Popular Articles (30d)
Understanding Zipline Motion: Height, Velocity, Maximum Speed and the Role of Calculus
Introduction A zipline provides an interesting real-world example of how gravitational potential energy is converted into kinetic energy. As a person travels along a cable connecting two elevated platforms, the person’s height changes continuously. Consequently, the speed also changes throughout the journey. One particularly interesting question arises: Why does the person reach maximum speed at…
Linear Approximation: Estimating Both Function Values and Input Values
Introduction Linear approximation is generally introduced as a method of estimating the value of a function near a known point. However, the same mathematical relationship can also be used in the opposite direction: estimating the input required to produce a particular function value. These are two different applications of the same underlying idea: using a…
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Understanding the Equation of Motion: The Connection Between Velocity, Acceleration, Displacement, and Energy
Introduction One of the fundamental equations in classical mechanics is: 
Why Newton’s Method Converges Quadratically: The Hidden Connection Between Linear, Quadratic, and Higher-Order Approximations
Introduction One of the most interesting features of Newton’s method is that it uses a linear approximation to achieve quadratic convergence. At first, this might seem surprising. Newton’s method estimates the root of a function by drawing a tangent line and finding where that tangent intersects the x-axis. A tangent line is a linear approximation,…
Why Finding the Root of a Function Is So Important: The Hidden Connection Between Equations, Equilibrium, Optimization, and Real-World Problems
Introduction At first glance, finding the root of a function might seem like a relatively narrow mathematical exercise. Given a function, the objective is simply to determine the value of x for which the function becomes zero. For example: 
Linear Approximation and Concavity: How the Second Derivative Sharpens Your Estimates
Tangent lines are one of the quickest ways to estimate a function near a known point. But a tangent line is only as good as the curve is straight. This article shows how concavity tells you whether your estimate is too high or too low, and how the second derivative gives you a better approximation.…
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