Module 10_ APCalcBC 8-4 Taylor Polynomials.pptx - 8-4...

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8-4 Taylor Polynomials Fri Feb 2 Do Now Find the linearization of y = x^2 at x = 1
Do Now Find the first 4 derivatives of sinx at x = pi
HW Review
First order approximations The linearization method L(x) from Ch 4 is referred to as a ‘first order’ approximation because L(x) and f(x) have the same first derivative at x = a First order approximations are only effective in a small interval around x = a
Higher order approximations We would like to use higher order approximations to approximate f(x). Our approximation will need to agree to order n with f(x) This means derivatives of both our approx. and f(x) are equal up to order n at x = a
Taylor polynomial The Taylor polynomial centered at x = a is defined as By dividing by factorials, each Taylor polynomial agrees with f(x) to any order n at x = a
Maclaurin Polynomial The Maclaurin Polynomial is a Taylor polynomial centered at x = 0 (a = 0)
Ex Find the first and third order Maclaurin polynomials for f(x) = e^x. Graph all 3 functions and compare.
Ex Compute the 3 rd order Taylor Polynomial centered at x = 3 for f(x) = (x+1)^(1/2)
Ex-Finding a general formula Find a general Taylor polynomial of f(x) = ln x centered at x = 1
Ex Find Maclaurin polynomials of f(x) = cos x

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