Laboratory Project: Taylor Polynomials: Question 5: Proof

In this video I go over Question 5 of the Laboratory Project: Taylor Polynomials and this time derive the general formula for an n-th degree Taylor Polynomial approximating the function f(x) and centered about x = a. In the previous parts of this series, I went over linear and quadratic approximations, but these are in fact still technically Taylor Polynomials but with degrees 1 and 2, respectively. The Taylor Polynomial formula can thus be viewed as a more generalized polynomial approximation to a function centered about a point. In this video I show that when we take the starting point of the quadratic approximation formula and then extend it to include n constants, we can then start to see a pattern when we enforce the conditions that the approximation and its derivatives at the value of x = a is set to be equal to the function we are approximating, and its derivatives, again at x = a. This pattern is such that the general k-th constant is equal to the k-th derivative of f(x) at x = a divided by k factorial (k!). Thus if we want to obtain a higher order polynomial approximation we can simply add more terms and determine the resulting constants from the derivatives of the function we are approximating. This is a very good video in understanding how we can derive formulas through pattern recognition as well as in understanding one of the most useful methods of approximating complicated functions, the Taylor Polynomial; so make sure to watch this video!

Video notes and playlist:

- PDF notes: https://1drv.ms/b/s!As32ynv0LoaIh48ZZpMnLhcVNwO7rg
- HIVE notes: https://peakd.com/mathematics/@mes/laboratory-project-taylor-polynomials-question-5-proof
- Laboratory Project: Taylor Polynomials playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GMTZJWnswRe2BYAL-Y2CeB

Related Videos:

Laboratory Project: Taylor Polynomials: Question 4: Approximating Square Roots: https://youtu.be/IFtudCiIe5s
Laboratory Project: Taylor Polynomials: Question 3: (x - a) Approximation Form: https://youtu.be/V_u_SHVbTdc
Laboratory Project: Taylor Polynomials: Question 2: Approximation Accuracy: https://youtu.be/MRSs0Qofd_M
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
Differentials Notation in Linear Approximation: http://youtu.be/s0adatWiZg4
Newton's Method of Linear Approximation - Introduction: http://youtu.be/aT4b_5l50RI
Factorials - i.e. 4! = 4*3*2*1 = 24: http://youtu.be/SWnrhDOLQgA .

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