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Taylor and MacLaurin Series. Lesson 9.7. 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

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taylor maclaurin polynomials

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)
linear approximations
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 approximations4
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
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 approximations7
Quadratic Approximations
  • So
  • Now how close are these?

View Geogebra demo

taylor polynomial degree 2
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
higher degree taylor polynomial
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
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
  • Try for c =  / 3
assignment
Assignment
  • Lesson 9.7A
  • Page 658
  • Exercises 1 – 4 all5 - 29 odd