The number of paths algorithm is a method to calculate the number of possible paths between two points on a grid, considering specific movement directions. This guide provides detailed steps and examples to help you understand and apply this algorithm effectively. Example 1: Counting Paths Between A and E Output: There are 4 paths connecting…
Differentiation by First Principles Differentiation by first principles is an algebraic approach to finding the derivative (gradient function) of a function. The method calculates the gradient of a function as the limit of the difference quotient. The general formula is: f'(x) = lim (h → 0) [f(x + h) – f(x)] / h This formula…
Understanding double angle formulas in trigonometry is crucial for solving complex equations and simplifying expressions. This guide provides a complete overview of the double angle formulas, their derivations, and practical applications. What are the Double Angle Formulas?. Let us derive the double angle formula(s) of each of sin, cos, and tan one by one. Double…
Dominance matrices are used to represent and analyze the outcomes of round-robin competitions, where each competitor faces every other competitor. This guide will show you how to create dominance matrices and use them to rank competitors effectively. What are Dominance Matrices? A dominance matrix is a numerical representation of which competitor dominates others within a…
What is the Average Rate of Change? The average rate of change of a function f(x) over an interval [a, b] is calculated using the formula: This formula represents the slope of the secant line, connecting the points (a, f(a)) and (b, f(b)). The secant line cuts through the curve, illustrating how the independent variable…
What is Sketching the Derivative? (Simple Explanation) Sketching the derivative means drawing the graph of a function’s derivative based on its original graph. The derivative of a function represents its rate of change or slope at any given point. By observing the shape and slope of the original function, you can create a new graph…
Volume of a Spherical Cap The volume of a spherical cap, a portion of a sphere cut off by a plane, can be calculated using specific formulas. This guide will explain the formula and provide examples to help you understand and apply it effectively. Formula for the Volume of a Spherical Cap The volume V…
Variance Calculation Guide Variance is a measurement of the spread or variability within a set of data. It represents how far each data point in the set is from the mean. This guide will walk you through calculating both sample and population variance, with detailed examples and explanations. What is Variance? Variance is the average…
Finding the Vector Between Two Points Finding the vector between two points is a fundamental concept in vector mathematics. This guide will walk you through the steps to calculate the vector, providing detailed explanations and examples to help you understand and apply this concept effectively. What is a Vector? A vector is a quantity that…
Finding a Vector Perpendicular to a Plane A vector perpendicular to a plane, also known as the normal vector, is crucial in many fields, including geometry, physics, and computer graphics. Here’s how you can find such a vector using different methods. 1. Using Three Points Given three points, A(x1, y1, z1), B(x2, y2, z2), C(x3,…