31/08/2023
Did you know? (Day 2)
Imagine you're trying to find out what happens to a ball as it gets closer and closer to a certain point, like touching the ground. The concept of limits in math helps us understand this idea.
Think of the limit as a way to predict where the ball will be just before it touches the ground. As the ball gets closer and closer to the ground, its position approaches a specific point, which we call the "limit." This is similar to how we might estimate where the ball will land by observing its trajectory.
The origin of limits comes from the need to solve tricky problems that involve things like dividing by zero or dealing with infinitely small or infinitely large numbers. Mathematicians developed the concept of limits to handle these situations in a more organized and systematic way. It's like creating a rulebook for how to handle situations that might seem a bit strange at first.
In the ball example, the limit helps us "see" where the ball is headed even if it hasn't quite reached the ground yet. Just like a guess that gets better and better the more information you have, limits help us predict the outcome when things get really close to a certain point, even if they never quite reach it.
Also, there are two possible ways (in 2d coordinates) for a value to "come from" before it approaches to the specific point (attached in the image below).
Source of image: https://calcworkshop.com/limits/finding-limits-graphically/
28/08/2023
Did you know? (Day 1)
Riemann sum is a fundamental concept in calculus used to approximate the area under a curve by dividing it into smaller segments and summing up the areas of those segments.
The concept is named after the German mathematician Bernhard Riemann, who introduced it in the mid-19th century. Riemann's work on integrals and sums contributed significantly to the development of modern calculus.
The basic idea behind Riemann sums is to approximate the area under a curve using rectangles (or other shapes) that closely approximate the curve's behavior. By dividing the interval over which you're integrating into smaller subintervals and evaluating the function at specific points within each subinterval, you can calculate the area of the corresponding rectangle and then sum up these areas to get an approximation of the total area under the curve.
As the number of subintervals increases and the width of each subinterval approaches zero, the Riemann sum becomes more accurate and approaches the actual value of the definite integral. This is essentially the foundation of the definite integral in calculus, where we take the limit as the number of subintervals approaches infinity to obtain the exact area under the curve.
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Source of photo: https://www.math.net/riemann-sum
28/08/2023
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