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Root approximation is a fundamental concept in mathematics and numerical analysis, which seeks to find an estimate for the zeroes of a function—that is, the points where the function's output is zero. Newton's method, an iterative technique for finding the roots of real-valued functions, is a classic example of an effective root approximation procedure. By starting with an initial guess and then successively improving it, the method zooms in on the true root with each iteration, until successive approximations are within a desired degree of accuracy. In our exercise, we used Newton's method to approximate a root of the function f(x) = tan(x) - 2x, and we refined our estimate until the difference between two successive approximations was less than 0.00001.

The tangent function, commonly denoted as tan(x), plays a critical role in trigonometry and calculus. This function, which can be thought of as the ratio of the sine to cosine functions, exhibits a repeating pattern and has its own distinctive wave-like graph. In calculus, especially when applying Newton's method, understanding the behavior of the tangent function and its graph is essential. When combining it with another function, such as -2x in our example, we create a new function whose roots are the points of intersection between the graph of tan(x) and the linear function 2x.

Derivative calculation is the process by which we determine the rate at which a function changes at any given point. It is a cornerstone of differential calculus and is crucial for many applications, including Newton's method. In our problem, we computed the derivative of the function f(x) = tan(x) - 2x to obtain f'(x) = sec^2(x) - 2. This derivative tells us about the slope of the tangent line to the graph of f at any point x, which is essential in determining the next guess for the root using Newton's method. Without this step, the method cannot progress.

Iterative methods are algorithms that solve mathematical problems through repeated rounds of approximation, each one building on the previous. Such methods are highly valuable when exact solutions are infeasible or too complex to find analytically. Newton's method is one of these iterative methods, starting with an initial guess and producing better approximations for the root with each iteration. This procedure is particularly useful when dealing with non-linear equations where roots cannot be derived algebraically.

Successive approximations refer to the technique of refining an estimate step by step, each time using the information gleaned from the previous estimation. In the case of Newton's method, each successive approximation is expected to be closer to the actual root than the last. The importance of this approach lies in its ability to converge to an accurate solution, given a good enough initial guess and a function that behaves nicely. In the exercise provided, we can observe the use of successive approximations where each new estimate for the root is derived from the last until the desired precision is reached.

Numerical analysis is the branch of mathematics that develops, analyzes, and implements algorithms to solve problems involving continuous variables. It encompasses topics like root approximation, iterative methods, and much more. Numerical methods are indispensable in scenarios where an analytical solution is not available or is difficult to obtain. They allow us to tackle complex equations by providing approximate solutions with controllable levels of accuracy. Newton's method is a classic example within numerical analysis, known for its power and efficiency in finding the roots of a function.