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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
antiderivatives (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 areaapproximating 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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