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Taylor Polynomials

Dec 13, 2017

Purpose of this blog:

To illustrate how Taylor Polynomial approximations work with an "interactive visualization"

Illustration

Before we discus the concept in detail, I would encourage you to interact with this application. Here the goal is to tune value of n, such that we can closely approximate the green curve. Our approximation is shown in purple.

Use slider to change value of n It might take few secods to load our interactive

What is definition of Taylor Series?

According to Wikipedia, "The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number a is the power series"
In our example below,
"real or complex-valued function f(x)" is sin(x)
"at a real or complex number a " is point 'a',at which function is centered (eveluated)
f(a) is value of function evaluated at point a.
f'(a) - first derivative of f(x) at a.
""infinitely differentiable" simply means that the curve, say sin(x) is discontinuous- there are no breaks

How to create such interactive?

Watch this video and checkout geogebra.org

How can I change the target function within the interactive?

Instructions are provided in this video

What is difference between a Taylor series and Maclaurin series?

Taylor series of a function centered at 0 is given a special name: Maclaurin series.

What are it's applications?

Taylor series show up in problems involving optimization. One can approximate e^x or pi using Taylor Series.

How is Taylor series used in Fixed Income (Bonds) analysis?

In bond portfolio management,

the first two terms of the Taylor expansion series are used to approximate the change in an option-free bond’s value when interest rates change. I will write a blog post on bond risk management in near future.

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