Taylor and Maclaurin Polynomials

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Taylor and Maclaurin Polynomials
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Why Taylor and Maclaurin Polynomials?

• It is easier to calculate with polynomials than
  it is with sines or exs etc. These series
  provide a way to approximate a function that is
  not a polynomial with one that is.

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Polynomial for ex

• Lets find a polynomial of degree 4 for ex.
• P(x) a0 a1x a2x2 a3x3 a4x4
• Note the following for ex
• F(0) 1 F(0) 1 F(0) 1
• F(0) 1 fiv(0) 1

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Polynomial for ex

• If P(x) is to be equivalent to ex then the
  derivatives of each should have the same value at
  the same point. Therefore
• P(0) 1 P(0) 1 P(0) 1
• P(0) 1 Piv(0) 1

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Polynomial for ex

• This means that
• P(0) a0 a10 a202 a303 a404
• a0 1
• P(0) a1 2a20 3a302 4a403
• a1 1
• P(0) 2a2 32a30 43a402
• 2a2 1
• P(0) 32a3 432a40
• 32a3 1
• Piv(0) 432a4 1

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Polynomial for ex

• P(x) 1 1x x2/2 x3/(32) x4/(432)
• Or
• P(x) x0/0! x1/1! x2/2! x3/3! x4/4!
• P(x) 1 x x2/2! x3/3! x4/4!

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Polynomial for sin(x)

• Lets find a polynomial of degree 4 for sin(x).
• P(x) a0 a1x a2x2 a3x3 a4x4
• Note the following for sin(x)
• F(0) 0 F(0) cos(0) 1
• F(0) -sin(0) 0 F(0) -cos(0) -1
• Fiv(0) sin(0) 0

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Polynomial for sin(x)

• If P(x) is to be equivalent to ex then the
  derivatives of each should have the same value at
  the same point. Therefore
• P(0) 0 P(0) 1 P(0) 0
• P(0) -1 Piv(0) 0

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Polynomial for sin(x)

• This means that
• P(0) a0 a10 a202 a303 a404
• a0 0
• P(0) a1 2a20 3a302 4a403
• a1 1
• P(0) 2a2 32a30 43a402
• 2a2 0
• P(0) 32a3 432a40
• 32a3 -1
• Piv(0) 432a4 0

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Polynomial for sin(x)

• P(x) 0/1 (1/1)x (0/2)x2 (-1/32)x3
  (0/(432)x4
• Or
• P(x) x - x3/3! OR
• P(x) f0(0)x0/0! f1(0)x1/1! f2(0)x2/2!
  f3(0)x3/3! f4(0)x4/4!

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Polynomial for any function?

• P(x) f0(0)x0/0! f1(0)x1/1! f2(0)x2/2!
  f3(0)x3/3! f4(0)x4/4!

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Nth Taylor or MacLaurin Polynomial

• Let f be a function with derivatives of all
  orders throughout some open interval containing
  0. Then the Taylor Series generated by f at x0
  is
• Pn(x) ? fk(0)xk/k!
• The sum goes from k0 to kn.
• This is the Taylor polynomial of order n for f at
  x0.

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Nth Taylor Polynomial

• Let f be a function with derivatives of all
  orders throughout some open interval containing
  a. Then the Taylor Series generated by f at xa
  is
• Pn(x) ? fk(0)(x-a)k/k!
• The sum goes from k0 to kn. This is the Taylor
  polynomial of order n for f at xa.

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Polynomial Approximations

• Note that the higher the degree that the
  polynomial approximation has, the farther out the
  polynomial has a good approximation for the
  function. The a value tells where the
  approximation is centered and so the closer you
  are to a the better the approximation as well.
• http//en.wikipedia.org/wiki/Taylor_series