Calculus Explained
A simplistic overview of Calculus
Mom, my friend’s brother is working on Calculus. What is this Calculus thingie ?
Calculus? There is a long time to get there.
Mommy.. You are an Engineer.. Just tell me a little bit.. It seems a cool buzz word..
OK.. Give me some time.. Let me think how to explain.. What does ‘Caaal’culus sound like?
Calculate, Calculator?
Yup.. In Sanskrit, ‘Kal’ means pebble or stone and was a measure of counting.. ‘Kaal’ became a count of time.. And the greek ‘Kalendar/Calendar’ kept a count of months.. CALCULUS is about counting and calculations..
Like 1, 2, 3..?
A little more advanced.. When we are little kids, we start with numbers and arithmetic. Then we move to equations where we describe using math. Have you ever wondered if the the branches of a tree could be written as an equation?
Equation of a tree? Well..it starts of straight.. Then the branch forks and curves.. And further forks and curves.
Awesome..You got a feel for it.. You were able to differentiate a tree from a factory a house... The curves give a sense of continuity and shape. Think of this as a equation of a single branch.
Think of a meandering river. It swirls left and right like a snake and that is its path and equation. The curviness kind of defines the river.
In Calculus, we try to see patterns around us and make estimations and predictions.. Look at the graphs.. See the best and worse conditions
Predictions? That’s cool..
If I need to go to Fremont and my speed is about 60 miles per hour, we can make a rough prediction that it takes 30 minutes for 30 miles right?
But real life is not always like a constant car drive.. It is more like a Roller Coaster ride.. There are slopes, there are ups and downs.. high speed and acceleration.. Drops. With Calculus one can design a safe roller coaster allowing a smooth transition.
The motion of Earth, Moon and Planets are governed by laws of Physics. With Calculus, we could check at what speed should space vehicle be launched in order to escape from Earth’s gravity? When is the next Lunar eclipse when the Moon is closest to the Earth?
That’s cool. It sounds like a lot of physics. Do we use it in Math?
Let’s look into some geometry.. You have a circle so very symmetric of radius ‘r’. We know that the perimeter is 2πr. How do we deduce the area?
I know this.. We can visualise the circle as many many tiny radial wedges.. Then interlace the wedges to form a rectangle of length r and width ‘πr’..
Perfect.. This is calculus.. We say the width of the wedges is ‘limiting’ or tending to zero and the height is tending to ‘r’.
Alternatively, we could also think of the circle as many concentric circles.. With Calculus, we can add up and get Areas, Volumes.. We are integrating small bits to form a bigger surface or volume.. Get the picture?
So Calculus is about adding and integration..
There is another part which is just the opposite.. In a roller coaster ride, the speed would keep changing time to time.. When Usain Bolt runs in a 100m sprint.. his speed every second matters.. It keeps changing.. When you freethrow your basket ball.. the speed keeps changing.. So calculus is also about rate and how things change with time..
OK.. Calculus is also about dividing. Do we have other examples of ‘Limits’?
Real world has infinite shades..infinite scenes.. But our phones and TV display has a limited pixels as our eye cannot differentiate.. And we have 30 frames per second in video as this is good enough.. This limit is good for our eye..
Similarly, a single photograph has so many pixels of information to store.. But we limit and store only the portions that are sensitive to our eyes.. In the phone conversation that we make, we again sample our voice and send.. The mobile phone is a perfect example of how we use ‘Limits’ to our advantage..
In the days of Artificial Intelligence, AI is all about calculus..
That sounds amazing.. AI & Calculus?
Well, like the tree example, we try to come up with equations. A self driving car is trained like a kid with examples of roads.. how to identify pedestrains and cars..
When it is on the road, it applies these equations and makes decisions based on calculus.. A calculus intelligence instead of the human intelligence you see.
That’s so cool. Calculus is so important for AI.
What is the feel you get about calculus after hearing this story?
