Laboratory Project: Taylor Polynomials: Question 3: (x - a) Approximation Form


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    Published on Dec 04, 2020
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    In this video I go over Question 3 of the Laboratory Project: Taylor Polynomials, and this time revisit the quadratic approximation but instead use a slightly different notation. In Question 1 I illustrated how the quadratic or parabola or 2nd order polynomial approximation P(x) = A + Bx + Cx2 can be used to approximate a function f(x) at x = a, with the conditions that P(a) = f’(a), P’(a) = f’(a), and P’’(a) = f’’(a). But in this video I show that it is often preferable to use a slightly different notation and instead use P(x) = A + B(x – a) + C(x – a)2. The only difference in using this form is that the constants A, B, C will not necessarily be the same. This notation has the benefit in that determine the constants with the 3 conditions listed above is fairly easy because when we input x = a into P(x) or its derivative, most terms vanish because a – a = 0.

    When we solve for the constants, I show that we obtain the function P(x) = f(a) + f’(a)(x – a) + f’’(a)(x – a)2, which is a very convenient form to determine the constants, just from f(x) and its derivatives at x = a. Furthermore, this is the basis for Taylor Polynomials which I will be illustrating in further parts of this Laboratory Project, so make sure to watch this video and fully understand this concept!

    UPDATE: Error at the end which I wrote f"(a)(x-a) instead the correct term of f'(a)(x-a).

    Download the notes in my video:!As32ynv0LoaIh45zx3rzhpzXg-omew

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    Related Videos:

    Laboratory Project: Taylor Polynomials: Question 2: Approximation Accuracy:
    Laboratory Project: Taylor Polynomials: Question 1: Quadratic Approximation:
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    tan(x) = sin(x) = x and cos(x) = 1 near x = 0: Linear Approximation in Physics:
    Differentials Notation in Linear Approximation: [

    Newton's Method of Linear Approximation - Introduction: .


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