I want to understand in the simplest terms why it is crucial for conputer science and if there is such a thing as a simple calculus exercise to be explained?
In: Mathematics
> why it is crucial for conputer science
Signal processing, machine learning, etc.
You think you can program ChatGPT just using Algebra?
Now, depending on what job you land, you may not actually use much calculus, but other jobs you will, or at least you need to know it in order to understand what’s going on (if an error occurs, you have to be able to recognize the wrong output is wrong and be able to trace the code and understand where it went wrong).
Calculus is a type of math that studies how things change and accumulate. It helps you figure out how fast something is moving or changing at a specific moment (differentiation) and then add up those changes to find the total result (integration). It’s like tracking how a car’s speed changes and then calculating the total distance it travels. This makes it useful in physics, computer science, and other fields.
For instance, if you want to simulate the movement of objects in a video game, like a car speeding up or slowing down, calculus provides the tools to calculate the car’s position, speed, and acceleration at any given time. Without calculus, it would be difficult to model these smooth, continuous changes.
Or e.g. in machine learning, calculus plays a big role in optimizing algorithms. When computers try to “learn” patterns from data, they need to adjust certain parameters to minimize errors. Calculus helps by figuring out how to make tiny changes to these parameters so that the machine can improve its predictions.
For a very simple exercise, imagine a car that starts from a stop and accelerates at a constant rate of 2 meters per second squared. If you want to know how fast the car is going after 5 seconds, you can use calculus. Since the acceleration is constant, the car gains 2 meters per second of speed every second. After 5 seconds, the speed is 10 meters per second, which you know by integrating the acceleration (basically summing) over time. Similarly, by integrating the speed over time you can get the distance that the car travelled since the start.
Once you’re into it, calculus mostly deals with handling “rates of change”… like, your velocity is the rate at which your position changes. Similarly, “acceleration” is how much your velocity changes.. like as your car goes from stopped (speed/velocity is 0) to highway speeds.
Very simple physics question: A car accelerates from a stop to 60 miles/hour smoothly over 30 seconds, and then maintains that speed until a total of 5 minutes have passed. Exactly how far did they travel at that 5 minute mark?
There are formulas you might be able to look up to tell you the answer, but it’s calculus that produced those formulas and prove them correct.
And I suppose that’s the main answer: physics is just full of calculus. And for computers to do higher level math for simulations.. video games, or physics applications, understand calculus is absolutely essential. And even for other things, calculus comes in really handy for some types of math like trigonometry. You can go a decent way without it, but at some point you’ll be glad you know some calculus.
Hmm, I’ll do my best here…
Computer science is, in many ways, the study of efficiency. It’s laziness. How can I do something and have it cost me the least amount of time and effort? To know that, we study algorithms, data structures, network traffic, etc.
Calculus allows us to understand how quickly things change. Acceleration is a great example. It’s the rate of change in velocity. We cannot know the precise rate of change without calculus. Calculus gives us the methods we need in order to determine lows and highs, volume, intersections. This is necessary in order to understand efficiency.
Calculus is the kind of math you use when you have to calculate something that’s *changing*.
For example, you can use calculus to figure out how much money will be in a savings account that earns interest – because the interest that you earn will change the amount of money in the account, which will change the amount of interest you earn next, which will change the amount of money in the account, which will change the amount of interest you earn next.
The calculus question I recall is
It’s snowing at a rate of 3 inches per hour.
Plowman Jack can clear five miles of road per hour.
There are 128 miles of road for Plowman Jack to clear.
If Plowman Jack starts plowing right away, he’s losing money.
If Plowman Jack starts plowing too late, he’s losing money.
When is the optimal time for Plowman Jack to begin plowing?
In general for computer science, it isn’t important. The majority of jobs will never require it.
But there are a few areas where it is a necessity. For example, video game development generally requires simulating movement and interactions between physical objects, calculus will be critical here ad it will help you calculate things like speeds and acceleration.
It is so critical in robotics where, again, you are often working with movement and position.
Finally it is important in the advanced statistical models of things like AI and machine learning.
So there are quite a few jobs that require it and you wouldn’t be prepared for them without it. That’s why a general CS degree will make you take calculus.
But there are lots of jobs like databases or web development where you will likely never use it in your job.
at its most basic, calculus is how to usefully multiply and divide by 0.
For example, how fast is the car moving RIGHT NOW? speed is [distance travled]/[time taken] so without calculus you can only say what your average speed over a time was, if you try to get the speed of right now (time taken is 0) that just doesnt work, division by 0. Calculus lets you say “Great, but what is it anyway?” and provides you the tools to figure out the answer by “approaching 0”. This is a derivative
Same for the area under a curve, area has a height and a width, you can divide up a shape into more smaller and smaller areas but with calculus we can say “but what if the width was 0, but we had an infinite number of areas? whats the area then?” and it works by approaching 0 and infinity.
There are a lot of applications of this, but for computer science it isnt really that important. There are a few things you NEED calculus for, but there are some things you NEED group theory for, none of those things are things the average programmer needs to worry about. What is more important is ways of thinking. Simply by learning calculus you learn some ways of thinking that absolutely lock in algebra (which you DO use) to an extent that just learning algebra doesnt. You also learn how to think about numbers approaching things instead of being things, which is useful for algorithm efficiency analysis (which EVERY programmer should be thinking about) where you need to consider how your algorithm scales as its inputs go to infinity.
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