Taylor and MacLaurin Series

1
Taylor and MacLaurin Series

• Lesson 9.7

2
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)

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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
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Linear Approximations

• Taylor polynomial degree 1
• Approximating f(x) for x near 0
• Consider
• How close are these?
• f(.05)
• f(0.4)

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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

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Quadratic Approximations

• Since
• We have

7
Quadratic Approximations

• So
• Now how close are these?

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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

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Higher Degree Taylor Polynomial

• For approximating f(x) for x near 0
• Note for f(x) sin x, Taylor Polynomial of
  degree 7

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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

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Assignment

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