the summation of a subset of terms from a sequence or series. In the context of sequences and […]
What Is the Difference Between Equation vs Expression The main difference between an expression and an equation is that an expression represents a mathematical “phrase” comprising numbers, variables, and operations without an equal sign, while an equation is a mathematical “sentence” asserting the equality of two expressions, always including an equal sign. Both “equation” and […]
The pentagonal prism is a captivating and versatile geometric shape that sparks interest in both the realms of mathematical theory and real-world application. As a three-dimensional figure, the pentagonal prism belongs to the family of prisms, characterized by two congruent bases and rectangular sides connecting the corresponding vertices. In the case of a pentagonal prism, its bases are pentagons, a five-sided […]
As one delves into the diverse and fascinating world of geometry, the Nonagon emerges as a figure of intrigue, brimming with unique properties and applications. One such figure is the Nonagon, a polygon defined by nine equal sides. Although it may not be as commonly recognized as the triangle or square, the Nonagon carries its […]
At the heart of geometry lies the humble triangle, a shape of utmost importance due to its structural stability and its inherent presence in many forms around us. Of all triangle types, the isosceles triangle stands out for its notable properties of symmetry. Characterized by two sides of equal length, known as the legs, and a distinct third side called […]
One of the fundamental concepts associated with matrices is the ‘inner product,’ a generalization of the dot product in vector spaces. In this article, we delve into the intricacies of the inner product of matrices, exploring their definitions, properties, and applications. Definition of Inner Product of Matrices The inner product of matrices is defined as the […]
The extreme value theorem states that a function has both a maximum and a minimum value in a closed interval $[a,b]$ if it is continuous in $[a,b]$. We are interested in finding the maxima and the minima of a function in many applications. For example, a function describes the oscillation behavior of an object; it […]
In the vast realm of mathematics, one of the foundational principles every student must grasp is the order of operations. Two mnemonic devices, BODMAS and PEMDAS, stand as sentinels guiding this uniformity. This article delves into the distinctions and similarities between BODMAS and PEMDAS, offering insights into their regional influences and core objectives. What is […]
The absolute value of $-8$ is $8$. The absolute value of any number is represented as | |. For example, we will represent the absolute value of $-8$ as $|-8|$, and the answer would be equal to $8$. The absolute value of $|8|$ is also $8$, hence the absolute value of $|-8|$ = $|8$| = […]