Christ Anecdotes by Db

A Retrospect to Begin with... In calculus, the concepts of limits and continuity form the foundation upon which differentiability is built. A limit helps us understand the behavior of a function as it approaches a specific point, while continuity ensures that a function has no abrupt jumps or breaks. Differentiability takes these ideas further, exploring how a function changes at an infinitesimal level. What is Differentiability? Differentiability is essentially defined as the ability to derive the function. A derivative refers to the slope of the tangent line to a curve defined at a point on the curve, or alternatively, it can define the function's instantaneous rate of change. To understand this better, consider the car analogy . If you travel 100 kilometers within 2 hours, the average speed can be calculated as 50 km per hour. Nevertheless, this does not tell whether the car was going really fast or with less speed or was really slow. The derivative describes this aspect by show...

Recap of Limits: A Quick Refresher In our last exploration, we delved into the magic of limits—how they allow us to understand the behavior of functions as they approach specific points, even when direct evaluation fails. Limits allow us to manage situations such as undefined values and serve as the foundation of many calculus concepts. Another important structural principle derived from this foundation is: continuity.  What is Continuity? In plain language, a function is continuous if you can draw its graph without lifting your pencil. It’s like a smooth, unbroken trail with no jumps, gaps, or sudden stops. Mathematically, we define continuity at a point as:  A function f(x) is said to be continuous at x=a if :  lim x → a f ( x ) = f ( a ) Also, a function is said to be continuous on the interval [a,b] if it's continuous at each point in the interval.  This means three things must be true for a function to be continuous at : 1. The function is define...

The Tease: How Close is Close Enough? Visualize creeping up on a buddy just for the purpose of surprising him. You're closing the gap, moving with deliberate caution until you're only a breath away from your friend. Alternatively, picture a basketball circling the rim, inching closer and closer to falling in. What is happening in those moments? You're approaching—but not quite reaching—something important. And that brings us to a fascinating question: what happens when we get infinitely close to something? In the world of mathematics, this has a name: limits. The Basic Idea of a Limit A limit is, in essence, about what value a function gets arbitrary close to when you approach a given number in the input. There is a caveat, though; it need not reach that number. Think of a graph that has a hole at a certain place. Even though the function isn’t defined there, the limit helps us make sense of what’s happening as we inch closer and closer to that point. For example, consider ...