How/when is calculus used in Computer Science?

Question Detail: 

Many computer science programs require two or three calculus classes.

I'm wondering, how and when is calculus used in computer science? The CS content of a degree in computer science tends to focus on algorithms, operating systems, data structures, artificial intelligence, software engineering, etc. Are there times when Calculus is useful in these or other areas of Computer Science?

Asked By : Victor

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/55462

Answered By : Jay

I can think of a few courses tht would need Calculus, directly. I have used bold face for the usually obligatory disciplines for a CS degree, and italics for the usually optional ones.

• Computer Graphics/Image Processing, and here you will also need Analytic Geometry and Linear Algebra, heavily! If you go down this path, you may also want to study some Differential Geometry (which has multivariate Calculus as a minimum prerequisite). But you'll need Calculus here even for very basic things: try searching for "Fourier Transform" or "Wavelets", for example -- these are two very fundamental tools for people working with images.
• Optimization, non-linear mostly, where multivariate Calculus is the fundamental language used to develop everything. But even linear optimization benefits from Calculus (the derivative of the objective function is absolutely important)
• Probability/Statistics. These cannot be seriously studied without multivariate Calculus.
• Machine Learning, which makes heavy use of Statistics (and consequently, multivariate Calculus)
• Data Mining and related subjects, which also use lots of Statistics;
• Robotics, where you will need to model physical movements of a robot, so you eill need to know partial derivatives and gradients.
• Discrete Math and Combinatorics (yes!, you may need Calculus for discrete counting!) -- if you get serious enough about generating functions, you'll need to know how to integrate and derivate certain formulas. And that is useful for Analysis of Algorithms (see the book by Sedgewick and Flajolet, "Analysis of Algorithms"). Similarly, Taylor Series and calculus can be useful in solving certain kinds of recurrence relations, which are used in algorithm analysis.
• Analysis of Algorithms, where you use the notion of limit right from the start (see Landau notation, "little $o$" -- it's defined using a limit)

There may be others -- this is just off the top of my head.

And, besides that, one benefits indirectly from a Calculus course by learning how to reason and explain arguments with technical rigour. This is more valuable than students usually think.