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 + Cx^2 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!
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Related Videos:
Laboratory Project: Taylor Polynomials: Question 2: Approximation Accuracy:
Laboratory Project: Taylor Polynomials: Question 1: Quadratic Approximation:
Taylor Polynomials - Introduction and Derivation:
Linear Approximation - Introduction and Examples:
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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I don't always use quadratic approximation but when I do I prefer to use the (x - a) notation because the constants align better with the function being approximated ;)
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/laboratory-project-taylor-polynomials-question-3-x-a-approximation-form
Critical math video