Laboratory Project: Taylor Polynomials: Question 2: Approximation Accuracy

In this video I go over Question 2 of the Laboratory Project: Taylor Polynomials and this time look at determining the accuracy of the quadratic approximation determined in Question 1. Recall from Question 1 that I showed that the quadratic or parabola approximation P(x) = 1 – (x^2)/2 was much more accurate in approximating f(x) = cos(x) than a simple Linear Approximation. In this video I look at that determining the values of x in which the given accuracy is within 0.1. In other words the absolute value of the difference f(x) – cos(x) is within 0.1. To determine this I first graph the functions P(x), f(x), as well as f(x) + 0.1 and f(x) – 0.1. This allows us to visually see the region in which P(x) is accurate to within 0.1. Zooming into the intersection I show that it is equal to +/- 1.26.

Also shown in this video is a look at a numerical online solver approach to determine a more accurate value for the range. But when dealing with inequalities and numerical calculations it is important to understand that rounding up or rounding down can affect the given inequality, so always keep this in mind! This is a very useful video in understanding the accuracy associated with graphical and numerical approximations, so make sure to watch this video!

Video notes and playlist:

- PDF: https://1drv.ms/b/s!As32ynv0LoaIh44lr-8yFszdUmViLQ
- HIVE: https://peakd.com/mathematics/@mes/laboratory-project-taylor-polynomials-question-2-approximation-accuracy
- Laboratory Projects playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GPz_diCZTkiVs6Q4mA_KHb

Related Videos:

Laboratory Project: Taylor Polynomials: Question 1: Quadratic Approximation: https://youtu.be/8bpF3vccvEU
Taylor Polynomials - Introduction and Derivation: http://youtu.be/p2EkXwkbflk
Linear Approximation - Introduction and Examples: http://youtu.be/bXEK8bkWTtM
tan(x) = sin(x) = x and cos(x) = 1 near x = 0: Linear Approximation in Physics: http://youtu.be/TPtZIxICa3Q
Differentials Notation in Linear Approximation: http://youtu.be/s0adatWiZg4
Newton's Method of Linear Approximation - Introduction: http://youtu.be/aT4b_5l50RI .

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