In this week’s blog post, we continue our discussion of the mathematics of aerospace engineering (and all engineering fields in general). We will combine fundamental concepts in differential calculus with mathematical structures called sequences and series to create nice functions that can allow engineers and mathematicians to approximate the values of weird and wonky functions.

It is important to start with some definitions.

First, we will call a list of numbers a sequence. For example, {1,2,3, π, 5,8,27, -80, 27} can be thought of as a sequence. More often than not, sequences have a pattern and go on forever. For example, {2,4,6,8,10,…(on and on forever)} has the pattern of every subsequent term being the next multiple of 2. For notation, we can label the first term of a sequence a_{1}, the second term, a_{2}, the third term a_{3}, and so on until we decide to label the “n-th” term a_{n}. We then call the entire sequence (a_{n}), where (a_{n}) = {a_{1}, a_{2}, a_{3}, …}.

Second, we will define a series to be the sum of a portion of (or all of) the numbers in a sequence. The reason why it is useful to sum elements of a sequence will become more clear in the next few paragraphs. Oftentimes, we will use the capital Greek letter sigma, ∑, to denote the sum of a lot of numbers. For the purposes of this blog post, we will define ∑ a_{n} to be the sum of all of the elements of the sequence (a_{n}), where (a_{n}) could potentially have infinitely many terms.

We are now ready to combine calculus with series. The basic problem we want to solve is to find an easier way to find the values of weird functions. For example, we may have some function f(x) = e^{-x} + sin(3πx) + 1/(1-x). It can be cumbersome to find the actual value of this function at, say, x = 1/10. Therefore, it would be really nice if there was some easier function, say P(x), where P(1/10) ≈ f(1/10). In this case, we could plug x = 1/10 into the nice function instead of plugging x = 1/10 into the complicated one.

To create a nice function that approximates a complicated one, we will look at polynomials. A polynomial is a function that is the sum of a whole bunch of constants multiplied by powers of x. This elegant structure makes polynomials convenient to work with. For example, y = 3x+1, y = 2x^{8} + 6x^{7} + 132.75x^{5} – x^{17}, and y = -8 – x^{101} are all polynomials. We can compactly write a polynomial P(x) as P(x)= ∑ a_{n}x^{n}, where a_{n} is the constant in front of the “n-th” power of x.

If we want to use a polynomial to approximate an arbitrary function f(x), it is important to anchor that polynomial around a certain point (x_{0}, f(x_{0})). Indeed, if we know that a polynomial lines up with a function at one particular point, then we can use information about f(x) at the anchor point to build a polynomial that approximates f(x) in the *vicinity* of x_{0}.

The key idea from calculus that we will bring in right now is that we want our polynomial P(x) to have the same value as f(x) at x = x_{0}, have the same slope as f(x) at x = x_{0}, and bend in the same way as f(x) at x = x_{0}.

If we want P(x_{0}) = f(x_{0}), then we can first put:

P(x) = f(x_{0}) + {other terms that we don’t know}.

It is our job now to determine these other terms. To begin this process, we can remember that we want P(x) to have the same slope as f(x) at x = x_{0}. This means that P ’(x_{0}) = f ’(x_{0}). We can force this to be the case by putting:

P(x) = f(x_{0}) + f ’(x_{0}) (x – x_{0}) + {other terms}.

If we ignore the other terms, we have created a linear approximation for f(x) at x = x_{0}, because P(x_{0}) = f(x_{0}) + f ’(x_{0}) (0) = f(x_{0}) and simultaneously P ’(x_{0}) = f ’(x_{0}).

To make P(x) to curve in the same way as f(x) at x = x_{0}, we can let:

P(x) = f(x_{0}) + f ’(x_{0}) (x – x_{0}) + f ’’(x_{0})/2 (x – x_{0}) + {other terms}.

Then, P ’’(x_{0}) = f ’’(x_{0}).

This works because higher order derivatives give information about how a function curves.

We can ultimately continue this process for infinitely many derivatives of f(x) until we arrive at the Taylor polynomial approximation for f(x), defined as follows.

P(x) = ∑ f^{ (n)}(x_{0}) (x – x_{0})^{n} / n!. (This is a sum of infinitely many weighted powers of x.)

We thus define the coefficient of the term with x to the power of n in the Taylor polynomial to be the “n-th” derivative of f(x) evaluated at the anchor point x = x_{0}, divided by n! = n*(n-1)*(n-2)*…*4*3*2*1. This n! part may at first seem random, but it comes about in order to guarantee that the “n-th” derivative of P(x) at x = x_{0} is equal to the “n-th” derivative of f(x) evaluated at x = x_{0}

These Taylor series can be quite powerful beyond just approximating a function near a point. They can simplify the process of analyzing the behavior of a function, and can be used to develop advanced computer algorithms (such as the Runge-Kutta Method) that numerically solve differential equations that would otherwise be unsolvable. I hope to continue exploring the power of Taylor series and other methods of approximating functions in future posts. Until next week, take care.

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