Calculus
3 Dr. David Tall, from the University of Warwick, describes an interesting way to teach calculus here, he describes it thus: "The fundamental ideas are those of change, the rate of change, and the accumulation due to change. Symbolically these are represented by the function concept, the derivative and the integral." If I had learned higher math like this then I would have picked it up much faster.
Even though Numerical Methods are used to do calculations, not calculus, the ideas underlying calculus are interesting, particularly the concept of infinitesimal, which blew my mind when I first heard of it.
An infinitesimal is some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity. A change in this quantity is called a differential, usually notated as dx, if x is the variable. The useful thing about infinitesimals is that they can be used to determine totals via integrals (from the Latin 'integratus' to 'make whole').
The way this works is that, when a differential is one side of a rectangle (dx), and a larger line is the other side of the rectangle (x), a very thin rectangle is created. The area of this rectangle is of course x times dx, usually just written xdx. A larger rectangle can be created by summing many xdx, using the integral character ∫, as in ∫xdx. This is extremely useful for calculating the areas under curves. If the x value is a value of a function, call it f(x) or 'f of x', and that function is the function describing a line such as f(x)=2x, then the area under that line (to the x axis) is a sum of the rectangles denoted by ∫f(x)dx. Integrals have many special features, and to evaluate an integral requires looking them up in a table. For the equation f(x)=2x, the area under the curve works out to be x^2 + C, where C is a constant. If we want to know the area of the curve f(x)=2x from x=1 to x=2, we enter in the values of x into the equation for the area and subtract:
(2^2 + C) - (1^2 +C) = 3, so the area under the curve (to the x axis) from 1 to 2 of the line f(x)=2x is equal to 3.
Many problems that require determining the total value of something can be solved using integrals.
A great resource for calculating integrals can be found here: integration
Calculus can be used to create equations for model systems, which are then calculated on computers and compared to the real result. In these cases integrals within limits can be useful, but even here the integrations of various equations are listed in tables so there is no need to memorize them. In addition, as with all math, calculus only approximates reality.
Calculus also allowed for the creation of useful tools, such as the Fourier Transform. Understanding the concept of Fourier transformation is useful, but going through the math of it also seems a waste of brain space.
The ideas behind each of the tools of calculus will be described, but detailed math behind is left for others.
