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Calculus Problems - some ideas and approaches with Matlab

Calculus problems are a branch of mathematics focused on limits, functions, derivatives and integrals.

This very broad topic constitutes a major part of modern education. It has two major branches, differential and integral calculus,
which are related due to the fact that integrals are the anti-derivatives (so to speak).... We use Matlab to fastly experiment with some important concepts.

Sequences and Series
Sequences and series are very related: a sequence of

numbers is a function defined on the set of positive integers (the numbers in the sequence are called terms)...

Infinite and Harmonic Series
In this article we are going to experiment on geometric and harmonic series. We review geometric series, the alternating harmonic series, the Leibnitz formula to find pi and...

Factorials
Factorials of positive integers n, are the product of all positive integers less than or equal to n. Factorials are denoted by n!...

Lucas Series
The French mathematician, E. Lucas, found a sequence of numbers (named the Lucas series) similar to the sequence found in the Fibonacci numbers.

Derivative
The derivative of a function f(x) at x = c is the slope of the tangent line to f at x = c. You can find the value of the derivative using the difference quotient, which is a formula...

Gradient
In our case, a numerical gradient is like having a 3D derivative. We get partial derivatives along x, y and z axes, and that's our gradient. It's convenient to follow the method shown to calculate derivatives...

Definite Integrals
A numerical integration can be performed by a number of algorithms that calculate the approximate value of definite integrals...

Trapezoidal Rule
The Trapezoidal Rule is a way to approximate area beneath a curve. Instead of constructing rectangles, this method uses small trapezoids. In effect, these trapezoids look the same as rectangles near their bases, but different at the top...

Simpson's Rule
Another area-approximating tool is the Simpson’s Rule. Geometrically, it creates parabolas to get closer to the function we’re approximating. The formula is similar to the trapezoidal rule, but you can only use an even number of subintervals...

Maclaurin Series
Maclaurin series are fast approximations of functions, and they offer more accurate function approximations than just linear ones. You have to consider only one general formula and you can...

Taylor expansion
Taylor series are another type of fast approximations of functions and are very similar to Maclaurin's. You have to consider only one general formula and you can...

Sine / Cosine series
The sine and cosine functions are two of the basic functions in trigonometry. In this article, we’re going to explore a number of ways to calculate the sine series without actually using the sine (or cosine) functions...

Fourier Analysis - intro
This field began with the study of the way periodic or general functions that might be represented by summations of simpler trigonometric functions (sine or cosine series). Fourier analysis has many applications in science...

Fourier Series - (analysis cont.)
Fourier series is an infinite sum of sines used to solve special types of problems in electronics. The series consists of an infinite sum of sines (and cosines) that repeats over fixed intervals...


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