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Taylor and MacLaurin Series

Learn how to approximate functions using Taylor and Maclaurin polynomials centered at a point c, exploring linear, quadratic, and higher-degree approximations with helpful Geogebra demos.

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Taylor and MacLaurin Series

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  1. Taylor and MacLaurin Series Lesson 9.7

  2. Centered at c or expanded about c 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)

  3. 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

  4. 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

  5. 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

  6. Quadratic Approximations • Since • We have

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

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

  9. 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

  10. 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

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

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