Taylor and MacLaurin Series

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Presentation transcript:

Taylor and MacLaurin Series Lesson 9.7

Taylor & Maclaurin Polynomials Consider a function f(x) that can be differentiated n times on some interval I Our goal: find a polynomial function M(x) which approximates f at a number c in its domain Initial requirements M(c) = f(c) M '(c) = f '(c) Centered at c or expanded about c

Linear Approximations The tangent line is a good approximation of f(x) for x near a True value f(x) Approx. value of f(x) f'(a) (x – a) (x – a) f(a) a x

Linear Approximations Taylor polynomial degree 1 Approximating f(x) for x near 0 Consider How close are these? f(.05) f(0.4) View Geogebra demo

Quadratic Approximations For a more accurate approximation to f(x) = cos x for x near 0 Use a quadratic function We determine At x = 0 we must have The functions to agree The first and second derivatives to agree

Quadratic Approximations Since We have

Quadratic Approximations So Now how close are these? View Geogebra demo

Taylor Polynomial Degree 2 In general we find the approximation of f(x) for x near 0 Try for a different function f(x) = sin(x) Let x = 0.3

Higher Degree Taylor Polynomial For approximating f(x) for x near 0 Note for f(x) = sin x, Taylor Polynomial of degree 7 View Geogebra demo

Improved Approximating We can choose some other value for x, say x = c Then for f(x) = sin(x – c) the nth degree Taylor polynomial at x = c

Assignment Lesson 9.7 Page 656 Exercises 1 – 5 all , 7, 9, 13 – 29 odd