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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!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh48ZZpMnLhcVNwO7rg

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/laboratory-project-taylor-polynomials-question-5-proof

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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